In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.
The mean number of arrivals per minute is four. Find the probability that
a. three, four, or five customers will arrive during the third minute.

Verified step by step guidance

1. Step 1: Recognize that this problem involves a Poisson distribution. The Poisson distribution is used to model the number of events (e.g., customer arrivals) occurring in a fixed interval of time or space, given a known average rate (mean) of occurrence. Here, the mean number of arrivals per minute (λ) is 4.
2. Step 2: Write the formula for the Poisson probability mass function (PMF): P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of arrivals, k is the number of arrivals, and e is the base of the natural logarithm (approximately 2.718).
3. Step 3: To find the probability of three, four, or five customers arriving during the third minute, calculate the individual probabilities for k = 3, k = 4, and k = 5 using the Poisson PMF formula. Specifically, calculate P(X = 3), P(X = 4), and P(X = 5).
4. Step 4: Add the probabilities calculated in Step 3 to find the total probability: P(3 ≤ X ≤ 5) = P(X = 3) + P(X = 4) + P(X = 5).
5. Step 5: Substitute λ = 4 into the formula for each k value and simplify the expressions. Remember to compute the factorials (e.g., 3! = 3 × 2 × 1) and powers of λ. Then sum the results to get the final probability.

Key Concepts

Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. It is particularly useful for modeling the number of arrivals in a fixed period, such as customers arriving at a grocery store. In this scenario, the mean arrival rate is four customers per minute.

Mean (λ) in Poisson Distribution

In the context of the Poisson distribution, the mean (denoted as λ, lambda) represents the average number of occurrences in a specified interval. For this problem, λ is equal to four, indicating that, on average, four customers arrive at the checkout per minute. This parameter is crucial for calculating the probabilities of different numbers of arrivals.

Calculating Probabilities

To find the probability of a specific number of events occurring in a Poisson distribution, the formula P(X=k) = (e^(-λ) * λ^k) / k! is used, where P(X=k) is the probability of k events, e is Euler's number, and k! is the factorial of k. In this case, to find the probability of three, four, or five customers arriving, you would calculate P(X=3), P(X=4), and P(X=5) and sum these probabilities.