5 ms·

How to create a biased coin and prove it with math

66 points by jackpirate 15y ago

leeoniya 15y ago
i don't think you need to go to such visible lengths to bias a coin. if one face of the coin was weighted more (by using a heavier alloy), it would likely bias it simply because when/if it bounces, it would be more likely to land with that face down. maybe?
- carbocation 15y ago
  Yes, you could bias the coin in many ways. But the author chose an inexpensive method available to readers without experience in metallurgy.
  - jackpirate 15y ago
    Also, based on how big of a bending angle you actually need to bias the coin, I doubt anything you could do would bias the coin without being very noticeable.
- Codayus 15y ago
  It's actually hard - and perhaps impossible - to bias a coin by weighting one face. It's not a very well studied area, but according to what info I can find[1], as general rule, you should assume that if some shady stranger in a bar is offering to bet money on the outcome of a coin flip, the coin is going to be perfectly fair. (Also, you can always work around a possibly biased coin by instead flipping the coin twice, and repeating the process until you get HT or TH, and then using the first element of the pair as the result.) On the other hand, it is well established that the process of flipping a coin is biased, and a skilled coin flipper can produce whichever result he wants with a probability approaching unity. And there's NO way to work around that bias. The lesson here is probably "don't wager money on the outcome of coin flips". [1]: http://www.stat.berkeley.edu/~nolan/Papers/dice.pdf http://www.stat.berkeley.edu/~nolan/Papers/dice.pdf
- carbocation 15y ago
  Codayus, if you notice this, you appear to be hellbanned (all of your comments for the past 10 days are dead, including the one in this thread). I looked over your comment history and I can't understand why this would be, so I'm alerting you in case you want to ask PG for a reprieve.
  - bobds 15y ago
    Easy way to check if you are hellbanned: Visit the .org website and you won't be logged in. http://news.ycombinator.org/submitted?id=YOUR_USERNAME_HERE http://news.ycombinator.org/submitted?id=YOUR_USERNAME_HERE If all your latest comments are grey, shed a tear for the time you've wasted.
  - tptacek 15y ago
    I'm not a "having conversations with 'pg over email" kind of guy, but I've found him pretty responsive on "it doesn't look like this person should be banned" mails. Next time, you might find it less effortful to bang out a quick mail to pg@ycombinator.com to fix stuff like this. (I don't know what the deal is with accidental hellbanning, but it seems to happen with some regularity). I mailed him already for this user, though.
    - carbocation 15y ago
      Thank you. I'll try that approach next time.
- wvoq 15y ago
  This fun article by Andrew Gelman and Deborah Nolan shows that it's practically impossible to create a coin that will demonstrate a bias when flipped, unless the coin is allowed to bounce: http://www.stat.columbia.edu/~gelman/research/published/diceRev2.pdf http://www.stat.columbia.edu/~gelman/research/published/dice...
carbocation 15y ago
In this case, the graphs don't seem to add any more clarity beyond what is already provided by the counts (e.g., 47 heads, 53 tails). I would be more interested to understand to what extent each coin appears to statistically differ from a fair coin, and in fact that's what I imagined the article would cover, given the title.
- jackpirate 15y ago
  That's exactly what the graphs were trying to convey. Maybe it could have been explained more clearly?
  - carbocation 15y ago
    Plotting it out forces the reader to try to visually calculate the integral. Wikipedia gives a nice treatment of how to get a numerical answer to "What is the probability that this coin is fair:" http://en.wikipedia.org/wiki/Checking_whether_a_coin_is_fair#Posterior_probability_density_function http://en.wikipedia.org/wiki/Checking_whether_a_coin_is_fair...
    - jackpirate 15y ago
      I actually tried including an analytic solution as well, but it took up a lot of space and seemed like clutter. I figured most people would prefer the graphical solutions anyways. Thanks for the link, I should include that inside the post somewhere. EDIT: Done.
yuvadam 15y ago
Reminds me of to achieve a fair coin toss result with a biased coin - Toss the coin twice, if result is HH or TT, repeat. If result is HT or TH the result is either H or T (apriori selection of either the first or second toss, WLOG).
- burgerbrain 15y ago
  http://en.wikipedia.org/wiki/Randomness_extractor#Von_Neumann_extractor http://en.wikipedia.org/wiki/Randomness_extractor#Von_Neuman...
polyfractal 15y ago
It's easier to just practice flipping coins. If you start on the same side, you can train yourself to flip the same height/force/rotations each time. This allows you to know what side it usually lands on.
- pavel_lishin 15y ago
  Wouldn't it be more rewarding to practice faking flipping a coin? Give it some wobble, but make sure it doesn't actually flip over.
thelema314 15y ago
I have a problem with the idea behind this - there's no proof that the coin is biased; even flipping it a hundred times and getting the same result is possible with an unbiased coin. All that statistics gives you is the probability of having a result at least this extreme with an unbiased coin. If this probability is low enough (often 1 in 20 or 1 in 100, but sometimes as low as 1 in 5), the "hypothesis" of the coin being unbiased is rejected, but this rejection will happen erroneously 1 in 20 or 100 or 5 times. As exemplified by the XKCD cartoon: http://xkcd.com/882/ http://xkcd.com/882/
- defen 15y ago
  > All that statistics gives you is the probability of having a result at least this extreme with an unbiased coin. If you subscribe to frequentist statistics and use the null hypothesis of "coin is not biased", yes. But if you're a Bayesian you can compute the probability that the coin bias is greater than a given amount.
  - thelema314 15y ago
    Granted. But in either case, there's no proof - only a probability that you haven't reached an incorrect conclusion. I associate the concept of proof with logical certainty, and no statistical method will give you that.
mathattack 15y ago
Thanks for sharing. I think it's a better math lesson than coin tampering lesson. Hard to imagine the wrench is state of the art. :-) One of the subtle lessons here is just how many tosses you have to make to get a high confidence level. In business, how often do we get that many rolls of the dice or tosses of the coin before having to decide? And what's that say about how confident we should be? (Perhaps this is way so many HN folks have pivoted)
losethos 15y ago
http://www.losethos.com/text/BIBLE.TXT http://www.losethos.com/text/BIBLE.TXT 100000 lines 17 bits (coin tosses) Riddle: How many does it take to prove God. You answer and I'll tell you if you're right.

