CONDITIONAL PROBABILITY HELP Q A box contains 8 fair coins a

CONDITIONAL? PROBABILITY HELP?

Q: A box contains 8 fair coins and a fake (two-headed coin), both of its sides are heads. A randomly chosen coin was flipped 5 times. Suppose that the coin landed heads on each time. What is the probability the fake coin was chosen?

Is the probability the fake coin was chosen always going to be 1/9? Since 9 total coins in the box and 1 fake coin.

OR

Is it conditional probability?

WANT: P ( coin = fake | coin = H)

= (1/9) * 1 / [ (1/9)*1 + (8/9)* .5] = 1/3

Not sure which method to go with.

Solution

Let

F = fair coin
B = biased coin
H = lands heads 5 times in a row
h = lands heads

Thus,

P(H) = P(F) P(h|F)^5 + P(B) P(h|B)^5 = (8/9)*(1/2)^5 + (1/9)*(1^5) = 0.138888889

Now,

P(B|H) = P(B) P(h|B)^5 / P(H) = (8/9)*(1/2)^5 / 0.138888889 = 0.2 [ANSWER]