Question

Draw an appropriate tree diagram, and use the multiplication principle to calculate the probabilities of all the outcomes. HINT [See Example 3.]

Your auto rental company rents out 30 small cars, 22 luxury sedans, and 48 slightly damaged "budget" vehicles. The small cars break down 12% of the time, the luxury sedans break down 8% of the time, and the "budget" cars break down 40% of the time.

P(Small and breaks down) = P(Small and does not break down) = P(Luxury and breaks down) = P(Luxury and does not break down) = P(Budget and breaks down) = P(Budget and does not break down) =

Answer 1

Let S shows the event that car is small, L shows the event that it is luxury sedan and B shows the event that it is slightly damaged "budget" vehicles.

Total number of vehicles = 30+22+48 = 100

So,

P(S) = 30/100 = 0.30

P(L) = 22/100 = 0.22

P(B) = 48/100 = 0.48

Let D shows the event that car is break down. So

P(D|S) = 0.12, P(D|L) = 0.08, P(D|B) = 0.40

Let D' shows the event that car is not break down. By the complement rule,

P(D'|S) = 1 - P(D|S) = 0.88, P(D'|L) = 1 - P(D|L) = 0.92, P(D'|B) = 1- P(D|B) = 0.60

Therefore,

P(Small and breaks down) = P(S and D) = P(D|S)P(S) = 0.12 * 0.30 = 0.036

P(Small and does not break down) = P(S and D') = P(D'|S)P(S) = 0.88 * 0.30 = 0.264

P(Luxury and breaks down) = P(L and D) = P(D|L)P(L) = 0.08 * 0.22 = 0.0176

P(Luxury and does not break down) = P(L and D') = P(D'|L)P(L) = 0.92 * 0.22 = 0.2024

P(Budget and breaks down) = P(B and D) = P(D|B)P(B) = 0.40 * 0.48 = 0.192

P(Budget and does not break down) = P(B and D') = P(D'|B)P(B) = 0.60 * 0.48 = 0.288

Following is the tree diagram: