[HN Gopher] Fair coins tend to land on the same side they started
___________________________________________________________________
Fair coins tend to land on the same side they started
Author : fbartos
Score : 344 points
Date : 2023-10-10 09:06 UTC (13 hours ago)
(HTM) web link (arxiv.org)
(TXT) w3m dump (arxiv.org)
| cabirum wrote:
| 50 authors in a paper about flipping coins?
| netrus wrote:
| By flipping thousands of coins for this paper, I am sure their
| participation exceeds that of many 5-author-papers.
| shusaku wrote:
| I guess that's one way to prevent p-hacking: you can't
| exclude/include someone's flips without them getting mad!
| bunderbunder wrote:
| Make me sit around flipping coins 10,000 times and recording
| the results, and you damn well better at least put my name on
| the paper.
| slingnow wrote:
| Then I guess we should include every study participant as an
| author across all disciplines. 200 participant study in
| psychology? 200 authors on the paper.
| bunderbunder wrote:
| Not necessarily? The difference is that the people flipping
| the coins are not just experimental subjects, they're
| actually implementing the experimental protocol.
|
| The kinds of participants you're talking about are not just
| left off the paper out of lack of interest. It's often the
| ethically preferable option. They often have a vested
| interest in remaining unnamed for privacy reasons, and
| derive no tangible benefit from being listed as authors.
|
| A big author list isn't totally unheard of. The paper where
| they announced the discovery of the Higgs boson had an
| author list that spanned 8 densely-packaged pages.
| jb1991 wrote:
| Sounds fair to me. Better than if it was only 49, or 51.
| blantonl wrote:
| [flagged]
| boomboomsubban wrote:
| Blind guess, it's cheaper and easier to recruit volunteers to
| flip a coin 7000 times if you promise them credit on the paper.
| fbartos wrote:
| You are absolutely right :)
| magicalhippo wrote:
| This reminds me of the opening scene of Rosencrantz &
| Guildenstern Are Dead[1]. Didn't get to see the play, but loved
| the movie.
|
| [1]: https://www.youtube.com/watch?v=gOwLEVQGbrM
| omneity wrote:
| This got me thinking, is it physically feasible to slightly bias
| the coin against the starting side to get closer to fairness?
| fbartos wrote:
| About a year ago, we embarked on a quest to answer one of the
| most intriguing questions:
|
| If you flip a fair coin and catch it in hand, what's the
| probability it lands on the same side it started?
|
| Today, we are finally ready to share the results. Thanks to my
| friends, collaborators, and even strangers from the internet, we
| collected flippin 350,757 coin flips. We ran several "Coin
| Tossing Marathons" (e.g., https://youtu.be/3xNg51mv-
| fk?si=o2E3hKa-ReXodOmc) and spent countless hours flipping coins.
|
| In short, we found overwhelming evidence for a "same-side" bias
| predicted by Diaconis, Holmes, and Montgomery 2007: If you start
| heads-up, the coin is more likely to land heads-up and vice
| versa. How large is the bias? In our sample, the mean estimate is
| 50.8%, CI [50.6%, 50.9%].
|
| We also found considerable variance in the same-side bias between
| our 48 tossers. The bias varied with a standard deviation of
| 1.6%, CI [1.2%, 2.0%], in our sample. The variation could be
| explained by a different degree of "wobbliness" between our
| tossers.
|
| If you bet a dollar on the outcome of a coin toss 1000 times,
| knowing the starting position of the coin toss would earn you 19$
| on average. This is more than the casino advantage for 6-deck
| blackjack against an optimal player (5$) but less than that for
| single-zero roulette (27$).
|
| The manuscript is at arXiv: https://arxiv.org/abs/2310.04153 And
| the open data, code, and video recordings at OSF:
| https://osf.io/pxu6r/.
|
| Diaconis, P., Holmes, S., & Montgomery, R. (2007). Dynamical bias
| in the coin toss. SIAM Review, 49(2), 211-235.
| https://doi.org/10.1137/S0036144504446436
| kzrdude wrote:
| Fun that you're showing your own work. Since this is your work,
| it could have Show HN in the title; I guess.
| gala8y wrote:
| No, not really.
|
| https://news.ycombinator.com/showhn.html
| input_sh wrote:
| If there isn't a product one can try, it's not a Show HN
| material.
| kzrdude wrote:
| I see, it's a bit more narrow than I knew
| fouronnes3 wrote:
| What is the physical explanation for this bias?
| [deleted]
| rstarast wrote:
| I would guess it's rather mathematical. Each coinflip has
| some number of half-flips. Now analyze the distribution of
| that number. If this distribution were to start at its
| maximum with 0 half-flips and decay as it increases, summing
| over the even values (same side up) clearly gives more than
| summing over the odd values. Now the distribution isn't going
| to be like that, but I expect that it's generally "front-
| loaded" in a way that causes a similar effect.
| mewpmewp2 wrote:
| Yeah, and seeing that the bias occurs only in some people,
| perhaps it occurs in people who do as little rotation as
| possible. Not sure if this study has a graph including
| amount of rotations occurred in general. E.g. you could
| take all coin flips where 0-5 rotations occurred and
| compare them to 6-11 rotations.
| bradrn wrote:
| It's in the paper:
|
| > The standard model of coin flipping was extended by Persi
| Diaconis [12] who proposed that when people flip a ordinary
| coin, they introduce a small degree of 'precession' or wobble
| --a change in the direction of the axis of rotation
| throughout the coin's trajectory. According to the Diaconis
| model, precession causes the coin to spend more time in the
| air with the initial side facing up. Consequently, the coin
| has a higher chance of landing on the same side as it started
| (i.e., 'same-side bias').
|
| [12] Diaconis P, Holmes S, Montgomery R. Dynamical bias in
| the coin toss. SIAM Review 2007; 49(2): 211-235.
| Philip-J-Fry wrote:
| After reading this the first thought I had was how do you
| stop people flipping the same way? Like, give me a baton and
| I could throw it at varying heights and control which side I
| caught it on. In theory the same applies to coin flipping.
| You can get quite consistent with your positioning and power.
|
| You could probably control for it by making people alternate
| which side was face up before the flip.
|
| That's my intuition anyway.
| aspenmayer wrote:
| Your comment reminded me of two-up.
|
| > Two-up is a traditional Australian gambling game,
| involving a designated "spinner" throwing two coins,
| usually Australian pennies, into the air. Players bet on
| whether the coins will both fall with heads (obverse) up,
| both with tails (reverse) up, or with a head and one a tail
| (known as "Ewan"). The game is traditionally played in pubs
| and clubs throughout Australia on Anzac Day, in part to
| mark a shared experience with diggers (soldiers).
|
| https://en.wikipedia.org/wiki/Two-up
| pirates wrote:
| Two-up sounds pretty fun. Your comment in turn made me
| think of Cho-han, which is somewhat similar but involves
| rolling dice instead of flipping coins.
| https://en.m.wikipedia.org/wiki/Ch%C5%8D-han
| aspenmayer wrote:
| It's wild and raucous and usually takes place outside
| pubs and local workers' clubs, lawn bowl clubs etc. The
| spinner puts the coins on a flat stick made for the game,
| but basically a popsicle stick but wider like a tongue
| depressor. They toss the coins up and flick the stick to
| tumble the coins. If you called the two matching coins
| correctly, you double your bet in winnings. Each game
| takes like 30s-1m and they go on from mid-morning til
| early afternoon ish.
|
| The losers indirectly pay the winners based on your call
| of two heads or two tails and if there's a split, the
| house wins that game.
|
| Most of the time is spent drinking your beverage of
| choice, yelling and cursing your own luck and talking
| smack to the coins, the spinner being booed or cheered
| for a well run game or a bad string of luck, but the bets
| are typically fairly low in my experience, although it's
| up to each individual player what they bet, but the
| spinner or venue sets the bet amount per section or per
| spinner. Each spinner usually takes a set amount, like
| $5/$10/$20/$50, even $100 in some cases. It's all cash
| and pretty much the honor system in that they aren't
| handing out receipts or tickets with your bet, so that
| sets an upper limit on how many bets each spinner can
| keep straight. No one wants to see someone lose their
| shirt, so most folks play against their friends/mates for
| fun, and ultimately you're all playing your own game
| because you decide your bet and it's actually fairly
| unpredictable due to the crowds milling around the
| spinners, of which there will be many, all taking bets
| and running games independently and simultaneously, and
| players can place bets in any or all of the games around
| them if they want.
|
| It's a wild affair. Highly recommended.
| JKCalhoun wrote:
| Fair coin, unfair flip.
| gala8y wrote:
| This was my intuition in childhood. If you choose tails to be
| yours and start with tails then catch it, it is most likely to
| be tails. I came up with this observation myself. Weird.
| bryanbuckley wrote:
| Noticed as a kid I could flip a quarter with a certain
| consistency, so I experimented a bit and quickly got to be
| >90% accurate with an ordinary (controlled) flip.
|
| Pretty simple. In fact I just picked up a quarter and
| practiced (20+ years out of practice) and have some
| observations: 1) harder than when I was a kid, my fingers are
| lot bigger + stronger so it's not as precise from the start.
| A bigger and heavier coin would help. 2) the timing factor is
| bigger than I recalled.. essentially you can watch the coin
| flipping and get a subconscious/automatic/predictable sort of
| count/feedback to it. You can bring your hand up to the coin
| in the air at a precise moment pretty easily and "tell" (>90%
| accuracy today of the flips I just did that I considered
| successful before looking at the result) if the flip was
| predictable. Hand eye coordination, spatial awareness is very
| correlated to this skill, I suppose. 3) it really is the same
| side that comes up.. again I think because of the automatic
| watching/count/completion of full rotations, i.e. catching
| the coin at the end of a full rotation instead of a partial.