66 points by jackpirate 15y ago

i don't think you need to go to such visible lengths to bias a coin. if one face of the coin was weighted more (by using a heavier alloy), it would likely bias it simply because when/if it bounces, it would be more likely to land with that face down. maybe?

carbocation 15y ago
Yes, you could bias the coin in many ways. But the author chose an inexpensive method available to readers without experience in metallurgy.
- jackpirate 15y ago
  Also, based on how big of a bending angle you actually need to bias the coin, I doubt anything you could do would bias the coin without being very noticeable.
Codayus 15y ago
It's actually hard - and perhaps impossible - to bias a coin by weighting one face. It's not a very well studied area, but according to what info I can find[1], as general rule, you should assume that if some shady stranger in a bar is offering to bet money on the outcome of a coin flip, the coin is going to be perfectly fair. (Also, you can always work around a possibly biased coin by instead flipping the coin twice, and repeating the process until you get HT or TH, and then using the first element of the pair as the result.) On the other hand, it is well established that the process of flipping a coin is biased, and a skilled coin flipper can produce whichever result he wants with a probability approaching unity. And there's NO way to work around that bias. The lesson here is probably "don't wager money on the outcome of coin flips". [1]: http://www.stat.berkeley.edu/~nolan/Papers/dice.pdf http://www.stat.berkeley.edu/~nolan/Papers/dice.pdf
carbocation 15y ago
Codayus, if you notice this, you appear to be hellbanned (all of your comments for the past 10 days are dead, including the one in this thread). I looked over your comment history and I can't understand why this would be, so I'm alerting you in case you want to ask PG for a reprieve.
- bobds 15y ago
  Easy way to check if you are hellbanned: Visit the .org website and you won't be logged in. http://news.ycombinator.org/submitted?id=YOUR_USERNAME_HERE http://news.ycombinator.org/submitted?id=YOUR_USERNAME_HERE If all your latest comments are grey, shed a tear for the time you've wasted.
- tptacek 15y ago
  I'm not a "having conversations with 'pg over email" kind of guy, but I've found him pretty responsive on "it doesn't look like this person should be banned" mails. Next time, you might find it less effortful to bang out a quick mail to pg@ycombinator.com to fix stuff like this. (I don't know what the deal is with accidental hellbanning, but it seems to happen with some regularity). I mailed him already for this user, though.
  - carbocation 15y ago
    Thank you. I'll try that approach next time.
wvoq 15y ago
This fun article by Andrew Gelman and Deborah Nolan shows that it's practically impossible to create a coin that will demonstrate a bias when flipped, unless the coin is allowed to bounce: http://www.stat.columbia.edu/~gelman/research/published/diceRev2.pdf http://www.stat.columbia.edu/~gelman/research/published/dice...

