If it rains on game day there is a 32% chance the Patriots will win. If it does

Question

If it rains on game day there is a 32% chance the Patriots will win. If it does not rain on game day there is a 29% chance the Patriots will win. There is a 41% chance that it will rain on game day.
a. What is the probability the Patriots will win? b. What is the probability it rains on game day and Patriots win? c. What is the probability that it rains on game day given that the Patriots win? If it rains on game day there is a 32% chance the Patriots will win. If it does not rain on game day there is a 29% chance the Patriots will win. There is a 41% chance that it will rain on game day.
a. What is the probability the Patriots will win? b. What is the probability it rains on game day and Patriots win? c. What is the probability that it rains on game day given that the Patriots win?
a. What is the probability the Patriots will win? b. What is the probability it rains on game day and Patriots win? c. What is the probability that it rains on game day given that the Patriots win?

Explanation / Answer

P(W|R) = 0.32
P(W|R') = 0.29
P(R) = 0.41 and P(R') = 0.59

(A)
P(W) = P(W|R)*P(R) + P(W|R')*P(R') = 0.32*0.41 + 0.29*0.59 = 0.3023

(B)
P(R and W) = P(W|R)*P(R) = 0.32*0.41 = 0.1312

(C)
P(R|W) = P(W|R)*P(R)/(P(W|R)*P(R) + P(W|R')*P(R')) = 0.1312/0.3023 = 0.434

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