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.

Key Concepts

Poisson Distribution

The Poisson Distribution is a probability distribution used to model or describe the number of events that might happen within a specified time period or area. If you are wondering when this particular distribution is handy, think of situations where events are happening independently and at a constant mean rate, like the number of accidents at an industrial plant.
To better understand, consider the expected number of accidents is known to be 5 per week. The beauty of a Poisson Distribution is that it enables us to calculate the probability of a given number of events happening over a fixed interval if we know the average number of times the event occurs. It uses the parameter \( \lambda \), which in this scenario is 5, representing the average frequency. The general formula for a Poisson probability is:

• \( P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} \)

Here, \( k \) is the number of events we want to find the probability for, and \( e \) is the base of the natural logarithm.Though in our exercise we don't need to calculate specific probabilities, understanding that the Poisson Distribution models the number of weekly accidents can deepen your comprehension of the setup.

Independent Random Variables

When dealing with multiple random variables, like the number of accidents and the number of injuries per accident, it's important to know whether they are independent. Independent Random Variables are variables whose outcomes do not affect each other. In statistical terms, the expected value or mean of their product is simply the product of their expected values.In our exercise, two independent random variables are featured:

• The number of accidents in a week \( E[A] = 5 \)
• The mean number of workers injured per accident \( E[W] = 2.5 \)

These variables being independent tells us that knowing the number of accidents doesn't improve predicting the number of injuries per accident and vice versa. Consequently, calculating the overall expected number of workers injured within a week is straightforward. Just multiply the individual expected values:

• \( E[Total ext{ }Workers ext{ }Injured] = E[A] \times E[W] \)

This independence simplifies problems that might otherwise be complex to solve, offering neat solutions using basic arithmetic operations.

Probability Theory

Probability Theory is the mathematical framework that helps us quantify uncertainty. It provides tools to calculate and interpret probabilities of various outcomes. In contexts like the industrial plant's accidents and injuries, probability theory allows us to determine expected values. The expected value or mean is a concept rooted in probability theory. It represents the average outcome we'd expect if an experiment (like counting accidents) were repeated many times. It is calculated as the sum of all possible values each multiplied by its probability, or as a simply deduced result from given independent expected values. Moreover, by leveraging probability theory, we can assess the relationship between different random variables, such as accidents and worker injuries. Probability Theory helps in

• Analyzing the frequency of events.
• Interpreting results such as expected injury counts.
• Dealing with randomness and uncertainty.

The exercises in this domain may seem straightforward, yet understanding and applying probability theory would equip you to tackle more complex real-life situations.