Originally posted by Anthem
First some notation:
Call the event of flipping heads 3 times in a row A, and the event that the coin you chose is counterfeit B. P(A|B) is the probability that A happens given that B happens (i.e. the posterior probability). P(B|A) is thus what we want to find. P(A^B) is the probability that A and B both occur (I'm using ^ in the way that the intersection s ...[text shortened]...
P(A) = 1/8 + 1/8 + 1/3 = 7/12 (coquette's answer)
So P(B|A) = (1*1/3)/(7/12) = 4/7 = 57.1%
Here is some text from the review of "The Theory That Would Not Die," author Sharon Bertsch McGrayne, reviewed by John Allen Paulos, a math prof at Temple U.
"Bayes theorem states ... that the posterior probability of a hypothesis is equal to the product of (a) the prior probability of the hypothesis and (b) the conditional probability of the evidence given the hypothesis, divided by (c) the probability of the new evidence." ... "The answer to this question... is 4 in 5."
The only thing I can think is that I stated the poser incorrectly.