Consider randomly selecting a single individual and having that person test drive 3 different vehicles. Define events A1, A2, and A3 by A? 1 =likes vehicle #1 A? 2 =likes vehicle #2 A? 3 =likes vehicle #3 Suppose that P(A1)= .55,P(A2) = .65,P(A3 )= .70,P(A1 ?A2) = .80, P(A2 ?A3) = .40, and P(A1 ?A2?A3)=.88. a.? ?What is the probability that the individual likes both vehicle #1 and vehicle #2? b.? ?Determine and interpretP(A2|A3) . c.? re A2 and A3 independent events? Answer in two different ways. d. ?If you learn that the individual did not like vehicle #1, what now is the probability that he/she liked at least one of the other two vehicles?

Solution: Step 1: Given an experiment that randomly selecting a single individual and having that person test drive 3 different vehicles. 3 events are defined by A1: likes vehicle #1 A2: likes vehicle #2 A3: likes vehicle # 3 Here it is also given that P(A1)=.55 P(A3)= .70 P(A2 A3) = .40 P(A2)= .65 P(A1 A2)=.80 P(A1 A2 A3)=.88 Step 2 : a) We have to find the probability that individual likes both vehicle # 1 and vehicle # 2. Since all the events are already defined. By the definition of probability. P(A1 A2)= P(A1)+P(A2)-P(A1 A2) So P(that individual likes both vehicle #1 and vehicle #2) =P( A1 A2)= P(A1)+P(A2)-P(A1 A2) = .55+.65-.80 = 0.4 b) We have to interpret P(A2/A3), by the definition of conditional probability we know that Which is the probability that the selected individual likes vehicle#2 given that he/she likes vehicle# 3. So P(A2/A3)= P(A2A3) P(A3) Since it is given that P(A2)=.65 P(A2 A3 )= .40 P(A3)=.70 P(that the selected individual likes vehicle#2 given that he/she likes vehicle# 3) P(A2/A3)= 0.40/0.70 =4/7 = 0.571