|
| Came in handy occasionally.. if I knew I was going to be
| wrong (other person usually waits to call mid-flip) I could
| catch the coin a little lower to give myself a chance, or
| punk them by not putting it on the back of my hand as is more
| standard (they might demand a re-flip.. kind of like if you
| are playing rock paper scissors and one person goes on 3 and
| the other on 4).
| kqr wrote:
| Yeah, I think you were just lucky that your superstition
| happened to be true.
|
| Trying not to be disrespectful but I don't believe you
| intuited a 50.8 % bias. So nothing weird going on at all.
| gala8y wrote:
| I know it's not on par with any real stats, but still... I
| just had this strong conviction that worked this way. The
| really interesting stuff is why it is so. I would think
| along lines of brain timing tossing and catching, eye-
| brain-timing rather than gravity and coin itself. Added:
| Yeah, now I remember actually manipulating timing of catch
| to achieve this.
|
| > Trying not to be disrespectful
|
| No worries, it is just me using high context communication
| style, where I assume that you know that I know this and I
| just share what was my experience in childhood (it was not
| like a single thought).
| nomilk wrote:
| It becomes clear why there's a same-side bias when watching the
| video: https://www.youtube.com/watch?v=3xNg51mv-fk
|
| These are fairly gentle coin tosses; barely going a foot into
| the air!
|
| When I think of a coin toss, I think high and spinning fast
| (like the ones before sports games, where the coin goes into
| the air and lands on the ground, usually rolls a short way, and
| is collected on whatever side it landed). I would guess the
| 50.8% same-side bias would be much closer to 50% if the coins
| were tossed this way in the experiment.
| gowld wrote:
| Bouncing erases the bias due to flipping.
| glandium wrote:
| With a 1 foot to 2 feet toss, landing in the hand, I can get
| the same side as the starting side more than 90% of the time,
| without even trying. I wouldn't trust such a toss to be fair.
| Landing on the floor would change the game.
| fbartos wrote:
| There was indeed a lot of variation in the height of the
| tosses. I however disagree with the conclussion: two of my
| friends at the video had the most different height of tosses
| (one tossed thrice as hight as the other one), yet both of
| them had exactly the same bias (0.505). The amount of spin is
| unfortunatelly very misleading from the 30fps videos--the
| coins often seem like not spinning at all but that's just a
| result of the poor video quality.
| sixstringtheory wrote:
| How do you control for bias coming from the same coin
| flipper? Do they usually flip their coin from the same
| starting height (whatever comfortable arm positioning they
| have, which I assume would also introduce bias by how they
| catch it as well) and to the same arc peak height? Or were
| they encouraged to try a different body position, strength
| and angle of launch for each flip?
| alkonaut wrote:
| > This is more than the casino advantage for 6-deck blackjack
| against an optimal player (5$)
|
| I have seen that figure (roughly 0.5% edge) but that has to
| depend on how deep the shoe is dealt? I remember playing only
| the last hands with dealers playing down to between 1.0 and 0.5
| decks left. That meant you could play hands where you knew
| almost all remaining cards were suited. I guess the average
| edge assumes constant bet and doesn't include betting
| strategies based on counting at all? (And those strategies
| obviously wouldn't work in any real casino because it's
| "frowned upon").
| noobermin wrote:
| >If you bet a dollar on the outcome of a coin toss 1000 times,
| knowing the starting position of the coin toss would earn you
| 19$ on average. This is more than the casino advantage for
| 6-deck blackjack against an optimal player (5$) but less than
| that for single-zero roulette (27$).
|
| This sounds like the plot of a western where a man travels from
| town to town and gleans a little cash from the local waterhole
| a little every time. I did the math though, in order to get
| just the $19, assuming you played a modest 20 times a day, it'd
| take 10 weeks (not including weekends), and by that point
| people would definitely figure out your trick. In order to make
| any profit quickly, you'd have to distribute the strategy,
| after which your secret would explicitly be out there. Even
| assuming perfectly honest colleagues, having that many parallel
| people using the same strategy in the open means that before
| you turn any real profit, people will find out. It's a fun idea
| to fantasize about though.
|
| Anyway, cheers on the paper! Pretty cool result that you guys
| put the effort in in implementing.
| cortesoft wrote:
| You also have to factor in your time cost. Your hourly rate
| is going to be really low, better off just getting a job.
| thih9 wrote:
| > in order to get just the $19, assuming you played a modest
| 20 times a day, it'd take 10 weeks (not including weekends)
|
| What if one play session consisted of 10 coin tosses (each an
| independent $1 bet)? I guess 20 games like this per day would
| still be doable. Would that mean $19 per week?
|
| Next we up the bet to $10 per throw.
| kqr wrote:
| The upshot is that as long as you only stake $1 at a time,
| you're unlikely to lose more than $50.
|
| On the other hand, /if/ you do, you'll have to play for 6000
| more flips until you can be fairly certain that you're even
| again.
|
| What's worse is if, after having lost $50, you're down to
| your last $50, there's almost a 1/5 chance you'll blow all of
| it trying to recover if you wager $1 each time.
|
| If you grow wise and start Kelly betting you'll get back to
| your starting $100 on average in 5000 flips, though. If you
| can take out a loan of $500 first, you can Kelly bet your way
| to even much faster, in an expected 700 flips. Whether this
| is worth the interest on the loan depends on how quickly you
| can find challengers to bet with.
| mewpmewp2 wrote:
| Make a deal with all the banks or systems that can print
| money that would allow you to take an infinite loan from
| them. Then just double the bet every time you lose.
|
| If they can print money, why not infinitely as you will
| always pay it back anyway, so you don't have to worry about
| introducing inflation. There will always be a point when
| you can just burn the money that you temporarily
| introduced.
| kqr wrote:
| ...from where does the money come that you pay back? Even
| if you have an unlimited stake, your winnings will be
| constrained by the counterparty eventually.
| mewpmewp2 wrote:
| Right, I forgot, you also need someone willing to take
| those bets. So I think what you should do is make bets
| against multiple casinos/institutions where you can
| develop an algorithm that will find you an optimal method
| of betting for reasonable 50/50 results, if it makes
| easier to think. So for example at some point you might
| want to go to a casino and put the max bet on a single
| number in roulette, but do it enough times that you would
| have 50% odds of winning.
|
| Once that is exhausted, you would have to become more
| creative, like trying powerball enough times, but I'm not
| sure how good the odds are there vs the reward. Maybe
| that wouldn't ever work.
|
| Actually, I forgot. You can just play with highly
| leveraged options. It's not infinite yet, but come back
| to me until you've multiplied enough times that even
| options are not enough.
|
| Forget everything I said before, just play with options
| and automated algorithm to buy more. And post here once
| you can't buy any higher cost options, and we'll figure
| something out together.
| andrewinardeer wrote:
| The Federal Reserve. They can print money arbitrarily.
| Semaphor wrote:
| > but less than that for single-zero roulette (27$).
|
| This reminds me of the casino I was at with a fair roulette.
| Betting Black/Red and getting a zero meant all bets stayed for
| the next spin ;)
|
| My winning strategy was to always bet the opposite color of a
| friend of mine.
| jonahx wrote:
| From the Method section:
|
| "In each sequence, people randomly (or according to an
| algorithm) selected a starting position (heads-up or tails-up)
| of the first coin flip, flipped the coin, caught it in their
| hand, recorded the landing position of the coin"
|
| Presumably if you instead allow the coin to land and bounce on
| a hard surface, the bias would disappear?
| ss1996 wrote:
| Thank you for this. I'll make sure to request a best of 350,757
| next time I'm deciding anything by coin toss.
| LadyCailin wrote:
| It's 50.8% bias, so you only need to do best of 1015!
| kqr wrote:
| ...to accomplish what? Wouldn't 1015 flips only give you a 70
| % probability of winning?
| tnzk wrote:
| fact(1015)?
| 1f60c wrote:
| How is that possible when both outcomes have a probability of
| exactly 50%?
| nottorp wrote:
| An ideal coin flipped by an unbiased super person would have
| that 50% probability. A real coin flipped by a real human no,
| as the study shows.
|
| Let's consider a spherical cow...
| kqr wrote:
| Wait, didn't Ed Thorp argue for the exact opposite - if you
| have a super-person that can guarantee lack of bias, then you
| also have absolute Newtonian predictability on virtue of the
| mechanical perfectitude. The randomness must come from
| somewhere, and it comes from imperfections which also
| incidentally introduce bias.
| tgv wrote:
| It's simpler: an ideal coin flip is simply assumed to be
| uniformly distributed, on the basis of there being two
| possible outcomes and no influence. Where the bias in reality
| comes from, doesn't matter.
|
| This also happens to be the great divide between frequentists
| and Bayesians.
| kqr wrote:
| Even simpler than that, actually. There's no requirement
| for any distribution at all. (And I would argue strongly
| against a uniform prior, but that is a separate
| discussion.)
|
| What's necessary to guess 50 % on the first toss is simply
| (a) complete ignorance about the bias, whatever it is, and
| (b) the hypothesis that the bias is just as likely to be
| negative as positive (i.e. a symmetric prior.)
| shawabawa3 wrote:
| because they don't have a probability of exactly 50%
| saalweachter wrote:
| How did you determine that both outcomes have a probability of
| exactly 50%?
| defrost wrote:
| But not running any large scale empirical studies and by
| ignoring any coins that landed on their edge.
| derbOac wrote:
| Playing games with my family, I've often wondered if there's
| something similar with dice.
| helsinkiandrew wrote:
| I think you can argue that the experiment wasn't representative
| of 'normal' coin flips.
|
| On average, each flipper in that experiment flipped a coin over
| 7000 times, after that amount many people will have learned to
| flip in a comfortable way with less variance between physical
| action and force they use. I'd imagine that in that case the coin
| would more likely land with the same orientation.
|
| I don't think this would be true if someone flipped without
| practice.
| fbartos wrote:
| I personally did 20,100 flips and I can assure you I have no
| clue how to control the flip. I centrally got much better at
| flipping and catching the coin in hand without dropping it---
| which takes some practice on its own.
