Problem 57 - A First Course in Probability

Question

Suppose that the expected number of accidents per week at an industrial plant is \(5 .\) Suppose also that the numbers of workers injured in each accident are independent random variables with a common mean of \(2.5 .\) If the number of workers injured in each accident is independent of the number of accidents that occur, compute the expected number of workers injured in a week.

Step-by-Step Solution

Answer

The expected number of workers injured in a week at the plant is 12.5. This is calculated by multiplying the expected number of accidents per week (5) by the mean number of workers injured per accident (2.5), as they are independent. \(E[Total Workers Injured] = 5 * 2.5 = 12.5\).

Step 1: Identify given information

We are given the following information: - Expected number of accidents per week (E[A]) is 5 - Mean number of workers injured per accident (E[W]) is 2.5

Step 2: Use the independence property

Since the number of workers injured in each accident is independent of the number of accidents that occur, we can directly multiply the expected values of both these variables to find the expected number of workers injured in a week. E[Total Workers Injured] = E[A] * E[W]

Step 3: Calculate the expected number of workers injured in a week

Substitute the given values into the equation from Step 2: E[Total Workers Injured] = E[A] * E[W] = 5 * 2.5 Calculate the result: E[Total Workers Injured] = 12.5 So, the expected number of workers injured in a week at the plant is 12.5.