Two rival game shows have developed similar (but not identical) ways of awarding

Question

Two rival game shows have developed similar (but not identical) ways of awarding prizes "randomly" to their contestants. Game show A awards prizes by selecting a number Z randomly from a standard normal distribution and awarding $500 to the contestant in the event that Z > 1.8. By using a computer or tables, a knowledgeable contestant knows that her chance of winning the prize is therefore 3.59%. The game show assistant is supposed to pass the value of Z secretly to the host but accidentally reveals that the value of Z is positive. How (if at all) does this affect the contestant's chances of winning? Bonus mark: Game show B awards prizes by generating a random number Y ~ Unif(10, 20) and awarding $100 to the contestant if Y

Explanation / Answer

For Game show A
Probability of winning, P(W) = 0.0359
Probability of positive score, P(P) = 0.5
Probability of positive and winnig, P(P and W) = 0.07186
P(W|P) = P(P and W)/P(P) = 0.07186/0.5 = 0.1437

Hence if it is known that number is positive, probability of winning is 0.1437

For Game Show B
Probability of winning, P(W) = (1/10)*1.5 = 0.15
Probability that score is less than 14, P(L) = (1/10)*4 = 0.4
Probability that socre is less than 14 and winning P(W and L) = 0.15/0.4 = 0.375

P(W|L) = P(L and W)/P(L) = 0.375/0.4 = 0.9375

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