|
| (I know that there are techniques for adding the wobble to the
| toss, but I didn't study them and I have no clue how to do
| them. I think it is safe to say you don't discover them
| intuitevelly.)
| SamBam wrote:
| I think this is a great point. One flipper 7000 times is quite
| different than 7000 flippers one time, if the aim is to see
| whether there is an underlying bias.
| misja111 wrote:
| I bet that with enough practice, you can learn to toss the coin
| so that you get a much bigger than 51% chance of landing it on
| the same side.
| rootusrootus wrote:
| I think you'd need more than practice. Most people would need
| a teacher. Just doing something over and over isn't
| automatically going to make you better. E.g. The old "10000
| hours of practice makes you an expert" rule assumes
| _deliberate_ practice. And even then, it 's incorrect.
| havnagiggle wrote:
| I've always caught the coin and flipped it onto the back of my
| hand to display the result. So I guess I have an opposite-side
| bias.
| quietbritishjim wrote:
| That's the usual way of doing it.
| rootusrootus wrote:
| Huh. I thought everyone did it my way. Flip it onto the
| floor, hunt it down, and see which side came up.
|
| I'm mostly serious. I know that's not ideal. But I've never
| been able to master the art of flipping a coin onto the back
| of my hand, and even catching it mid-air is hit-or-miss for
| me. The vast majority of time I or anyone I'm with has
| flipped a coin it's ended up on the floor.
| iamflimflam1 wrote:
| I would have expected some variance between different currencies
| and denominations - but maybe that would need a lot more data.
| morelisp wrote:
| If you read the original Diaconis, Holmes, and Montgomery paper
| you'll see why it shouldn't depend on this.
|
| There's also the classic Gelman & Nolan
| https://www.tandfonline.com/doi/abs/10.1198/000313002605 "You
| Can Load a Die, But You Can't Bias a Coin", in case you're
| imagining more complex dynamical behavior.
| [deleted]
| goindeep wrote:
| [dead]
| jbandela1 wrote:
| Von Neumann described a very elegant way to get fair results from
| a biased coin.
|
| 1. Flip the coin twice
|
| 2. If you get the same result both times, goto 1
|
| 3. Now that you have different results for your pair of flips,
| use the first element of the pair of flips as your result.
|
| https://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_...
| 1980phipsi wrote:
| You can load a dice, but you can't bias a coin:
| http://www.stat.columbia.edu/~gelman/research/published/dice...
| [deleted]
| esafak wrote:
| Basically _rejection sampling_ , which is about obtaining
| variates from one distribution using variates from another. We
| have samples from p-coins (where p is the probability of
| flipping heads) and we wish to generate samples from 0.5-coins.
|
| https://www.newton.ac.uk/files/seminar/20100623134014301-152...
| oakwhiz wrote:
| Pocket change is Manchester encoded by default.
| bryanrasmussen wrote:
| ok but if the coin tends to land as starting then
|
| 1. starting from heads
|
| 2. flip heads
|
| 3. flip heads
|
| 4. starting from heads
|
| 5. flip heads
|
| 6. flip tails
|
| take 5 = heads?
|
| heads should still be more likely to occur than tails under
| this scenario, although, Zeno-like, with decreasing likelihood
| approaching zero over time?
|
| on edit: of course Von Neumann's process has more restrictions,
| leading closer to fairness.
| raphael_kimmig wrote:
| If you always start with heads the method works out. The key
| is that the first and the second toss need to be independent
| so that HT and TH have the same probability. If you influence
| the second toss based on the first one it no longer works.
| thaumasiotes wrote:
| That only works if the result of the coin is independent of
| whatever it showed on the prior flip.
|
| (You might say "of course it is!", but if that's your approach
| to the problem, you should be aware that biased coins don't
| exist...)
| ctenb wrote:
| That's really cool. The key insight is that "The reason this
| process produces a fair result is that the probability of
| getting heads and then tails must be the same as the
| probability of getting tails and then heads, as the coin is not
| changing its bias between flips and the two flips are
| independent." So you have to make sure you always start with
| the same side up.
| mucle6 wrote:
| This is very interesting, but it assumes a coin is biased the
| same way every flip.
|
| If a coin is more likely to land on the side it starts on, then
| the bias can change between flips. To fix this, we just need to
| make sure the coin starts the same side up before every flip.
| antisthenes wrote:
| > 2. If you get the same result both times, goto 1
|
| Well, yes. The wiki description basically states that you get
| to throw away results of coin tosses in some particular cases.
|
| In that sense, it's not really any different from just making
| up the results of the coin tosses entirely. There's 10000
| different ways to make your data garbage.
| ComputerGuru wrote:
| The same principle is used in embedded systems as the base of a
| prng when obtaining randomness from a possibly biased or
| imperfect source of noise (eg adc low bits or uninit memory
| values).
|
| I find it easier to understand if you say "instead of using the
| level/value as the source of randomness, use the transition
| from one level/state to the other as the bit of entropy."
| (edge-based instead of level-based) I.E. instead of head is 0
| and tails is 1, head to tail is 0 and tails to head is 1, and
| the other transitions are disregarded.
| wslh wrote:
| With all due respect to Von Neumann, intuitively I would change
| it to use the information in the two coins: one for (X, Y) and
| another for (Y, X). Not the first.
| Schiphol wrote:
| Yes, and as the second coin carries no information (because
| we are focusing now on sets of two different consecutive
| outcomes) both your and JvN's protocols are equivalent.
| k7sune wrote:
| How come when the results are the same you have to go to 1 and
| flip twice again? Can you just toss another one and use the
| last two results?
| lalaithion wrote:
| Ah, but this assumes a fixed unfairness; with this result, you
| could pretend to do the von neumann method but change starting
| sides on each flip, giving a biased result.
| ChocMontePy wrote:
| If anyone wants to test it, someone wrote a short code that
| simulates doing that 100,000 times:
|
| https://www.techiedelight.com/generate-fair-results-biased-c...
|
| The coin is biased to come up TAILS 80% of the time, but using
| Von Neumann's method in the program I got HEADS 50.035%, TAILS
| 49.965%.
| dfxm12 wrote:
| You should change your name to ChocMonte _Carlo_ Py :)
| 098799 wrote:
| Why would you test it?
|
| Probability of two heads: p*p
|
| Probability of two tails: (1-p)*(1-p)
|
| Probability of head followed by tails: p*(1-p)
|
| Probability of tails followed by heads: (1-p)*p
|
| It's not difficult to notice that if you remove the first
| two, the last two form a 50/50 distribution
| [deleted]
| toxik wrote:
| I think about it this way
|
| p(th) = p(t) p(h)
|
| p(ht) = p(h) p(t)
|
| Hence p(th) = p(ht) regardless of coin imbalance as long as
| both events actually will happen. QED.
| aqme28 wrote:
| Yeah this is simpler. You're just throwing away every
| pair that isn't a TH or HT
| [deleted]
| nofinator wrote:
| Ironically, this reminds me of a story (folk tale?) about
| Von Neumann himself.
|
| A colleague told him about the Two Trains Problem
| (https://mathworld.wolfram.com/TwoTrainsPuzzle.html), and
| Von Neumann replied with the correct answer. When his
| colleague said, "Ah! You figured out he trick!", Von
| Neumann replied, "What trick? I just summed up the
| distances in my head!"
| someone7x wrote:
| > Why would you test it
|
| Is this a wrong way to get a right answer?
| voidfunc wrote:
| Some of us suck at math.
| Natsu wrote:
| Part of the problem is that the basic statistical model
| simply neglects to differentiate between observing and
| doing, which changes the odds. This is very important
| when trying to reason about causality. When you observe
| an association like your thermometer shows a high number
| when it's warm out it's one thing, but when you set your
| thermometer to a high number you won't get any warmer.
| Whereas if you warm the room, your thermometer will rise.
| This symmetry breaking is captured by something called do
| calculus.
| williamstein wrote:
| Math is often much more fun and compelling for some
| people when you both theoretically prove something works
| and then also convince yourself of the same thing via a
| numerical experiment. I'm pretty good at math proofs
| (pure math PhD, wrote some books and papers), but I still
| love to do numerical experiments. It's fun, and you also
| set yourself up to be able to easily ask different
| questions that may be very hard to answer theoretically.
| foobarian wrote:
| For me it's a case of, "see, what I do is powerful after
| all!" after a 5 minute analytical proof matches 3 hours
| of simulation work. :-)
| User23 wrote:
| Some of you might have just suffered from poor math
| education. I don't believe anyone capable of learning to
| program competently lacks the cognitive horsepower to do
| math competently with more or less equivalent ease. Many
| do however lack the training.
| HideousKojima wrote:
| Also certain unintuitive things in math/statistics (like
| the Monty Hall problem) because a lot clearer when you
| write up a quick simulation.
| cantrevealname wrote:
| > _Why would you test it?_
|
| I recall conversations on Usenet decades ago about the
| Monty Hall problem[1] in which people gave elementary
| proofs that probabilities don't change by opening a door.
| Even from mathematicians and statisticians. People were
| very insistent that the analytical solution was simple and
| obvious and that switching doors didn't change anything.
|
| The _only_ thing that changed some people 's minds was a
| program that simulated the Monty Hall problem. This was
| needed to get people to reconsider their proof when the
| claim was highly counterintuitive.
|
| [1] https://en.wikipedia.org/wiki/Monty_Hall_problem
| CuriouslyC wrote:
| You can demonstrate the Monty Hall problem solution
| analytically with Bayesian statistics using prior
| probabilities, no need to go all the way to Monte Carlo
| methods.
| TOMDM wrote:
| As a proof the math is entirely sufficient, but for those
| who may struggle with it the simulation is persuasive.
| selimthegrim wrote:
| This was how I convinced someone (RIP) by really
| stressing every element in the definition of Bayes'
| theorem and probability space.
