A certain sports car comes equipped with either an automatic or a manual transmi

Question

A certain sports car comes equipped with either an automatic or a manual transmission and the car is available in one of four colors. Relevant probabilities for various combinations of transmission type and color are given in the accompanying table. Let A = (automatic transmission), N = {black}, and C = {white}. a. Calculate P(A) and P(A intersection B). b. Calculate both P(A|B) and P(B|A). An automobile service facility specializing in engine tune-ups knows that 50% of all tune-ups are done on four-cylinder automobiles, 40% on six-cylinder automobiles, and 10% on eight-cylinder automobiles. Let X = the number of cylinders on the next car to be tuned. a. Construct the probability distribution of X? b. Calculate the expected value of the number of cylinders on the next car to be tuned.

Explanation / Answer

1) a) P(A) = (0.13 + 0.10 + 0.11 + 0.11) = 0.45

P(A B) = 0.11

b) P(B) = 0.11 + 0.15 = 0.26

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

             = 0.11 / 0.26

             = 0.42

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

             = 0.11 / 0.45

             = 0.24

2) a) P(X = 4) = 0.5

P(X = 6) = 0.4

P(X = 8) = 0.1

b) E(X) = 4 * 0.5 + 6 * 0.4 + 8 * 0.1 = 5.2 or 5 (approx)

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