You and your friend just rented a car for an 8,000 mile cross-country road trip. Your rental car
may be one of three different types: new (N), nearly one year old (O), and old (L). If the car you
receive is brand new, it will break down with probability 0.08. If the car is nearly one year old, it
will break down with probability 0.09. And if the car is old, it will break down with probability
0.5. The probability that the rental company gives you a new car, a nearly one year old car, and
an old car is 0.5, 0.25, and 0.25, respectively. What is the probability that your car breaks down
on your road trip? Use a probability tree to support your answer.
To solve this problem, we can create a probability tree to visualize the different possibilities.
Step 1: Determine the probabilities of receiving each type of car:
- P(New car) = 0.5
- P(Nearly one year old car) = 0.25
- P(Old car) = 0.25
Step 2: Determine the probabilities of each car breaking down:
- P(Breakdown | New car) = 0.08
- P(Breakdown | Nearly one year old car) = 0.09
- P(Breakdown | Old car) = 0.5
Step 3: Calculate the probabilities of your car breaking down on the road trip:
- P(Breakdown) = P(New car) * P(Breakdown | New car) + P(Nearly one year old car) * P(Breakdown | Nearly one year old car) + P(Old car) * P(Breakdown | Old car)
Calculating the probabilities:
P(Breakdown) = (0.5 * 0.08) + (0.25 * 0.09) + (0.25 * 0.5)
= 0.04 + 0.0225 + 0.125
= 0.1875
Therefore, the probability that your car breaks down on your road trip is 0.1875 or 18.75%.
To solve this problem, we can use a probability tree to visualize and calculate the probabilities.
First, let's represent the event of getting a new car as event N, getting a nearly one year old car as event O, and getting an old car as event L. The probabilities of these events are given as follows:
P(N) = 0.5 (probability of getting a new car)
P(O) = 0.25 (probability of getting a nearly one year old car)
P(L) = 0.25 (probability of getting an old car)
Now, let's consider the probabilities of the car breaking down given the type of car. We have:
P(B|N) = 0.08 (probability of the car breaking down given a new car)
P(B|O) = 0.09 (probability of the car breaking down given a nearly one year old car)
P(B|L) = 0.5 (probability of the car breaking down given an old car)
To calculate the probability that the car breaks down on the road trip, we need to consider all the possible scenarios using the probability tree:
(P=B) (P~B) (P~B) (PL) (PL)
.08 .02 .02 .125 .125
____|____ | |____ |_____
| | |
(P~B) (P=B) (P~B) (PL)
.09 .91 .045 .045
___|____ ____|____ ____|____
| | | |
(P~B) (P~B) (P~B) (P=B)
.5 .5 .048 .452
Let's calculate the probabilities step by step:
P(B) = P(N) * P(B|N) + P(O) * P(B|O) + P(L) * P(B|L)
P(B) = (0.5 * 0.08) + (0.25 * 0.09) + (0.25 * 0.5)
P(B) = 0.04 + 0.0225 + 0.125
P(B) = 0.1875
Therefore, the probability that your car breaks down on your road trip is 0.1875 or 18.75%.