| jackfoxy wrote:
| The Monty Hall problem is a fun one to code up, and yeah,
| there are otherwise smart people who refuse to believe
| it.
|
| I coded it up in F#
| https://github.com/jackfoxy/LetsMakeADeal to convince one
| of the founders of a start-up I worked for. He just
| grunted and walked away. Pretty sure he still doesn't
| want to hear about Bayes' Theorem.
| mensetmanusman wrote:
| Just start with an infinite number of doors and move
| backwards from that.
| bscphil wrote:
| I think the great advantage of "simulation", for the
| programming-literate, is not that you can simulate your
| way to a correct answer, but that the _process_ of
| creating a simulation is likely to show you the error in
| your reasoning.
|
| As a young teenager, I encountered the Monty Hall problem
| for the first time, and I didn't believe that the
| "analytical" answer was correct. I decided to simulate it
| by programming. In the 20 minutes it took me to write a
| simulation, I went from complete incomprehension to a
| full understanding of why I got the results I got.
| Programming a simulation of the problem _forces_ you to
| write out the algorithmic significance of "Monty reveals
| one of the goats".
| lcnPylGDnU4H9OF wrote:
| Something that makes this a lot more intuitive is to
| increase the number of doors. If there's 100 doors, _the
| host will open all remaining doors except 1_ , and _they
| will never open the door with the car behind it_ , then
| one has a 1% chance of winning the car if they don't
| switch doors and it would happen only because they
| initially chose the door with the car.
| tshaddox wrote:
| I'm curious. When those people see the simulation, do
| they then go back to the analysis and uncover their
| mistaken reasoning? Or do they just continue to reject
| the analysis but begrudgingly accept the outcome of the
| simulation? The analysis of the Monty Hall problem is so
| very simple I find it very odd to staunchly reject it
| _but then_ be persuaded by the simulation.
| [deleted]
| I_Am_Nous wrote:
| The Monty Hall Problem is a fun one because you can try
| to approach it from a purely analytical perspective and
| get one answer, while incorporating the whole situation
| (especially the fact that the final probability is not
| natural as they force the final decision into far fewer
| doors than originally present) and testing you can find a
| different answer entirely.
|
| I suppose this is an interesting corollary with
| discoveries made by deep theoretical mathematics. While
| something may seem possible because "the math checks out"
| it could be only theoretically possible as it relies on
| some unnatural value to "be" possible in the first place.
|
| Testing is where hopeful theories are smashed by reality
| until all that remains is the verifiable truth. Truly,
| why wouldn't we test?
| pmontra wrote:
| That's how it went when I was solving problems at the
| Statistics course at university. I modeled the problem
| perfectly, got the wrong result. Changed assumptions, got
| the wrong result. Checked the solution, its reasoning
| didn't make much sense anyway. Run a simulation, got an
| approximate result close to the correct solution.
| dotancohen wrote:
| This sounds like the classic "tweak the model until the
| results fit with our preexisting conclusion". Very common
| across all industries unfortunately.
| zh3 wrote:
| Also known in a derogatory fashion [0] as "adding
| epicycles" (after the Ptolemaic view of the heavens).
|
| [0] https://en.wikipedia.org/wiki/Deferent_and_epicycle#B
| ad_scie...
| naniwaduni wrote:
| Fundamentally, the trouble with the Monty Hall problem
| isn't that analysis comes to the wrong answer, it's that
| people often come to the wrong _model_ when reasoning
| about it informally.
|
| It's not any harder to do the "correct" analysis than to
| write up a simulation. It's mostly just easier to
| convince yourself that the simulation matches the problem
| description when it reaches the unintuitive result.
| s1artibartfast wrote:
| Fundamentally, I think the real trouble with the Monty
| Hall problem is that the assumptions of the game are not
| clearly stated. Because of this, people come up with
| different models.
| alexdowad wrote:
| That's absolutely right; further, if you explicitly model
| the behavior of the game show host, you can exhibit
| models under which "it's better to switch" and models
| under which "it doesn't matter if you switch or not".
| vince3455 wrote:
| >models under which "it doesn't matter if you switch or
| not".
|
| Could you provide an example? It seems obvious that a
| switcher wins exactly when a non switcher looses, which
| is 2 / 3 ?
| alexdowad wrote:
| Take a game show host who lets you choose a door,
| randomly reveals what is behind one other door, and then
| gives you an opportunity to change your choice. This game
| show host CAN (randomly) reveal the prize; he has equal
| probability of revealing ANY of the unchosen doors.
|
| Say you are playing the Monty Hall game with this host.
| You choose your door, he opens another door, and it
| happens (purely by chance) that there is no prize there.
| Do you still believe that you have a 2/3 chance of
| winning if you switch to the other unopened door?
| Natsu wrote:
| This modelling ambiguity is resolved by do calculus,
| which makes a clear distinction between intervention and
| observation: https://arxiv.org/pdf/1305.5506.pdf
| Natsu wrote:
| This isn't so mysterious once you learn a bit of do
| calculus and realize that observation and intervention
| are fundamentally different things.
| chankstein38 wrote:
| This was how I built an intuition into the Monty Hall
| problem as well! Wrote a little app that simulated it a
| decade or so ago when I was discussing with friends!
| dfxm12 wrote:
| The Monty Hall problem is especially unintuitive if
| you've ever watched Let's Make a Deal, since the problem
| set up is oh so close to, but not exactly, the set up of
| the Big Deal in the show. It's too easy to conflate the
| rules of the show with the math problem, which will lead
| to confusion.
|
| I think seeing the results of a simulation also
| elucidates the set up of the math problem vs reading a
| proof.
| cortesoft wrote:
| Yeah, I feel like the Monty Hall confusion goes away if
| you are explicit about the rules:
|
| "Hall will always open one of the two non-chosen doors
| and will never reveal the prize"
|
| I think most people who don't understand the problem miss
| that critical detail.
| bigstrat2003 wrote:
| No, I don't think that detail makes it any easier. I know
| that but I still really can't accept the correctness of
| the Monty Hall strategy (I have to basically just take it
| on faith and stop trying to understand it). I was trying
| to put my finger on why, and I think it's this.
|
| After Monty eliminates one of the three doors, then the
| prize is behind one of the two. If someone were to come
| in this point, with no prior knowledge whatsoever, their
| chance of picking the correct door at random is 1/2. And
| that is still true even if they pick the door which our
| contestant is being asked whether or not to switch from!
| This is a real mind fuck to try to accept, that the same
| state of what's behind each door leads to different odds
| of making a correct random choice, depending on when you
| make the choice.
|
| I honestly don't think I'll ever be able to "get" the
| Monty Hall strategy. I think I get why it works (choosing
| to switch means you're going from a 1/3 probability to
| 1/2), but it makes no sense at all. It seems like even if
| you choose to stay on the same door, your probability is
| 1/2 (the same as if Joe came in off the street and chose
| the same door as you). Like I said, I just have to take
| it on faith.
| dfxm12 wrote:
| Maybe this will help understand it intuitively. You have
| a choice between doors 1 2 3. You pick door 1. You know
| the odds of the car being in door 1 is 1/3. The odds of
| the car being in door 2 _or_ door 3 are 2 /3.
|
| Monty opens door 3, showing a zonk. You _knew_ there was
| a 2 /3 chance of the car being in door 2 or 3, but _now
| you know_ there 's a 2/3 chance of the car being in door
| 2 (since you know it is not in door 3).
|
| All this didn't change anything you _know_ about door 1.
| It has the same 1 /3 chance it started with. Probability
| is all about what you _know_ in the moment.
|
| The math involves understanding the rules, that Monty
| will never open the door you picked and will never open
| the door with the car behind it. This is why one can't
| look above and say "well, there is a 1/2 chance of the
| car being behind door 1 after door 3 was opened and there
| wasn't a car there". This would only be true
| mathematically if the door Monty opened was random, but
| we _know_ the door Monty picks isn 't random. In fact,
| the pool of doors that could be opened depends on your
| initial pick. Monty was _never_ going to open door 1 (the
| door that you picked), even if it was a zonk & Monty was
| never going to open the door with the car, therefore one
| can't make that assertion.
| cortesoft wrote:
| I have fun trying to explain this problem. Let me see if
| I can give an explanation that will help you.
|
| So lets say you have just picked a door in the beginning.
| You know you have a 1/3 chance of being right.
|
| If I then tell you, "I will give you two options... you
| can either bet you are right, or bet that you are wrong"
|
| You would obviously choose to bet you are wrong, correct?
| Because you know you only have a 1/3 chance of being
| right with your guess, which means you have a 2/3 chance
| of being wrong. The smart bet is that your original guess
| was wrong.
|
| This is actually what is happening in the game if you
| think about it. You pick a door and it has 1/3 chance of
| being the right one; since we know Monty is only going to
| ever reveal a goat and never the prize, we don't even
| NEED Monty to reveal the door at this point - we know he
| is going to reveal a goat, no matter what. We don't even
| have to wait to see which door he reveals, since that
| isn't going to give us more information (it is going to
| be a goat, no matter what). So when he asks you if you
| want to switch doors, he isn't asking you to switch to
| ONE of the other two doors, he is asking if you want to
| switch to having BOTH other doors as your choice. Whether
| he reveals the goat before or after you choose to switch
| doesn't matter, because you know it will always be a
| goat.
|
| If that is still not clear, lets just write out all the
| options:
|
| There are three doors, A B C. One has a prize, the other
| two have goats. Let see what happens with your two
| options (switch or dont switch).
|
| In our first example, you pick door A and you are going
| to switch.
|
| 1/3 of the time the prize is behind door A. If the prize
| is behind door A, and you switch, you lose. This is 1/3
| of the time, and you lose for switching.
|
| 1/3 of the time the prize is behind door B. You picked
| door A, so Monty reveals door C. You switch to the
| remaining door (B) and you win.
