PROBABILITY AND STATISTICS FOR SCIENCE AND ENGINEERING WITH EXAMPLE Brian is int

Question

PROBABILITY AND STATISTICS FOR SCIENCE AND ENGINEERING WITH EXAMPLE Brian is interested in the result of a baseball game whether the ace starting pitcher of his team pitches or not. He is given the following probabilities. The probability that the pitcher pitches and his team wins is 0.18, the probability that his team wins is 0.55. The probability that the pitcher pitches in a game is 0.25. a. Given that Brian's team wins, what is the probability that the pitcher pitched? b. Given that the pitcher pitched, what is the probability that Brian's team wins? c. What is that probability that Brian's team loses or the pitcher did not pitch? d. What is the probability that the pitcher did not pitch and Brian's team loses? 2.23

Explanation / Answer

Let A denote probability that pitcher pitches and B that the team wins.

Given P(A) = 0.25 P(B) = 0.55 P(A B) = 0.18

a. P(A|B) = P(A B) / P(B)

= 0.18 / 0.55 = 0.3273

b. P(B|A) = P(A B) / P(A)

= 0.18 / 0.25 = 0.72

c. P(Ac U Bc) = P(A B)c by De-Morgan's law

= 1 - P(A B)

= 1 - 0.18

= 0.82

d. P(A U B) = P(A) + P(B) - P(A B)

= 0.25 + 0.55 - 0.18 = 0.62

=> P(Ac Bc) = P(A U B)c

= 1 - P(A U B) = 1 - 0.62 = 0.38.

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