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
b. more than four customers will arrive during the first minute.

Verified step by step guidance

1. Step 1: Recognize that this problem involves a Poisson distribution because it deals with the number of events (customer arrivals) occurring in a fixed interval of time (one minute). The mean number of arrivals per minute (λ) is given as 4.
2. Step 2: Recall the formula for the Poisson probability mass function (PMF): P(X = k) = (λ^k * e^(-λ)) / k!, where X is the random variable representing the number of arrivals, λ is the mean, 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 'more than four customers arriving,' calculate the complement of the cumulative probability for X ≤ 4. This means you need to compute P(X > 4) = 1 - P(X ≤ 4).
4. Step 4: Compute P(X ≤ 4) by summing the probabilities for X = 0, 1, 2, 3, and 4 using the Poisson PMF formula. Specifically, calculate P(X = 0), P(X = 1), P(X = 2), P(X = 3), and P(X = 4), then add these probabilities together.
5. Step 5: Subtract the cumulative probability P(X ≤ 4) from 1 to find P(X > 4). This result represents the probability that more than four customers will arrive during the first minute.

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) represents the average number of occurrences in a specified interval. For this question, λ is equal to four, indicating that, on average, four customers arrive at the checkout counters each minute. This parameter is crucial for calculating probabilities related to the number of arrivals.

Calculating Probability

To find the probability of more than four customers arriving in the first minute, we can use the cumulative distribution function (CDF) of the Poisson distribution. Specifically, we calculate the probability of zero to four customers arriving and subtract this from one. This approach allows us to determine the likelihood of observing more than the average number of arrivals in a given time frame.