A hat contains 100 coins where 99 are fair but one is double

A hat contains 100 coins, where 99 are fair but one is double-headed (always landing Heads). A coin is chosen uniformly at random. The chosen coin is flipped 7 times, and it lands Heads all 7 times. Given this informtion, what is the probability that the chosen coin is double-headed? (of course, another approach here would be to look at both sides of the coin- but this is a metaphorical coin).

Solution

Note that

P(7 consec heads) = P(fair) P(7 consec heads|fair) = P(not fair) P(7 consec heads|not fair)

= (99/100)(1/2)^7 + (1/100)(1^7)

P(7 consec heads) = 0.017734375

Now,

P(7 consec heads and not fair) = P(not fair) P(7 consec heads|not fair) = (1/100)(1^7) = 0.01

Thus, as

P(not fair|7 consec heads) = P(7 consec heads and not fair) / P(7 consec heads)

= 0.01 / 0.017734375

= 0.563876652 [ANSWER]