Question

The three most popular options on a certain type of new car are a built-in GPS (A), a sunroof (B), and an automatic transmission (C). If 37% of all purchasers request A, 46% request B, 63% request C, 53% request A or B, 71% request A or C, 74% request B or C, and 77% request A or B or C, determine the probabilities of the following events. [Hint:"A or B" is the event that at least one of the two options is requested; try drawing a Venn diagram and labeling all regions.]

(a)

The next purchaser will request at least one of the three options.

(b)

The next purchaser will select none of the three options.

(c)

The next purchaser will request only an automatic transmission and not either of the other two options.

(d)

The next purchaser will select exactly one of these three options.

Answer 1

The three most popular options on a certain type of new car are a built-in GPS (A), a sunroof (B), and an automatic transmission (C). If 37% of all purchasers request A, 46% request B, 63% request C, 53% request A or B, 71% request A or C, 74% request B or C, and 77% request A or B or C, determine the probabilities of the following events. [Hint:"A or B" is the event that at least one of the two options is requested; try drawing a Venn diagram and labeling all regions.]

P( A) = 0.37   P( B) =0.46   P( C) = 0.63.

P( A or B) = 0.53 P( A or C) = 0.71 P( B or C) = 0.74

P( A or B or C) = 0.77

Using this identities,.

P(A U B) = P(A) + P(B) - P(A ∩ B).

P (A U B U C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)

We get,

P( A and B) = 0.30 P( A and C) =0.29 P( B and C) = 0.35

P( A and B and C) = 0.25

Venn diagram ( in probabilities).

(a)

The next purchaser will request at least one of the three options.

P=0.77

(b)

The next purchaser will select none of the three options.

P=1-0.77=0.23

(c)

The next purchaser will request only an automatic transmission and not either of the other two options.

P=0.24

(d)

The next purchaser will select exactly one of these three options.

P=0.03+0.06+0.24

=0.33