|
| 1/3 of the time the prize is behind door C. You picked
| door A, so Monty reveals door B. You switch to the
| remaining door (C) and you win.
|
| Add up all those choices, and 2 out of the 3 times you
| win.
|
| Now lets imagine that we DON'T switch.
|
| 1/3 of the time the prize is behind door A. Monty reveals
| one of the other doors, but you don't switch. You win.
|
| 1/3 of the time the prize is behind door B. Monty reveals
| door C, but you don't switch from A. You lose.
|
| 1/3 of the time the prize is behind door C. Monty reveals
| door B, but you don't switch. You lose.
|
| So in this not switching world, you win 1/3 of the time.
|
| In summary, switching wins 2/3rds, not switching wins
| 1/3.
|
| Does that help at all?
| benchaney wrote:
| The best probability estimate you can make is constrained
| by the information you have available. The new person
| showing up has less information than the existing
| constant, so it makes sense that their best estimate
| would be less precise. Similarly, if someone with x-ray
| vision walked up in the middle of the game, they could
| pick the car 100% of the time, because they have access
| to more information than either of the existing
| contestants.
|
| Your last paragraph isn't correct though, By switching
| you go from a 1/3 probability to a 2/3 probability. Based
| on the information the original contestant has, switching
| gets the car 2/3 of the time.
| bigstrat2003 wrote:
| I don't see how a new contestant has less information,
| though? They know that one of the two doors contains the
| prize, which is all the previous contestant knows either.
| benchaney wrote:
| When one door was opened it revealed information about
| the other two doors.
| cortesoft wrote:
| The crucial bit of information that the new contestant
| doesn't have is that there was a door that was ineligible
| to be eliminated (the door chosen by the original
| contestant).
|
| If the game had different rules, it would work like you
| are imagining. Specifically, if Monty randomly eliminated
| one of the two doors, meaning there was a chance for
| Monty to reveal the prize instead of a goat. If Monty has
| the chance to eliminate the prize before giving the
| contestant a chance to switch, then switching does not
| give you an advantage.
| 4star3star wrote:
| The way the problem makes sense to me is this.
|
| Doors: Goat Goat Car
|
| You pick a door. Monty shows you a Goat. You switch or
| stay.
|
| Monty will never show you the Car before offering a
| switch. He always shows you a Goat. It doesn't matter
| which Goat he shows you - it's just "not the Car".
|
| If your first choice is a Goat, switching will win you
| the Car. If your first choice is a Car, switching will
| win you a Goat. You have a 2/3 chance of picking a Goat,
| so, effectively, you want to pick a Goat so that you
| switch to the Car.
| LoganDark wrote:
| This is the best explanation I've seen for the problem so
| far. Thank you
| lukasb wrote:
| Even simpler - you pick, knowing nothing, so there's a
| 2/3 chance you're wrong.
|
| If you're wrong, Monty points to toward the right door.
|
| So you should switch.
| aplusbi wrote:
| The best way that I've thought about it is like this:
|
| You pick a door, then Monty let's you switch to the two
| remaining doors and if the car is behind either of them
| you win.
|
| Obviously choosing the two remaining doors is better.
|
| The trick is to realize that Monty showing you the
| contents of one door and letting you choose the other one
| is identical to Monty letting you choose both the
| remaining doors.
| tacitusarc wrote:
| For sufficiently analytical folks that works, but for lay
| people it tends to still be confusing.
|
| The best way I've heard it explained to help people get
| it through intuition is by changing the number of doors
| and goats. Say there are 100 doors, and they all have
| goats except one, which has a car. You pick door 1. Monty
| then proceeds to open doors 2 through 48, skips door 49,
| and then opens the remaining doors. After all that, he
| stops and asks you, would you like to switch?
| thaumasiotes wrote:
| There's a better way to think about it.
|
| 1. You pick a door.
|
| 2. You get the offer "Do you want to keep that door, or
| choose both [all] of the other doors? In either case,
| you'll keep anything that isn't a goat."
|
| 3. Nobody opens any doors.
|
| Should you keep your one door, or switch to the two
| doors?
| tomrod wrote:
| This is the right way to think about it. Very clear and
| concise, thank you.
| matsemann wrote:
| I've never been happy with that explanation. I don't get
| why the host would not just open a single door, that's
| what the host does in the other scenario to me.
| cstrahan wrote:
| > that's what the host does in the other scenario to me.
|
| Is it, though? It seems apparent that, after the first
| guess, the host opens all but the last two doors, which
| just so happens to be 1 door.
|
| To check the math:
|
| Start with the $NUM_DOORS open doors. Now open all but
| the last two. So that's $NUM_DOORS-2, which is 3-2, which
| equals 1 open door.
| matsemann wrote:
| My interpretation of the host opening a single door and
| asking if you want to switch is equally valid when you
| expand it to 100 doors.
| joshuamorton wrote:
| Sure, it could be that the host only opens one door, or
| it could be that he opens all but one door. In _every_
| case, however, it is better to switch. The all-but-one
| example is hyperbolic but still follows precisely the
| same mathematical rules. Your interpretation is valid,
| but _so is_ the all-but-one example, and they all lead to
| the same result, it 's just more obvious when you open
| nearly all the doors.
| s1artibartfast wrote:
| I always feel like there is something fundamental missing
| from the examination of Monty Hall problems.
|
| I think it has to do with the difference between
| "probable outcome in reality" and "probably outcome based
| on personally known information".
|
| Lets say when you get down to doors #1 and #49, Monty
| brings in someone new, with no information and says pick
| a door. For that new person, standing right next to you,
| doors #1 and #49 have a 50-50% chance, while for you they
| are a 2% vs 98% chance.
|
| How can door #1 simultaneously have a 2% chance for you
| and a 50% chance for Bob? The answer is that the chance
| is not a single fixed property of the door itself- which
| is hard to wrap ones head around.
|
| And for that matter, Monty Hall himself knows one of the
| doors is 100% and the other is 0%.
| Natsu wrote:
| There is something missing: regular stats don't
| differentiate between _doing_ things and _observing_
| things and these two are not at all the same. If I have a
| digital thermometer and I observe it to show a high
| temperature, then I will note an association between that
| and feeling warm. But if I merely set the thermometer
| gauge to a high value artificially, it 's not going to
| make me feel any warmer.
|
| This ambiguity is resolved by something called do
| calculus - https://arxiv.org/pdf/1305.5506.pdf
| s1artibartfast wrote:
| I think it is more fundamental than that, and not even
| mathematical. I think the issue is that people conflate
| or blur the difference between reality and their models
| of reality.
|
| Your personal, information limited calculation of the
| chance a car is behind door #1 has no impact on if there
| is a car behind door #1. Reality is binary and constant.
| There was always a car there, or there always wasn't.
|
| Most people correctly intuit that of course the _real_
| probability that the car is behind door #1 cant change
| with reveled information. It isn 't a quantum car. They
| just get caught up on the fact that predictive chance is
| a attribute of the model, not the real door.
| Natsu wrote:
| I agree that the map is not the territory, but there is a
| better model here that captures the difference.
|
| The car isn't moving, as you say, but that intervention
| by the host lets us trade one door for both of the other
| doors.
| evouga wrote:
| The situation is now counterintuitive in the other
| direction: if Monty Hall had opened those 48 doors at
| random and they just happened to not contain the car,
| then there is no advantage to switching, though many
| people would insist otherwise.
| joshuamorton wrote:
| But the doors _weren 't_ opened at random. You know he
| won't open the car, because that's part of the rules of
| the game.
|
| Let's demonstrate with a slightly different construction:
| You're no longer playing with monty, but with a demon.
| This demon wants you to lose, but also picked a very bad
| game for themselves. You pick a door, then the demon
| opens all-but-one of the remaining doors. Then, you can
| pick _any_ door, open or closed, and you get what 's in
| it.
|
| If the demon opens doors at random, _nearly all the time_
| (with 100 doors) you 'll see the car and be able to pick
| it directly. In this situation, switching between the
| closed doors doesn't really matter, but you'll usually
| know exactly which door to pick, because you can see the
| car.
|
| So instead, the demon only opens doors that _don 't_ have
| a vehicle behind them. You only ever see goats. At this
| point, he's _not_ opening doors at random. If he were,
| you 'd see the car 98% of the time, but you never do. At
| this point, since he's using additional information, it
| is in your best interest to switch.
| Sohcahtoa82 wrote:
| > if Monty Hall had opened those 48 doors at random
|
| The fact that Monty Hall opens the doors
| deterministically (not randomly) is _KEY_.
|
| In the original problem, Monty ALWAYS opens a door with a
| goat. In using 50 doors, Monty would ALWAYS open doors
| containing goats, and not the car. It's not random.
|
| Knowing it's not random, it should be very intuitive.
| strangattractor wrote:
| The Monty Hall problem is small enough to enumerate all
| the out comes on paper. If you then test all the
| different scenarios (not many) and compare switching to
| not switching - switching produces a slight edge. So it
| can be done without a computer and not a great deal of
| effort. I am not a mathematician so it was the only way I
| could prove it to myself at the time.
| BurningFrog wrote:
| You're of course right, but maybe 1% of the population
| understands that, while 100% understands the practical
| test.
| gorjusborg wrote:
| > Probability of two heads: p _p > Probability of two
| tails: (1-p)_(1-p) > Probability of head followed by tails:
| p _(1-p) > Probability of tails followed by heads: (1-p)_p
| > > It's not difficult to notice that if you remove the
| first two, the last two form a 50/50 distribution
|
| Very nice way to illustrate why throwing out the duplicate
| sequences gets back to a 50/50 distribution.
| bob88jg wrote:
| Why would you not - analytical solutions are the rare
| occurrences might as well approach everything with
| simulation...
| kqr wrote:
| While I agree we should leave the correct answer to
| simulation, analytical approximations are often
| surprisingly close and have the benefit of being
| intuition-building.
