The question as it appears in the 10/23 edition of Specs & Techs from GlobalSpec:
Jack likes to flip his favorite coin to make difficult decisions, but started to suspect that the coin was biased on sunny days. After tracking results for a while, Jack finds that when the coin comes up heads, 70% of the time it's a sunny day, however when the coin comes up tails, only 45% of the time it's a sunny day. Jack considers days sunny and not sunny. What is the probability that Jack's coin will come up heads on a sunny day?
(Update: Oct 30, 8:47 AM EST) And the Answer is...
The probability that Jack's coin will come up heads on a sunny day is 60.9%. There are two ways to arrive at this answer.
1. You could notice that you have complete data for Sunny days. 45 instances the coin came up tails, 70 instances the coin came up heads. Thus the probability (based on this data) that the coin will come up heads on a sunny day is 70/(45+70)= 60.9%
If you have trouble seeing that this is the case, you can use Bayes' Theorem, noting the following :
The following probabilities are given or known; P(HI)=.5, P(TI)=.5, P(SH,I)=.7, P(ST,I)=.45. You wish to find P(HS,I).
Using Bayes' Theorem, and using the normalization condition P(HS,I) + P(TS,I) = 1, we get:
P(HS,I) = [P(HI) x P(SH,I) x (1 P(HS,I))] /[ P(TI) x P(ST,I)]
P(HS,I) = [ .5 x .7 x (1P(HS,I))]/ [.5 x .45]
P(HS,I) = .6086
So there is a 60.9% chance the coin will come up heads on a sunny day.

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