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 51% of all purchasers request A, 57% request B, 80% request C, 70% request A or B, 90% request A or C, 87% request B or C, and 94% 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.]

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

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

Answer 1

from above:

P(A)=			0.51
P(B)=			0.57
P(C)=			0.8
P(AnB)=P(A)+P(B)-P(A U B)=			0.38
P(BnC)=P(B)+P(C )-P(B U C)=			0.5
P(AnC)=P(A)+P(C)-P(A U C)=			0.41
P(AUB)=			0.7
P(BUC)=			0.87
P(AUC)=			0.9

P(A u B u C) =0.94

P(A n B n C) =P(A u B u C)-(P(A)+P(B)+P(C)-P(A n B)-P(A n C)-P(B n C))

=0.94-(0.51+0.57+0.8-0.38-0.5-0.41)=0.35

therefore:

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

=P(C)-P(A n C)-P(B n C)+P(A n B n C) =0.8-0.5-0.41+0.35 =0.24

b)

exactly sets =P(A)+P(B)+P(C )-2(P(AnB)+P(BnC)+P(AnC))+3*P(AnBnC) =

0.35

(below is Venn diagram for your reference:

0.07 0.03 0.04 0.35 0.15 0.06 0.24 0.06