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 calculate the probability that your car breaks down on the road trip, we can use a probability tree. The probability tree shows the different possibilities and their associated probabilities.
Let's start by drawing the probability tree. We have three different types of cars: new (N), nearly one year old (O), and old (L). The probabilities of receiving each type of car are 0.5, 0.25, and 0.25, respectively.
```
N (0.5)
/ \
0.08 0.92
/ \
Breakdown No breakdown
O (0.25)
/ \
0.09 0.91
/ \
Breakdown No breakdown
L (0.25)
/ \
0.50 0.50
/ \
Breakdown No breakdown
```
Now, let's calculate the probability of each outcome:
For a new car:
- The probability of a breakdown is 0.08 (given in the problem statement).
- The probability of no breakdown is 1 - 0.08 = 0.92.
For a nearly one year old car:
- The probability of a breakdown is 0.09.
- The probability of no breakdown is 1 - 0.09 = 0.91.
For an old car:
- The probability of a breakdown is 0.5.
- The probability of no breakdown is 1 - 0.5 = 0.50.
Now, let's calculate the overall probability of a breakdown:
```
Probability of breakdown = Probability of new car * Probability of breakdown for new car
+ Probability of nearly one year old car * Probability of breakdown for nearly one year old car
+ Probability of old car * Probability of breakdown for old car
```
Substituting the values:
```
Probability of 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%.