5 ms·

The Coin Flip: A Fundamentally Unfair Proposition? (2009)

41 points by gluxon 11y ago

raymondh 11y ago
See also: http://www.stat.columbia.edu/~gelman/research/published/diceRev2.pdf http://www.stat.columbia.edu/~gelman/research/published/dice...
jmount 11y ago
My small take: http://www.win-vector.com/blog/2015/04/i-still-think-you-can-manufacture-an-unfair-coin/ http://www.win-vector.com/blog/2015/04/i-still-think-you-can...
- plonh 11y ago
  If the coin has a substantial thickness, is it still accepted as a coin? I think not. How much density differential do you need to generate a 1% bias in a coin no thicker than any existing UN member currency?
  - jmount 11y ago
    Persi Diaconis did a lot of experiments- the biases are very low.
plonh 11y ago
A commenter explains that the most interesting claim (51% bias) is not explained correctly in the article. Humans flipping a coin add a bit of precession (spin) that biasses the toss. The "odd/even time in the air" analysis is incorrect -- that only matters if it is possible to stop the coin before the first flip.
- letstryagain 11y ago
  Yep. To see this, consider a coin flipped into the air so that it's tilted 45 degrees with the floor and spinning only around the vertical axis. To a casual observer it will appear as if the coin is tumbling but in fact one side faces up 100% of the time.
  - thret 11y ago
    This is a trick that can be learnt with practice. If you spend a lot of time around gamblers, don't ever accept a coin toss proposition.
    - jahnu 11y ago
      Doesn't the person not tossing the coin get to call though?
      - letstryagain 11y ago
        If you call tails, I catch the coin and show it to you. If you call heads, I'll catch the coin and invert it before showing it.
        jahnu 11y ago
        Ahh right. People I know usually call after the flipper catches and covers it. So they would have to have some technique for flipping it as they raise their hand to reveal the coin. Difficult but I guess not impossible.
heyitsnick 11y ago
"1% may not sound like a lot, but it's more than the typical casino edge in a game of blackjack or slots." The house edge for slots games is anywhere from 2% to 15%+ and is one of the worst bets you can make in the casino. Blackjack, video poker and some bets on craps are the only times in a casino where the house edge shrinks to below 1%.
- wozniacki 11y ago
  Blackjack, video poker and some bets on craps are the only times in a casino where the house edge shrinks to below 1%. Care to cite your source for this or are you a gaming industry insider?
  - PhantomGremlin 11y ago
    There's no need for a source. It's common knowledge. Many books, websites, and research papers have extensively studied those games. Edward Thorp is a mathematician who did a lot of analysis about blackjack over 50 years ago.[1] He's a seminal figure in this field. Here's one site I recall mostly because it has a memorable name: http://wizardofodds.com/ http://wizardofodds.com/ [1] https://en.wikipedia.org/wiki/Beat_the_Dealer#History https://en.wikipedia.org/wiki/Beat_the_Dealer#History
    - ChristianBundy 11y ago
      > There's no need for a source. It's common knowledge. Many books, websites, and research papers have extensively studied those games. It's obviously not common knowledge to the person that's asking the question. :~) Thanks for the source though.
      - mod 11y ago
        That one person doesn't know it doesn't mean it's not common knowledge. That said, the edge is not always below 1%. It varies based on the rules. Some games don't allow a surrender, for instance, which changes the edge pretty drastically.
zw123456 11y ago
The most interesting part to me was this "A coin will land on its edge around 1 in 6000 throws, creating a flipistic singularity." (https://en.wikipedia.org/wiki/Flipism https://en.wikipedia.org/wiki/Flipism) I assume that was as the result of using the tossing machine, but I really thought it would not have been that frequent.
- danieltillett 11y ago
  I have this happen a lot when tossing on wooden floors with a gap between the boards - basically the coin gets stuck in the gap. My brother and I got quite good at deliberately doing this. I have never had it happen on a flat surface and I must have tossed 10,000 of coins in my youth.
  - PhantomGremlin 11y ago
    I once attempted to hand a quarter to a convenience store clerk. Somehow we dropped it during the exchange. The coin rolled about 8 feet and went thru a tiny gap between two large cardboard boxes on the floor. I'm sure I could have stood there for an hour attempting to drop a coin onto the floor and have it roll into that gap, and I would have failed.
    - danieltillett 11y ago
      I had a similar situation in which a coin I tossed managed to end up stuck in a small gap in the celling. I remember having to get up on a chair on top of a table (I must have been under 10 at the time) to get it out. Good times :)
    - thret 11y ago
      The same is true for every possible outcome, that one just happened to be memorable.
  - aetherson 11y ago
    If the odds of landing on an edge are 1 in 6000, it's not particularly unbelievable that you wouldn't have seen it in 10,000 tries. The odds of not seeing it are about 18%. I saw a coin land on edge once.
qznc 11y ago
The paper ends with "The caveats and analysis also point to the following conclusion: For tossed coins, the classical assumptions of independence with probability 1/2 are pretty solid."
soup10 11y ago
Good to know! Relevant clip of some very entertaining gambling by two poker pros: https://www.youtube.com/watch?v=ZQSIx3CU9rA https://www.youtube.com/watch?v=ZQSIx3CU9rA
shuriu 11y ago
This video from Numberphile goes in depth about this: https://www.youtube.com/watch?v=Obg7JPd6cmw https://www.youtube.com/watch?v=Obg7JPd6cmw . The guy talking, is Persi Diaconis, is the main author of the paper this article is based upon. He also has done research on the randomness of shuffling cards: https://www.youtube.com/watch?v=AxJubaijQbI https://www.youtube.com/watch?v=AxJubaijQbI . Pretty cool stuff!
coldpie 11y ago
Mostly off topic, but boy do I enjoy articles that use the word "tosser" innocuously.
RogerL 11y ago
I think this link botches the explanation for 51%. 51% is because of precession, as explained in the paper[1]. With enough precession the coin will always come up heads if it starts out heads. That's how magicians control coin tosses. He tries to describe it in terms of HTHTHT. Coins do not have memory. They don't know if they've previously flipped 3 times, 4 times, or 1e308 times. If you draw from [HTHTHT..HT] randomly, you'll get 50% heads and 50% tails. Change that to [THTHTH..TH] and the answer is the same. With precession the answer changes because it stays in the initial state longer. As the paper points our "Keller showed that in the limit of large initial velocity and largerate of spin, a vigorous flip, caught in the hand without bouncing, lands heads half the time." Keller assumed no precession. [1] http://statweb.stanford.edu/~susan/papers/headswithJ.pdf http://statweb.stanford.edu/~susan/papers/headswithJ.pdf edit: I just saw that in the comments they point this flaw out. It gets hand waved away by the author as a "oversimplification" that he made. My opinion is that it is okay to simplify, but not to the point of being wrong. The explanation is not how probability works, at all.