| CuriouslyC wrote:
| You learn a lot more from generating an analytical
| solution than a simulation, so it's usually worth at
| least taking a stab at it analytically before jumping to
| monte carlo methods.
| taway_6PplYu5 wrote:
| (preface with "in today's world")
| thih9 wrote:
| > Why would you test it?
|
| Why not?
|
| The fact that you can show something with a mathematical
| equation doesn't make other demonstrations any less cool.
| sidkshatriya wrote:
| The thing about math is that you can do things in multiple
| ways.
|
| Theory is useful but so is experiment.
| jszymborski wrote:
| Some folks have more faith in their ability to derive
| proofs than write simulations and vice-versa.
| skrebbel wrote:
| > It's not difficult to notice that
|
| Look I found the mathematician
| chaorace wrote:
| Wow. As usual, Von Neumann makes it look easy
| nullc wrote:
| The VN debiaser is very simple but it's not very efficient-- it
| loses a lot of your randomness.
|
| Under the same IID assumption you can take N flips that
| returned M heads and map them to the N choose M possible ways
| that could have happened. The result will (under IID
| assumption, even in the presence of bias) be a uniform number
| on the range [0..N choose M). The ctz(N choose M) trailing bits
| can be used directly (as they will be uniform) but the rest
| would have to be converted to binary via something like an
| arithmetic coder or rejection sampling.
|
| The result is muuch more efficient.
|
| Less directly, VN debiasers can also be stacked. Each debiaser
| outputs three streams: the normal one, one that says if the
| normal one output anything, and one that says if it got HH or
| TT. Then run VN debiasers on those. Though it takes a fairly
| large tree to extract most of the entropy.
| dentalperson wrote:
| I love the math/stats history around gambling-related things,
| thanks for mentioning this. This method assumes the flipper
| can't introduce bias. ET Jaynes in his book Probability Theory
| also mentions that it is easy to learn to flip a fair coin in
| such a way that the result can be predetermined. I searched a
| tiny bit for this but couldn't find what he was referring to
| though.
| ryanar wrote:
| Lots of practice, use your thumb to hit the edge of the coin
| to give you more control, aim for a specific spot so you use
| the same amount of force and control the amount of times it
| flips. You can also use a surface that absorbs more so the
| coin is less likely to flip after hitting it.
| aspenmayer wrote:
| I've heard that, with many hours of practice, dedicated
| amateurs and many famous magicians are able to do this kind
| of thing. I wouldn't call it sleight of hand, but it is
| similar, although it may fall under that category broadly.
| I'm not a domain expert but I was taught some simple coin
| tricks as a child by my artist mom's artist friend who ran
| the local frame shop. I never tried or thought to try to
| favor the coin flip or introduce bias, but it's definitely a
| skill that can be acquired.
| kqr wrote:
| I remember at one point in my childhood learning a trick
| where it looks like you're flipping the coin, but you're
| really only causing it to rotate and wobble, meaning it's
| guaranteed to land on whatever side faced up as you tossed
| it. I don't remember how I did it though, and a few minutes
| of trying to recreate the effect has failed.
| stainablesteel wrote:
| you can do that all you want my coin is the same on both sides
| mewpmewp2 wrote:
| That's amazing, but I guess it won't help when the person can
| choose the bias?
|
| Because according to the study the person can choose the bias
| by choosing which side start up.
|
| So if the person wants tails based on what you've said, they
| should always
|
| 1. Do the first throw starting tails up.
|
| 2. If the first one is tails, then they now want to start
| second one heads up.
|
| 3. If the first one is heads, they will want to try and get
| heads again to dismiss the results. So they will do heads up.
|
| So assuming for example that they have an ability to control
| bias 75% vs 25%.
|
| Then there would be 75% chance of getting first as tails. After
| that 75% chance of getting heads.
|
| So they will have 56.25% chance of getting it right the first 2
| rounds.
|
| The worst case for them would be if they get heads first (25%
| chance), and then are unable to get heads again. Which would be
| another 25% chance so 6.25% odds to lose with the first round.
|
| So 56.25% chance of winning the first round of 2, or 6.25%
| losing and 37.5% of having to try again.
|
| And I think the odds would converge at somewhere around 90% to
| 10%. I didn't do full calculations here, but overall it seems
| this strategy would increase the bias even more.
| Retric wrote:
| The final calculation is easy 56.25/(56.25 + 6.25) = 90%,
| unless the persons skills change between rounds or something.
| mewpmewp2 wrote:
| Yeah, thought so as well, interesting how easily those
| numbers worked out, but then again it's because 75/25=3 and
| 3x3 = 9 so the final difference must be 9x between the
| probabilities or 100 / (9 + 1).
|
| I was still lucky with the numbers as for example with 80%
| vs 20% it would've been 4x4=16 and so 1 to 16 comes to 100
| / 17.
| wesleychen wrote:
| You can solve this easily by always flipping with the same
| side (doesn't matter which) facing up for all flips.
| mewpmewp2 wrote:
| Unless they can also introduce bias using
| strength/technique of the throw.
| havnagiggle wrote:
| There is skill to coin flipping. You'd need to blind the
| flipper, either physically blindfold or make it so they
| don't know which result is the positive outcome ahead of
| time.
| bee_rider wrote:
| Or ask the competitors to flip the coin in a manner they
| doesn't allow for skill, like put it in a Yahtzee cup and
| toss from there.
| xp84 wrote:
| I was imagining spinning the coin with a flick of the
| finger. That doesn't seem to be gameable to me, but I
| supposed you'd need to do a lot of flicks to see if
| flicking the head side or tails side matters. I'd think
| there's no way a coin can be more likely to spin an odd
| or even number of times before falling, but weirder
| things have happened.
| kqr wrote:
| > That's amazing, but I guess it won't help when the person
| can choose the bias?
|
| Alice writes on a piece of paper whether to use the result
| from the first or the second coin, Bob flips the coins
| however he likes, then once there are two different sides of
| the coins up, Alice turns over the paper and reveals to Bob
| which coin contains the result.
|
| Though I guess that unnecessarily complicates the procedure -
| maybe Alice can just write "heads" or "tails" on a note and
| then Bob flips without having seen the note. It essentially
| replaces the second coin with Alice's mind which hopefully
| doesn't suffer from the same known bias.
| mewpmewp2 wrote:
| Suppose Alice needs to take the coin first to herself, to
| use the aforementioned strategy without intentionally
| introducing bias, and then using result of that, which
| would determine whether the first or the second result from
| Bob would be used. Because otherwise Bob may be able to
| make psychological "guesses".
| c22 wrote:
| deg Put the coins in a cup
|
| deg Shake the cup vigorously and dump coins on the table
|
| deg If coins match, go back to step one
|
| deg If coins are opposed take the result of the
| southernmost coin
| amluto wrote:
| There are nice protocols like this that don't require
| anyone to visibly flip a coin. See, for example:
|
| https://en.m.wikipedia.org/wiki/Commitment_scheme
| sebzim4500 wrote:
| If you're going to go that way you can skip the coin flip
| entirely. Just get both of them to write heads or tails on
| a note and then compare. This technique is used in some
| crypto projects, except instead of writing on a note you
| share cryptographic commitments.
| mewpmewp2 wrote:
| But they need to remove the possibility of a
| psychological guessing game. E.g. Bob could've researched
| before hand that people are 55% likely to pick heads if
| they can pick by themselves.
| arijun wrote:
| That doesn't remove the possibility of a psychological
| guessing game, just makes it more convoluted. If Bob
| knows Alice will pick first, he can still bias the
| results.
| CuriouslyC wrote:
| Inconceivable!
| kibwen wrote:
| At this point you can just play odds and evens: one person
| picks odd, the other picks even, they both hold up either
| one or two fingers behind their back, reveal them at the
| same time, then sum the result. This prevents the
| randomness from being in any one actor's control. If you're
| worried that your brain's RNG can be gamed, then put an
| odd-denominated coin in one hand and an even-denominated
| coin in another, and mix them up so that even you don't
| know which hand has which.
| mewpmewp2 wrote:
| We still need a study then to confirm that when people
| try to mix the coins in their hands like that, it would
| be random enough. And that would take another year...
| InitialLastName wrote:
| I can feel the difference between denominations of my
| local coins no problem. What you need are a pair of coins
| with an odd year imprint and an even year imprint.
| asimpletune wrote:
| But wasn't the bias in the paper something like 50.5% vs
| 49.5%?
| mewpmewp2 wrote:
| Yes, but I used more extreme numbers for ease of
| calculation and to clearly indicate the direction of a
| probability.
| dataflow wrote:
| I'm confused, how does this help? If coins are biased to land
| same-side up, then don't I always have an advantage by guessing
| whatever side is up before the first throw?
| jpeterson wrote:
| The probability of [HEADS, TAILS] is always the same as the
| probability of [TAILS, HEADS], no matter how the coin is
| weighted.
| dataflow wrote:
| I get that but I don't see how it answers my question?
| arrowsmith wrote:
| You're flipping pairs of coins until you get either "HT"
| or "TT". So the only two possibilities are:
|
| 1. keep flipping until you get HT (and so you choose
| 'heads') 2. keep flipping until you get TH (and so you
| choose 'tails')
|
| Since HT and TH are equally likely, results 1 and 2 are
| equally likely, i.e. there's a 50% chance of choosing
| heads, 50% change of choosing tails.
| [deleted]
| kqr wrote:
| I see what you're getting at, and it's subtle! To simplify
| the discussion, let's assume we always start out with heads
| up.
|
| You're right that the first coin is more likely to end up
| heads. But so is the second coin, and if both occur, that
| would invalidate the pair of tosses. Now, imagine you guessed
| tails despite the coin starting on heads. If the first toss
| lands tails, the second coin is still more likely to land
| heads, which keeps the pair valid.
|
| In other words, whatever you gain by guessing the side that's
| up on the first coin, you lose on account of the second coin
| having that same higher probability of invalidating the pair.
