Question

An actuary studied the likelihood that different types of drivers have at least one collision during any one- year period. The results of the study appear below. Type Teen Young adult Middle aged adult Senior Percentage 22 55 12 Collision% 15 6 3 5 Find the probability that a driver with at least one collision in the past year is a Young adult driver.​

Answer 1

The probability that a driver with at least one collision in the past year is a Young adult driver is 0.4 or 40%.

To find the probability that a driver with at least one collision in the past year is a Young adult driver, you can use conditional probability. The conditional probability of an event A given that event B has occurred is denoted as P(A | B) and is calculated using the formula:

`\[ P(A | B) = (P(A \cap B))/(P(B)) \]`

In this case:

- Event A: Being a Young adult driver.

- Event B: Having at least one collision in the past year.

The information given in the study provides the following probabilities:

`\[ P(\text{Young adult}) = 0.55 \]`

`\[ P(\text{Collision}) = 0.15 \]`

`\[ P(\text{Young adult} \cap \text{Collision}) = 0.06 \]`

Now, you can use these probabilities to calculate the conditional probability:

`\[ P(\textYoung adult ) = \frac{P(\text{Young adult} \cap \text{Collision})}{P(\text{Collision})} \]`

`\[ P(\text Collision) = (0.06)/(0.15) \]`

`\[ P(\text Collision) = 0.4 \]`

So, the probability that a driver with at least one collision in the past year is a Young adult driver is 0.4 or 40%.