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. Here's how you can construct it:
Step 1: Start by drawing a tree diagram with three branches representing the three possible types of rental cars: new (N), nearly one year old (O), and old (L).
Step 2: Assign the probabilities of receiving each type of car: P(N) = 0.5, P(O) = 0.25, and P(L) = 0.25. These probabilities represent the chances of getting each type of car from the rental company.
Step 3: For each car type, add another set of branches to represent the possibility of the car breaking down or not breaking down. Assign the respective probabilities of each event.
For the new car (N), the probability of it breaking down is 0.08. Therefore, the probability of it not breaking down would be 1 - 0.08 = 0.92.
For the nearly one year old car (O), the probability of it breaking down is 0.09, so the probability of it not breaking down is 1 - 0.09 = 0.91.
For the old car (L), the probability of it breaking down is 0.5, and the probability of it not breaking down is 1 - 0.5 = 0.5.
The probability tree should now have three branches coming from each car type, with respective probabilities for breaking down and not breaking down.
Step 4: Multiply the probabilities along each branch to find the likelihood of each possible outcome. Multiply the probabilities of each car type with the respective probabilities of it breaking down or not breaking down.
Step 5: Sum up the probabilities of all branches that lead to a breakdown, i.e., add up the probabilities of the bottommost branches representing a breakdown (N breaking down, O breaking down, and L breaking down). This will give you the total probability that your car breaks down on the road trip.
I advise you to draw the probability tree as explained and compute the probabilities for each branch. Once you have done that, simply add up the probabilities of all the bottommost branches representing a breakdown to get the final answer.