Question

Patients arriving at the emergency room of a local hospital follow a poisson distribution with an average arrival rate of 31 per half hour. find the probability that between 38 and 41 patients (inclusive) will arrive at the emergency room within a half hour. round your answer to four decimal places, if necessary.

Answer 1

To find the probability that between 38 and 41 patients will arrive at the emergency room within a half hour, we can use the Poisson distribution formula. The average arrival rate is 31 per half hour. Using the cumulative probability, we find that the probability is approximately 0.0979.

Step-by-step explanation:

To find the probability that between 38 and 41 patients will arrive at the emergency room within a half hour, we can use the Poisson distribution formula. The Poisson distribution is used to model the number of events that occur in a fixed interval of time or space, given the average rate of events. In this case, the average arrival rate is 31 per half hour.

To calculate the probability, we need to find the cumulative probability of 38 patients or less and subtract it from the cumulative probability of 41 patients or less.

Using a calculator or statistical software, we find that P(X ≤ 38) ≈ 0.8254 and P(X ≤ 41) ≈ 0.9233. Therefore, P(38 ≤ X ≤ 41) = P(X ≤ 41) - P(X ≤ 38) ≈ 0.9233 - 0.8254 ≈ 0.0979.