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futureoptions
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Coin Toss Again

May 21st, 2012, 7:14 pm

Given a coin is flipped 100 times and it shows heads 100 times, what is the likelihood that the coin is fair?
 
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EscapeArtist999
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Coin Toss Again

May 22nd, 2012, 8:15 am

QuoteOriginally posted by: outrunBeta distribution f(0.5, 101, 1)this assumes that the prior is uniform though. Or have I missed something?
 
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Vanubis1
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Coin Toss Again

May 22nd, 2012, 9:29 am

I think you can advantage p=0.5 for your prior but in this precise example (as 0.5^100 is 8e-31), it doesn't matter except if you are sure the coin is fair.
 
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Vanubis1
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Coin Toss Again

May 22nd, 2012, 10:56 am

You can suppose before the throws that p=0.5 with a probability of 99% and uniform for the last 1% and use your formula by multiplying the p^100 by the prior probabilities.
 
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Traden4Alpha
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Coin Toss Again

May 22nd, 2012, 12:34 pm

QuoteOriginally posted by: Vanubis1You can suppose before the throws that p=0.5 with a probability of 99% and uniform for the last 1% and use your formula by multiplying the p^100 by the prior probabilities.The effect of the "p=0.5 with a probability of 99%" is negligible because the p^100 multiplier is so strong. Only if the prior evidence that "most coins are fair" were on the order of p=0.5 with probability (1-10^-28) would this bit of knowledge change the outcome substantially.Of course, this raises an interesting issue. Everyone thinks they have high sample sizes of experience with coins in everyday life, but the experience is largely cross-sectional and non-rigorous. How many of us have actually collected a large data set of flips with one coin to estimate whether that coin if fair or not? How many have done this test with multiple coins or even a small % of the coins they've seen in life? I'd bet that >>99.999% of the coins that we handle on a daily basis have never been tested for fairness. Perhaps 1/2 the coins are p=0.49 coins and 1/2 the coins are p=0.51 coins so that NONE of the coins are fair. Or, perhaps all of the coins are p=0.5001 coins due to mass distribution asymmetries in the manufacturing of the head side and tail side.
 
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Traden4Alpha
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Coin Toss Again

May 22nd, 2012, 1:08 pm

Yes, the design of the head-side and tail-side would affect three critical physical variables: 1) the center of mass of the coin; 2) the heights of the rims; 3) the signed rate of change of these variables as the coin stamping die wears down. Moreover, after the coin is made, the effects of everyday usage would also create p≠0.5 effects (e.g., a dent on the head-side rim that increases the chance of tails) so that the scatter of p would grow in time and virtually all coins would become increasingly unfair.Perhaps the only fair coins in the world are those that occupy statistics textbooks and brainteasers.
 
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foxkingdom
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Coin Toss Again

May 22nd, 2012, 1:29 pm

QuoteOriginally posted by: Traden4AlphaYes, the design of the head-side and tail-side would affect three critical physical variables: 1) the center of mass of the coin; 2) the heights of the rims; 3) the signed rate of change of these variables as the coin stamping die wears down. Moreover, after the coin is made, the effects of everyday usage would also create p≠0.5 effects (e.g., a dent on the head-side rim that increases the chance of tails) so that the scatter of p would grow in time and virtually all coins would become increasingly unfair.Perhaps the only fair coins in the world are those that occupy statistics textbooks and brainteasers.This thread suddenly becomes ultra-realistic.
 
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Traden4Alpha
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Coin Toss Again

May 22nd, 2012, 1:40 pm

QuoteOriginally posted by: foxkingdomQuoteOriginally posted by: Traden4AlphaYes, the design of the head-side and tail-side would affect three critical physical variables: 1) the center of mass of the coin; 2) the heights of the rims; 3) the signed rate of change of these variables as the coin stamping die wears down. Moreover, after the coin is made, the effects of everyday usage would also create p≠0.5 effects (e.g., a dent on the head-side rim that increases the chance of tails) so that the scatter of p would grow in time and virtually all coins would become increasingly unfair.Perhaps the only fair coins in the world are those that occupy statistics textbooks and brainteasers.This thread suddenly becomes ultra-realistic.Indeed! Given that the financial markets are "ultra-realistic" with the trillions of dollars/euros/drachmas and the lives of millions/billions at stake, that seems warranted.What's crucial here is the breaking of an unstated central assumption. In the academic world, all coins are fair unless stated or until proven otherwise. In the real world, no coins are fair until proven otherwise. And given the limits of finite sample sizes, one can never prove that a real-world coin is fair, only that it is unlikely to be unfair outside a confidence interval.
Last edited by Traden4Alpha on May 21st, 2012, 10:00 pm, edited 1 time in total.
 
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Traden4Alpha
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Coin Toss Again

May 22nd, 2012, 2:00 pm

QuoteOriginally posted by: outrunOn a serious note: is there anyone who can explain to me the Haldane prior?The Haldane prior makes sense for systems that have a deterministic binary outcome on an independent variable with a very wide dynamic range. If we ask, is material, M, a solid at temperature T, the answer will be almost always yes for an extremely wide range of cold temperatures (e.g., -273 to 0K for water) and no for an extremely wide range of high temperatures (e.g., 0 to ∞ for water). Systems that have Haldane priors might include the biological activity of a drug vs. concentration of a drug, physical state vs. temperature or pressure, whether chemical Y dissolves in solvent X, etc.If you imagine a sigmoid response function on a variable that has an extreme range of values, then the response function will look like a delta across that wide natural range.