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 a 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)^10 = 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