Question

Evidence shows that the probability that a driver will be involved in a serious automobile accident during a given year is .01. A particular corporation employs 100 full-time traveling sales reps. Based on this evidence, use the Poisson approximation to the binomial distribution to find the probability that exactly two of the sales reps will be involved in a serious automobile accident during the coming year.

Short answer

Answer: The probability that exactly two sales reps will have a serious automobile accident in the coming year is approximately 0.18394 or 18.39%.

Step-by-step solution

Write down the Poisson distribution formula

Recall the Poisson distribution formula: P(X = k) = (e^(-λ) * (λ^k)) / k! where λ is the mean number of accidents, k is the number of accidents we want to find the probability for, and e is the base of the natural logarithm.

Plug in the values

For this problem, λ = 1 (mean number of accidents), k = 2 (number of accidents we want the probability for), and e ≈ 2.71828. Plug the values into the formula: P(X = 2) = (e^(-1) * (1^2)) / 2!

Calculate the probability

Perform the calculations: P(X = 2) = (2.71828^(-1) * (1^2)) / 2 P(X = 2) ≈ (0.36788 * 1) / 2 P(X = 2) ≈ 0.18394 So, the probability that exactly two sales reps will be involved in a serious automobile accident during the coming year is approximately 0.18394 or 18.39%.

Key concepts

Understanding the Binomial Distribution

The binomial distribution is a common way to model situations where there are only two possible outcomes: success or failure. It's especially useful in scenarios where you're conducting an experiment several times, and each time the likelihood of success is constant.
In the original exercise, the scenario involves drivers who may or may not be involved in a car accident, and it can be seen as a binomial distribution. However, when dealing with a large number of trials (in this case, 100 sales reps) and a small probability of success (0.01, where success refers to having an accident), it becomes more efficient to use a different approach. This is where the Poisson distribution can approximate the binomial distribution very well.

• This approximation is valid when the number of trials is large, and the probability of success is small.
• The Poisson distribution helps simplify calculations that might otherwise be complex with the binomial distribution.

Understanding when and why to use these distributions is vital for solving real-world statistical problems efficiently.

Exploring Probability in the Context

Probability is the field of mathematics that deals with the likelihood or chance of different outcomes occurring. In the context of the Poisson approximation to the binomial distribution, it's all about estimating how often an event will occur in a given period.
For our exercise, the main task was to calculate the probability of exactly two sales reps being involved in a serious accident in one year, using the Poisson distribution.

• This involves knowing two key parameters: the mean number of events (accidents, in this case) that occur, represented as \( \lambda \), and the set number of events we want to calculate the probability for.
• Probability is a crucial component that enables accurate forecasting and decision-making in various fields.

By refining and understanding these probabilities, businesses and governments can make informed choices about resource allocation and risk management.

Mean Number of Events: Why It Matters

The term \( \lambda \) in the Poisson formula represents the mean number of events expected to occur in a fixed interval of time or space. It's essential because it simplifies how we calculate probabilities for rare events happening a given number of times.
In the example provided, the mean number of sales reps expected to be involved in serious car accidents is calculated by multiplying the total number of sales reps (100) by the probability of each being involved in an accident (0.01), thus \( \lambda = 100 \times 0.01 = 1.0 \).

• This mean number forms the backbone of the Poisson calculation, allowing us to plug in values and compute probabilities with ease.
• Understanding \( \lambda \) fully helps in anticipating the behavior of rare events across broader time frames and situations.

Effectively, it offers a lens through which statisticians and analysts can predict frequencies of occurrences efficiently, influencing how safety measures and preventive actions are considered.