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 find the probability that your car breaks down on your road trip, we can use a probability tree to visualize the different possibilities.
First, let's list the given probabilities:
- Probability of receiving a new car: P(New) = 0.5
- Probability of receiving a nearly one year old car: P(OneYearOld) = 0.25
- Probability of receiving an old car: P(Old) = 0.25
- Probability of a new car breaking down: P(Breakdown|New) = 0.08
- Probability of a nearly one year old car breaking down: P(Breakdown|OneYearOld) = 0.09
- Probability of an old car breaking down: P(Breakdown|Old) = 0.5
Now let's construct the probability tree:
|___________0.5 (New)___________|
| |
| |
| P(Breakdown|New) = 0.08 | P(No Breakdown|New) = 0.92
| |
| |
|___________________________|
| | |
| | |
0.25 (OneYearOld) | |___________|____________ 0.25 (Old)
| |
| |
| P(Breakdown|OneYearOld) = 0.09 | P(No Breakdown|OneYearOld) = 0.91
| |
|_____________________________|
| | |
| | |
|________________|____________|_________________
To find the probability that your car breaks down on your road trip, we need to consider all the possible paths in the probability tree and combine the probabilities accordingly.
P(Breakdown) = P(New) * P(Breakdown|New) + P(OneYearOld) * P(Breakdown|OneYearOld) + P(Old) * P(Breakdown|Old)
Substituting the respective values:
P(Breakdown) = 0.5 * 0.08 + 0.25 * 0.09 + 0.25 * 0.5
Now we can calculate:
P(Breakdown) = 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 answer this question, we can use a probability tree to visualize the probabilities and calculate the probability of the car breaking down.
First, let's represent the car types with N for new, O for nearly one year old, and L for old.
Now, let's construct the probability tree:
/ N (0.5)
/ \
/ \
/ \
/ \
| \
N (0.08) O (0.25) L (0.25)
/ \ \
/ \ \
(0.5) (0.25) (0.25)
\ /
\ /
\ /
| \
Breakdown Breakdown
(0.08) (0.09)
Now, to calculate the probability of the car breaking down, we multiply the probabilities along each branch, add up all the probabilities where the car breaks down, and the sum will be our final answer.
Probability of a new car breaking down = 0.5 * 0.08 = 0.04
Probability of a nearly one year old car breaking down = 0.25 * 0.09 = 0.0225
Probability of an old car breaking down = 0.25 * 0.5 = 0.125
Now, we sum up the probabilities of the car breaking down for each car type: 0.04 + 0.0225 + 0.125 = 0.1875
Therefore, the probability that your car breaks down on the road trip is 0.1875 or 18.75%.