|
| ----
|
| Using extreme numbers, in case that makes it more clear:
| imagine a coin that has a 99 % probability of ending up with
| the same side we start with, and - for simplicity of
| exposition - we always start with heads facing up before the
| toss.
|
| If you guess heads, and the first coin lands heads, then
| there is a 1 % chance that you win, namely that when the
| second coin lands tails.
|
| If you guess tails, and the first coin lands tails, then
| there is a 99 % chance that you win, namely that when the
| second coin lands heads.
|
| The two outcomes of the first coin (99 % and 1 %
| respectively) perfectly balance out the two valid outcomes of
| the second coin.
| dataflow wrote:
| Ah fascinating! Thank you!
| vikingerik wrote:
| Also interestingly, this extends beyond a two-sided coin, to
| any number of possible results, like a die with N sides.
|
| To get a fair result from a biased dN: Roll it N times. If you
| don't get all N distinct results, restart. If you do, then the
| first of those is your final result.
| aidenn0 wrote:
| That is only guaranteed to work if subsequent tosses are
| independent of each other. TFA suggests that they are not.
|
| [edit]
|
| If you always start with the same side of the coin face-up,
| then the tosses will be independent of each other, but if you
| e.g. always flip it once or always keep it the same before the
| next toss, then they are not.
| elijaht wrote:
| TFA does not suggest that. Even if a coin is biased towards
| the side it started on, this wouldn't carry over from one
| flip to the next.
|
| It would be important to start the flip on the same side, but
| that's doesn't make the second flip dependent on the first
| aidenn0 wrote:
| Hah, realized that after I posted and edited while you were
| commenting. It's a good point. My original assumption would
| you'd just pick it up and flip it; not attempt to put the
| same side face up each time.
| m3kw9 wrote:
| Flip till I get the side I wanted
| arkitaip wrote:
| I absolutely love that even with a corrupted system you can use
| properties of the system to ensure just outcomes.
| sph wrote:
| This works only if the bias is constant/independent of the
| result of the previous flip.
| mnw21cam wrote:
| https://www.giantitp.com/comics/oots0327.html
| cantrevealname wrote:
| I'm still looking for an intuitive or ELI5 explanation of the
| _mechanism_ for this bias.
|
| The original paper says:
|
| The standard model of coin flipping was extended by Persi
| Diaconis who proposed that when people flip a ordinary coin, they
| introduce a small degree of precession' or wobble--a change in
| the direction of the axis of rotation throughout the coin's
| trajectory. According to the Diaconis model, precession causes
| the coin to spend more time in the air with the initial side
| facing up. Consequently, the coin has a higher chance of landing
| on the same side as it started.
|
| Another coin toss experiment[1] site says this:
|
| The basic reason is that, instead of rotating around a horizontal
| axis as one might imagine, a typical tossed coin is rotating
| around a tilted axis which is precessing in 3-space, and this
| entails a certain degree of "memory" of the initial parameters.
|
| The Diaconis paper[2] has the definitive explanation but it's
| hardly intuitive. I got a feel for why it is, but I can't do an
| ELI5. The best I'm able to write is this: A human being is likely
| to introduce some precession in the coin toss. If there is
| precession, then the angular momentum vector is going to spend
| more time in the heads direction if starting from heads, and that
| accounts for the bias.
|
| What I think would work well for an ELI5 is an animation of a
| coin toss showing the angular momentum vector sweeping out a
| region during its flight, and showing it spends slightly more
| time pointing toward heads.
|
| [1] https://www.stat.berkeley.edu/~aldous/Real-
| World/coin_tosses...
|
| [2] http://epubs.siam.org/doi/10.1137/S0036144504446436
| Q_is_4_Quantum wrote:
| Perhaps it helps to imagine someone had a "screwy thumb" and
| the coin _only_ precesses when they "flip" it (in fact people
| can train themselves to do this, and its very difficult for
| you, the sucker, to see in the air that the coin is not
| rotating but just precessing!). Hopefully its obvious that
| whatever side is initially facing up will be the same one
| facing up when its caught?
|
| The next step is not at all intuitive to me, namely that even
| someone trying to do a fair flip causes some precession, and
| that this isn't decoupled from the rotation.
| dmarchand90 wrote:
| Excellent ig nobel prize contender here
| [deleted]
| hurtuvac78 wrote:
| This is incredibly puzzling.
|
| Is there a minimum number of rotations per coin flip to consider
| it valid?
|
| If looks like the bias was not evenly distributed across people.
| How did you protect your experiment from skilled bad actors who
| could influence the data with a few bad/skilled flips? Did
| strangers on the internet fare any differently than in-person
| attempts from trusted people?
| fbartos wrote:
| We told people that the coin has to flip at least once (which
| would bias it for the opposite site). Whenever instructing
| people, I tried to explaining that the coin flip should look
| like you were trying to determine an outcome of a bet. You can
| find the complete experimental protocol here:
| https://osf.io/hkv8p
|
| Also, I wish I had (any) budget to hire proffesional skilled
| tossers haha.
| cantrevealname wrote:
| I assumed that the 1% bias was entirely due to coins that did
| not undergo any rotation at all. However, reading that you
| told people that the coin has to flip at least once, I think
| I assumed wrongly. It sounds like the bias is due to coins
| that have undergone an integral number of 360-degree
| rotations (not zero rotations). But what exactly is the
| physical mechanism causing this bias? It's easy to understand
| why zero rotations would introduce a bias, but I can't easily
| picture a reason for a bias toward an integral number of
| 360-degree rotations. Is there a simple and intuitive way you
| can explain the physical reason?
| fbartos wrote:
| It's not about the number of rotations at all. I doubt that
| you can control it at all even after dozens of hours coin
| flipping (I did more than 20h and I can't eveb guess how
| many rotations the coin made) Diaconis, Holmes, and
| Montgomery (2007) proposed a physical model of coin
| flipping that introduces the bias as a result of wobblines
| (i.e., off-axis rotation in the flips).
|
| Diaconis, P., Holmes, S., & Montgomery, R. (2007).
| Dynamical bias in the coin toss. SIAM Review, 49(2),
| 211-235. https://doi.org/10.1137/S0036144504446436
| morelisp wrote:
| Why do you think this is puzzling? This bias has been
| analytically and dynamically predicted for years.
| hurtuvac78 wrote:
| Because I find it counter-intuitive. And because I am not
| aware of scientific development in this field.
|
| @bradrn on this thread kindly extracted the description of
| the proposed physical model from inside the paper, it is
| helpful: https://news.ycombinator.com/item?id=37830265
| brettwall wrote:
| Based my back-testing of stock market, I found a similar
| conclusion: if a stock rise yesterday, then today the probability
| of raise > the probability of fall. P(raise) is about 50.1%,
| P(fall) is about 49.9%. Vice versa. Having this theory means you
| can't rely on a single bet, you have to bets many many times to
| make profit from stock market. Even though I knew that, I am
| still working as a developer, I wish one day I have enough money
| to start stock career.
| mewpmewp2 wrote:
| Alert: It's not going to be as simple as this to make money
| even with large numbers. So don't fret about not having the
| money and missing some golden opportunity.
|
| There are enough actors out there trying to develop complex
| algorithms to find an edge, and so there wouldn't be any simple
| edges like this left anywhere as they would be arbitraged away.
|
| If there is a simple pattern, it is noticed and traded until
| the pattern disappears.
| kqr wrote:
| Has hmate9 pointed out, a simple pattern like the one
| described can persist indefinitely - but the risk of
| exploiting it is priced in already.
| mewpmewp2 wrote:
| Yeah, you could find a pattern that wins 99% of the time to
| yield 1% of what you risk, but you don't consider that
| there's always 1% odds of losing it all and it's priced in,
| it just hasn't happened to happen yet, but systems with
| more precise data have accounted for it.
|
| E.g. something like selling 0dte options sufficiently out
| of money might seem like a free money hack, but once
| something unexpected happens you are down to just "who
| could have possibly foreseen that event to occur, my
| decision making was solid.".
| saalweachter wrote:
| Also, if you do find an edge, you need to be prepared for
| what happens when everyone else discovers your arbitrage
| opportunity, and/or you tap out the potential of it.
|
| A number of failed finance companies have the story: find a
| legitimate arbitrage opportunity; take on a billion dollars
| in investment to exploit the opportunity; make bank; other
| people discover your arbitrage opportunity and jump on; stop
| making bank; try riskier and riskier investment opportunities
| to keep it going; engage in outright fraud to keep it going;
| go bankrupt and/or to jail.
| mewpmewp2 wrote:
| Stock market is such a slippery addiction slope. With
| casino a reasonable person would know, the odds are stacked
| against themselves, but with Stock Market, there's no
| guarantee and it's really easy to convince yourself that
| you have an edge, and sometimes it works for a while, and
| then it doesn't, but you are already used to that feeling
| of reward, and as you said you will start to engage in
| riskier and riskier opportunities to get that feeling back.
| sjamaan wrote:
| > Having this theory means you can't rely on a single bet, you
| have to bets many many times to make profit from stock market.
|
| If you take into account trading costs, you'll probably lose
| money this way even if the theory is correct.
| Karellen wrote:
| > you have to bets many many times to make profit from stock
| market
|
| You also have to take into account transaction fees and broker
| spread. (If you get a great deal on one of these, check the
| other very carefully!) I'd be quite surprised if the edge on
| your system is enough to cover those.
| drexlspivey wrote:
| Surely the armies of quants would have found this most trivial
| edge and exploited it if that was the case?
| hmate9 wrote:
| You have to take into account how much it rises and how much it
| falls too. It might be you win more often but when you lose you
| lose more.
| mucle6 wrote:
| I don't know enough, but I bet with options you could make
| something close to a double or nothing boolean bet
| fowlie wrote:
| Very interesting! Is there reason to believe that the outcome
| would be different if the experiment was re-run but replacing the
| humans with coin-flipping machines?
