# what is the probability that the next toss of that coin is also a head?

Q:

A jar has 1000 coins, of which 999 are fair and 1 is double headed. Pick a coin at random, and toss it 10 times. Given that you see 10 heads, what is the probability that the next toss of that coin is also a head?

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There are two ways of choosing the coin. One is to pick a fair coin and the other is to pick the one with two heads.

Probability of selecting fair coin = 999/1000 = 0.999

Probability of selecting unfair coin = 1/1000 = 0.001

Selecting 10 heads in a row = Selecting fair coin * Getting 10 heads  +  Selecting an unfair coin

P (A)  =  0.999 * (1/2)^5  =  0.999 * (1/1024)  =  0.000976

P (B)  =  0.001 * 1  =  0.001

P( A / A + B )  = 0.000976 /  (0.000976 + 0.001)  =  0.4939

P( B / A + B )  = 0.001 / 0.001976  =  0.5061

Probability of selecting another head = P(A/A+B) * 0.5 + P(B/A+B) * 1 = 0.4939 * 0.5 + 0.5061  =  0.7531

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