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
c. more than four customers will arrive during each of the first four minutes.

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Step 1: Recognize that this problem involves a Poisson distribution because we are dealing with the number of arrivals in a fixed interval of time (per minute) and the mean number of arrivals is given as 4. The Poisson probability mass function is given by 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.

Step 2: To find the probability of 'more than four customers arriving,' calculate the complement of the probability of 'four or fewer customers arriving.' This means we need to compute P(X > 4) = 1 - P(X ≤ 4).

Step 3: Compute P(X ≤ 4) by summing the probabilities for k = 0, 1, 2, 3, and 4 using the Poisson formula. Specifically, calculate P(X = 0), P(X = 1), P(X = 2), P(X = 3), and P(X = 4), and then sum them: P(X ≤ 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4).

Step 4: Once P(X ≤ 4) is calculated, find P(X > 4) for one minute using the complement rule: P(X > 4) = 1 - P(X ≤ 4).

Step 5: Since the problem asks for the probability that more than four customers will arrive during each of the first four minutes, raise the probability P(X > 4) to the power of 4 (because the events across the four minutes are independent). The final expression is [P(X > 4)]^4.

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주요 개념

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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 and Variance

In the context of the Poisson distribution, the mean (λ) represents the average number of events (customer arrivals) in a given time frame, while the variance is also equal to the mean. This property simplifies calculations, as both the mean and variance are four in this case. Understanding these concepts is crucial for calculating probabilities related to customer arrivals.

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a certain number. In this question, to find the probability of more than four customers arriving in the first four minutes, one would first calculate the cumulative probability of four or fewer arrivals and then subtract this from one. This approach is essential for determining the likelihood of exceeding a specific threshold.

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 thatc. more than four customers will arrive during each of the first four minutes.