| kortex wrote:
| Folks have made flipping machines than can achieve near perfect
| reliability. Same guy mentioned in the paper.
|
| https://www.npr.org/2004/02/24/1697475/the-not-so-random-coi...
| qup wrote:
| Sorry, does this mean perfect 50/50, or 100% controllable?
| rax0m wrote:
| I watched one of the 12-hour coin tossing marathons.
|
| They were sitting with their laptops and pressed a button for
| every result.
|
| I wonder if human error can explain (at least part of) the
| deviation from 50/50:
|
| * locations of the buttons they pressed on the laptops (they only
| pressed once per toss before enter, meaning the button
| represented same-side or other-side)
|
| * remembering what the coin started out as may be harder (or
| easier, but probably harder) when the result is other-side
|
| * other??
|
| Need to repeat this amount of tosses but with a higher degree of
| supervision to be sure of the result.
| fbartos wrote:
| That's actually not completelly accurate. The study protocol
| (https://osf.io/hkv8p) describes the procedure in greater
| detail.
|
| People were pressing one button for heads and another button
| for heads (which we deemend less error prone and less likely to
| be subcontiously influenced). The trick was that the next coin
| flip started the same side-up as the previous landed. Therefore
| there was no need to record the start (and we randomized the
| starting position of every 100th flip)
|
| We also did some auditing of the video recordings (trying to
| decode the outcomes from the videos) and they showed quite
| consistent degree of bias as the original responses.
| matteoraso wrote:
| I actually used to be really good at manipulating this as a kid.
| Basically, if you toss a large coin with a stiff arm, you can get
| it to flip exactly 1 and a half times before you catch it. I
| would always use this to win bets with my friends.
| gowld wrote:
| If you are allowed to cheat, you can also spin the coin to not
| flip at all.
| Q_is_4_Quantum wrote:
| Given access to repeated uses of a coin of unknown bias "p"
| (which is not 0 or 1) you can (eventually) always generate a new
| coin flip with bias given (exactly) by:
|
| 1. 1/2 (i.e fair - von Neumann)
|
| 2. p^2
|
| 3. p^2/(p^2+(1-p)^2)
|
| 4. sqrt(p)
|
| Number 4 really surprised me, I learned it from this paper:
| http://www.math.chalmers.se/~wastlund/coinFlip.pdf
|
| But you can never generate the biases:
|
| 5. 2p
|
| 6. 4p(1-p)
|
| Although... if you change the game to allow a quantum coin then
| 5. and 6. _are_ possible (a paper of mine:
| https://arxiv.org/abs/1509.06183)
| shapefrog wrote:
| Can someone who has access to a precision robot and controlled
| environment please check; if one applies the same force on the
| coin in flipping, with it landing on a (soft) surface at the same
| height it - will it land the same side every time.
| morelisp wrote:
| You could read the Diaconis paper?
| shapefrog wrote:
| It is paywalled so I can not.
|
| However, I was unaware that they already conducted said
| experiment. I am now left confused as to why they would not
| mention such in the abstract, and refer only to natural
| experiments and second order measurements.
|
| I shall aquire the paper and learn why.
| tgv wrote:
| Using Google Scholar (1), you can find PDF versions of the
| paper
|
| (1) https://scholar.google.com/scholar?cluster=168770038670
| 74896...
| tiagod wrote:
| Your comment sniped me into thinking of some over-engineered
| systems to measure this with the human factor included. Some
| sort of RFID coin balanced on all axis with an IMU chip that
| measures the force applied to the force (through acceleration)
| and the mode and number of rotations of the coin. Maybe a
| computer vision solution would also work
| shapefrog wrote:
| Moving hoop won't let you miss -
| https://www.youtube.com/watch?v=myO8fxhDRW0
|
| I guess in theory you could "catch" with the correct call by
| varying the height of the catching platfrom.
| p00dles wrote:
| Could it have anything to do with contact with the hand?
| Moisture/oil from the hand making the "same side they started on"
| a tiny bit heavier than the other side? Or, and I'm no physicist,
| static electricity or something?
| a_c wrote:
| I always tell people that result of coin flip is highly start
| state dependent. Imagine a sequence of H(ead), T(ail), H, T, H,
| T, ... if the sequence starts with H first, in no way can the
| number of T exceed that of H, but the number of H might be 1
| greater that that of T. I never tested my self, but I hypothesize
| that the propability will be more skewed if the number of
| revolutions is less, i.e. having a shorter Head-Tail sequence.
|
| Edit: The sequence was meant to represent the sequence of head
| and tail facing up during the rotation. A sequence of [H, T]
| denotes one full rotation, starting with head. [T, H, T, H]
| denotes two full rotation, starting with a tail, ends with a
| head. I didn't mean the result of a flip. So the result of a flip
| is the final element of the sequence.
| cabirum wrote:
| Can you toss a coin without it flipping at least once (changing
| state)? I find it quite unlikely to happen, so if the starting
| state was H, your sequence will be THTH...
| a_c wrote:
| I can't quite define what counts as start of sequence. Maybe
| a sequence should always have at least one element, and toss
| straight up is allowed. But if some flipping is mandatory,
| then the start of sequence would mean the other side of the
| coin. All I could deduce before this paper is the probabilty
| of coin flip is skewed.
| Adverblessly wrote:
| If I were to dig through my comment history I would find I have
| already responded to this exact sort of comment before, so let
| me regurgitate :)
|
| If the coin is resting on your hand waiting to be flipped, it
| is currently mid-way through being on side up. This is because
| the switch between being e.g. heads up to tails up is done when
| the coin is vertical. If it isn't a clear explanation, try
| imagining catching the coin and "flattening" it at different
| angles, while 50% of angles will match either side, at the
| moment the coin is flipped, it is already half-way through the
| angles representing the current side.
|
| This means that the correct sequence you describe it not THTHTH
| but rather THHTTHHTTHH. Taken at even intervals, both sides
| will appear the same number of times. Taken at odd intervals,
| at half of the intervals there are more Ts and the other half
| have more Hs.
| mucle6 wrote:
| I would have never considered this. Such an interesting way to
| think about the problem
| [deleted]
| roflmaostc wrote:
| Of course that is true, if you wait not significantly long
| enough.
|
| Imagine the situation that you flip the coin (starting at H)
| and you grab it in air immediately. Of course, you will get H
| as result.
|
| But let's say, the time to stop the coin can be a relatively
| long time T. Then, I think the probability is some kind of sum.
| Let's choose \Delta T= 10ms as time discretization:
|
| P(H) = 1 / T * (10ms-0ms) + (30ms-20ms) + (50ms-40ms) + ... =
| 1/T \sum_{i=0}{floor(T / (2 * \Delta T))} \Delta T
|
| P(T) = 1 / T ((20ms-10ms) + (40ms-30ms) + (60ms-50ms) + ... =
| 1/T \sum_{i=0}{floor(T / (2 * \Delta T)) - 1} \Delta T
|
| For T -> \infty P(H) and P(T) getting more similar.
|
| But, in practice you wouldn't wait equally distributed in time
| but more like a Gaussian distributed time period. Hence, each
| term of the sum would get weighted differently. And the
| variance and the offset of the Gaussian distribution can shift
| the probability in favor of H or T. It's really dependent of
| the concrete parameters. If you grab always after 35ms, then
| you'll always get T for example.
| alexmolas wrote:
| I would like to see how TianqiP flips the coin. This user have a
| ratio of 0.601 [0.582, 0.619] heads, which is a lot. This is the
| type of skill that can get you a couple of bucks if played
| strategically.
| jcoder wrote:
| Or perhaps their coin was biased
| cesaref wrote:
| I think the sporting approach of letting the coin drop to the
| ground rather than catching it is the answer to avoid bias in
| coin tossing.
| fbartos wrote:
| Not neccessarily because spinning and bouncing coins are often
| much more biased then flipped coins. (Unequal weight
| distribution on the side can bias a spinned coin while it doesn
| not bias a flipped coin. There are a couple of studies on it
| too.)
| sjducb wrote:
| How do I bias a coin flip?
|
| Based on the paper it looks like 55% chance that it will land on
| the same side it started is possible. This was the most extreme
| subject.
|
| The bias is caused by procession so I want my flip to process as
| much as possible. Maybe I offset my finger as far away from the
| center of the coin as possible. Also putting as much force into
| it as possible is probably a good idea.
|
| Finally I have to catch it in a way that the top side is facing
| up when I reveal it.
|
| Any thoughts?
| rootusrootus wrote:
| > procession [...] process
|
| Nitpick: Precession. This is one case where a minor mispelling
| really does throw off the meaning of the sentence. At least to
| me.
| curriculum wrote:
| Give it some spin with your index finger. Imagine putting a
| coin between your index finger and thumb, heads side up,
| resting on your middle finger below -- not too different than
| how most people start a coin flip. With your index finger,
| rotate the coin in its plane, so that it stays "heads up" but
| the head is rotating. Now try doing this and simultaneously
| flipping the coin by flicking it with your thumb at a point on
| the bottom, close to the edge. If you do it right, you'll
| impart a spin. If you impart a modest spin, the coin will never
| actually flip over, but will just wobble, and will therefore
| land heads up. An observer will likely not know what you did,
| because it is hard for the eye to tell the difference between a
| flip and a wobble at high speeds.
| harimau777 wrote:
| Supposedly dice tend to role higher if you roll them starting
| with the largest side up. I use that when I'm creating my
| character in D&D and it seems to work.
| mihaic wrote:
| Interestint, my intuition was inverse to the result: landing on
| the opposite side takes an odd number of flips and on the same
| side it would be an even number of flips.
|
| Since for every number of flips either the number of odds is the
| same as the number of evens, or higher by 1, the chances of
| getting an odd number of flips is higher. There seems to be more
| at work here, and only testing validates a model!
| m3kw9 wrote:
| What about if they were flipped by computer random functions
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