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# The probability of a bad coin

## Question

If you flip a coin n times and you get n-1 Tails from those flips, what is the probability that the coin is biased?

## 6 answers to this question

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In that case, I'll go ahead and just say what I consider the answer to be.

1) Suppose God tells you that the coin is fair, and after 1000 flips there are 999 tails. In that case, it probably really is a fair coin and it was just by chance that He tossed a bunch of tails.

2) Suppose someone off the streets in Vegas comes up to you with a coin and proposes a wager. Even if he flips the coin twice and gets one tail, I'd still think the coin is probably going to be rigged.

Bayesian probability comes into play. You need to have some prior estimate of the probability that the coin is fair, denoted F. If F is absolutely 1, then no number of flips will change your certainty. Only if there's some doubt to begin with can you be influenced by seeing an improbable outcome.

If you see n-1 tails out of n flips, then the probability that the coin is fair is

F*P(n|F) / [(1-F)*P(n|B) + F*P(n|F)]

where P(n|F) is the probability that you will get n-1 tails out of n flips if the coin is fair, and P(n|B) is the probability that you will get n-1 tails out of n flips if the coin is biased.

P(n|F) is the only one of those terms that can be calculated.

F depends on how likely you consider the coin to be fair before you start flipping, based on the context of the scenario like the cases I mentioned at the start.

P(n|B) would also have to be estimated, because we don't know how biased an unfair coin ought to be.

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If the question concerns the probability that a fair coin will

give n-1 tails in n tosses, the answer is p = n/2n.

But I don't think we can conclude that 1 - p is the probability the coin is biased.

If "fair" means p(T) lies inside a small interval centered on 1/2,

0 <= 0.5-x < p(T) < 0.5+x <= 1, then we need a value for x.

We could derive a confidence level for fairness.

Its complement then would be the assurance of bias.

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There is not enough information presented in the OP to be able to give an answer. The real question is: what more do you need to know in order to be able to answer it?

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hmmm you may both be right. I was asked such a question and couldn't answer myself.

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Alternatively, you could just do what biologists do. Calculate a p value, and if it's less than 0.05 then consider that to be proof that the coin is biased. I think p<0.05 for n>7 based on bonanova's formula.

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Yeah. What he said. Prior ... Something.

Bushindo explained Bayes to me once before, plasmid just did again,

but it's still not something I could explain to my grandson.

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