41 points by gluxon 11y ago

See also: http://www.stat.columbia.edu/~gelman/research/published/diceRev2.pdf http://www.stat.columbia.edu/~gelman/research/published/dice...

jmount 11y ago

My small take: http://www.win-vector.com/blog/2015/04/i-still-think-you-can-manufacture-an-unfair-coin/ http://www.win-vector.com/blog/2015/04/i-still-think-you-can...

plonh 11y ago
If the coin has a substantial thickness, is it still accepted as a coin? I think not. How much density differential do you need to generate a 1% bias in a coin no thicker than any existing UN member currency?
- jmount 11y ago
  Persi Diaconis did a lot of experiments- the biases are very low.

plonh 11y ago

A commenter explains that the most interesting claim (51% bias) is not explained correctly in the article. Humans flipping a coin add a bit of precession (spin) that biasses the toss. The "odd/even time in the air" analysis is incorrect -- that only matters if it is possible to stop the coin before the first flip.

letstryagain 11y ago
Yep. To see this, consider a coin flipped into the air so that it's tilted 45 degrees with the floor and spinning only around the vertical axis. To a casual observer it will appear as if the coin is tumbling but in fact one side faces up 100% of the time.
- thret 11y ago
  This is a trick that can be learnt with practice. If you spend a lot of time around gamblers, don't ever accept a coin toss proposition.
  - jahnu 11y ago
    Doesn't the person not tossing the coin get to call though?
    - letstryagain 11y ago
      If you call tails, I catch the coin and show it to you. If you call heads, I'll catch the coin and invert it before showing it.
      - jahnu 11y ago
        Ahh right. People I know usually call after the flipper catches and covers it. So they would have to have some technique for flipping it as they raise their hand to reveal the coin. Difficult but I guess not impossible.

heyitsnick 11y ago

"1% may not sound like a lot, but it's more than the typical casino edge in a game of blackjack or slots." The house edge for slots games is anywhere from 2% to 15%+ and is one of the worst bets you can make in the casino. Blackjack, video poker and some bets on craps are the only times in a casino where the house edge shrinks to below 1%.