carbocation 15y ago

In this case, the graphs don't seem to add any more clarity beyond what is already provided by the counts (e.g., 47 heads, 53 tails). I would be more interested to understand to what extent each coin appears to statistically differ from a fair coin, and in fact that's what I imagined the article would cover, given the title.

jackpirate 15y ago
That's exactly what the graphs were trying to convey. Maybe it could have been explained more clearly?
- carbocation 15y ago
  Plotting it out forces the reader to try to visually calculate the integral. Wikipedia gives a nice treatment of how to get a numerical answer to "What is the probability that this coin is fair:" http://en.wikipedia.org/wiki/Checking_whether_a_coin_is_fair#Posterior_probability_density_function http://en.wikipedia.org/wiki/Checking_whether_a_coin_is_fair...
  - jackpirate 15y ago
    I actually tried including an analytic solution as well, but it took up a lot of space and seemed like clutter. I figured most people would prefer the graphical solutions anyways. Thanks for the link, I should include that inside the post somewhere. EDIT: Done.

yuvadam 15y ago

Reminds me of to achieve a fair coin toss result with a biased coin - Toss the coin twice, if result is HH or TT, repeat. If result is HT or TH the result is either H or T (apriori selection of either the first or second toss, WLOG).

burgerbrain 15y ago
http://en.wikipedia.org/wiki/Randomness_extractor#Von_Neumann_extractor http://en.wikipedia.org/wiki/Randomness_extractor#Von_Neuman...

polyfractal 15y ago

It's easier to just practice flipping coins. If you start on the same side, you can train yourself to flip the same height/force/rotations each time. This allows you to know what side it usually lands on.

pavel_lishin 15y ago
Wouldn't it be more rewarding to practice faking flipping a coin? Give it some wobble, but make sure it doesn't actually flip over.

thelema314 15y ago

I have a problem with the idea behind this - there's no proof that the coin is biased; even flipping it a hundred times and getting the same result is possible with an unbiased coin. All that statistics gives you is the probability of having a result at least this extreme with an unbiased coin. If this probability is low enough (often 1 in 20 or 1 in 100, but sometimes as low as 1 in 5), the "hypothesis" of the coin being unbiased is rejected, but this rejection will happen erroneously 1 in 20 or 100 or 5 times. As exemplified by the XKCD cartoon: http://xkcd.com/882/ http://xkcd.com/882/

defen 15y ago
> All that statistics gives you is the probability of having a result at least this extreme with an unbiased coin. If you subscribe to frequentist statistics and use the null hypothesis of "coin is not biased", yes. But if you're a Bayesian you can compute the probability that the coin bias is greater than a given amount.
- thelema314 15y ago
  Granted. But in either case, there's no proof - only a probability that you haven't reached an incorrect conclusion. I associate the concept of proof with logical certainty, and no statistical method will give you that.

mathattack 15y ago

Thanks for sharing. I think it's a better math lesson than coin tampering lesson. Hard to imagine the wrench is state of the art. :-) One of the subtle lessons here is just how many tosses you have to make to get a high confidence level. In business, how often do we get that many rolls of the dice or tosses of the coin before having to decide? And what's that say about how confident we should be? (Perhaps this is way so many HN folks have pivoted)

losethos 15y ago

http://www.losethos.com/text/BIBLE.TXT http://www.losethos.com/text/BIBLE.TXT 100000 lines 17 bits (coin tosses) Riddle: How many does it take to prove God. You answer and I'll tell you if you're right.