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You toss a coin $n$ times; $m$ times you get heads. What is the probability that it is a fair coin?

My solution:

A fair coin will give us probability of $0.5$ of getting heads.

Assume: $\;\;\;n=100$ tosses, $\;\;m=50$ heads, $\;\;p=0.5$.

$$P(X=50)=\frac{100!}{(50!*50!)}*0.5^{50}*0.5^{50} = 0.079.$$

So there's a $7.9$% chance that the coin is fair.

Am I right?

Thank you!

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    $\begingroup$ $0.079 \neq 79\%$ $\endgroup$
    – David
    Commented Dec 4, 2021 at 0:01
  • $\begingroup$ Thanks I fixed it $\endgroup$
    – Sigal Cohen
    Commented Dec 4, 2021 at 0:26
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    $\begingroup$ There is literally no way to tell. It depends on the PDF of the random value $p,$ the probability of heads for a rand9m coin. For example, if all coins are fair, the probability that it is fair is $1.$ If half of them are fair and the other half have $p=m/n,$ then the probability is less than $1.$ If $p$ is uniform, you get another value. $\endgroup$
    – Thomas Andrews
    Commented Dec 4, 2021 at 0:31
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    $\begingroup$ You are confusing the probability of getting $m$ heads in $n$ tosses given a fair coin with the probability of having a fair coin given $m$ heads in $n$ tosses. You don't have a probability distribution defined for fair and unfair coins, so nothing can be said about the probability of a fair coin regardless of the experimental outcome. $\endgroup$
    – hardmath
    Commented Dec 4, 2021 at 1:16
  • $\begingroup$ Your calculation only tells you that a fair coin has a probability of approximately 0.079 of showing exactly 50 heads in 100 consecutive tosses. That does not by itself provide any information on judging the "fairness" of the coin. The probability would be exactly the same if 100 "fair" coins were tossed simultaneously and we wanted to find the probability of exactly 50 heads showing. $\endgroup$
    – user882145
    Commented Dec 4, 2021 at 1:18

1 Answer 1

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With known values of $n$ and $m$, you cannot derive a probability for $p$ to be an exact value, but you can derive a range so that $p$ is within this range (i.e. confidence interval CI) with a certain probability (i.e. confidence level CL).

For instance, when $n=100$ and $m=50$, $p$ will be within (0.4 ~ 0.6) with a probability of 95%.

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