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I have n coins, where n-1 are fair and one is biased, showing tail on both sides. I want to ascertain whether the coin is fair by throwing k-times. If I get k-times tail, I decide it is biased. The question is :

What is the probability that this way of determination is false?

W( determination is false ) C(biased coin) T(throwing T k-times)

The solution is :

P(W) = P(¬ C,T) +P(C, ¬ T)

I am not sure why do we care for the probability of not getting tail with a biased coin. Why don’t we determine the probability of W like this:

P(W) = P(¬ C,T) +P(C, T)

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You are wrong if you pick a fair coin and get tails $k$ times because you will declare it biased. You are also wrong if you pick the biased coin and do not get tails $k$ times because you will declare it to be fair. With your two tailed coin the second possibility cannot occur, but you might have a coin that shows tails $0.99$ of the time and use the same algorithm. The equation you were given correctly states that chance that your decision is wrong.

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