We have 100 coins, one of which may be double-headed with probability P(D)=0.5 (else all coins are fair). We randomly choose one coin and flip it 7 times, all of which are heads.
Q: What is the probability that the coin we have chosen is double headed?
So:
Let D be the event that there is a double-headed coin: P(D) = 0.5
Let H be the event that the chosen coin lands heads 7 times in a row.
Let F be the event that the chosen coin is fair.
(For a fair coin to land heads 7 times in a row the probability is 0.0078)
So I'm guessing what we are looking for is the sum of P(Fc|H, D) + P(Fc|H, Dc)
(Fc = F complement, Dc = D complement). But I can't even get anywhere with the first part, because:
$$P(Fc|H,D)= { P(D|Fc,H)P(Fc|H)\over P(D|H)}.$$
of which how do I get to P(D|Fc,H)? How do I get the probability that there is a dodgy coin given that 7 heads in a row and a dodgy coin have both occurred?