wozniacki 11y ago
Blackjack, video poker and some bets on craps are the only times in a casino where the house edge shrinks to below 1%. Care to cite your source for this or are you a gaming industry insider?
- PhantomGremlin 11y ago
  There's no need for a source. It's common knowledge. Many books, websites, and research papers have extensively studied those games. Edward Thorp is a mathematician who did a lot of analysis about blackjack over 50 years ago.[1] He's a seminal figure in this field. Here's one site I recall mostly because it has a memorable name: http://wizardofodds.com/ http://wizardofodds.com/ [1] https://en.wikipedia.org/wiki/Beat_the_Dealer#History https://en.wikipedia.org/wiki/Beat_the_Dealer#History
  - ChristianBundy 11y ago
    > There's no need for a source. It's common knowledge. Many books, websites, and research papers have extensively studied those games. It's obviously not common knowledge to the person that's asking the question. :~) Thanks for the source though.
    - mod 11y ago
      That one person doesn't know it doesn't mean it's not common knowledge. That said, the edge is not always below 1%. It varies based on the rules. Some games don't allow a surrender, for instance, which changes the edge pretty drastically.

zw123456 11y ago

The most interesting part to me was this "A coin will land on its edge around 1 in 6000 throws, creating a flipistic singularity." (https://en.wikipedia.org/wiki/Flipism https://en.wikipedia.org/wiki/Flipism) I assume that was as the result of using the tossing machine, but I really thought it would not have been that frequent.

danieltillett 11y ago
I have this happen a lot when tossing on wooden floors with a gap between the boards - basically the coin gets stuck in the gap. My brother and I got quite good at deliberately doing this. I have never had it happen on a flat surface and I must have tossed 10,000 of coins in my youth.
- PhantomGremlin 11y ago
  I once attempted to hand a quarter to a convenience store clerk. Somehow we dropped it during the exchange. The coin rolled about 8 feet and went thru a tiny gap between two large cardboard boxes on the floor. I'm sure I could have stood there for an hour attempting to drop a coin onto the floor and have it roll into that gap, and I would have failed.
  - danieltillett 11y ago
    I had a similar situation in which a coin I tossed managed to end up stuck in a small gap in the celling. I remember having to get up on a chair on top of a table (I must have been under 10 at the time) to get it out. Good times :)
  - thret 11y ago
    The same is true for every possible outcome, that one just happened to be memorable.
- aetherson 11y ago
  If the odds of landing on an edge are 1 in 6000, it's not particularly unbelievable that you wouldn't have seen it in 10,000 tries. The odds of not seeing it are about 18%. I saw a coin land on edge once.

qznc 11y ago

The paper ends with "The caveats and analysis also point to the following conclusion: For tossed coins, the classical assumptions of independence with probability 1/2 are pretty solid."

soup10 11y ago

Good to know! Relevant clip of some very entertaining gambling by two poker pros: https://www.youtube.com/watch?v=ZQSIx3CU9rA https://www.youtube.com/watch?v=ZQSIx3CU9rA

shuriu 11y ago

This video from Numberphile goes in depth about this: https://www.youtube.com/watch?v=Obg7JPd6cmw https://www.youtube.com/watch?v=Obg7JPd6cmw . The guy talking, is Persi Diaconis, is the main author of the paper this article is based upon. He also has done research on the randomness of shuffling cards: https://www.youtube.com/watch?v=AxJubaijQbI https://www.youtube.com/watch?v=AxJubaijQbI . Pretty cool stuff!

coldpie 11y ago

Mostly off topic, but boy do I enjoy articles that use the word "tosser" innocuously.

RogerL 11y ago

I think this link botches the explanation for 51%. 51% is because of precession, as explained in the paper[1]. With enough precession the coin will always come up heads if it starts out heads. That's how magicians control coin tosses. He tries to describe it in terms of HTHTHT. Coins do not have memory. They don't know if they've previously flipped 3 times, 4 times, or 1e308 times. If you draw from [HTHTHT..HT] randomly, you'll get 50% heads and 50% tails. Change that to [THTHTH..TH] and the answer is the same. With precession the answer changes because it stays in the initial state longer. As the paper points our "Keller showed that in the limit of large initial velocity and largerate of spin, a vigorous flip, caught in the hand without bouncing, lands heads half the time." Keller assumed no precession. [1] http://statweb.stanford.edu/~susan/papers/headswithJ.pdf http://statweb.stanford.edu/~susan/papers/headswithJ.pdf edit: I just saw that in the comments they point this flaw out. It gets hand waved away by the author as a "oversimplification" that he made. My opinion is that it is okay to simplify, but not to the point of being wrong. The explanation is not how probability works, at all.