Final answer:
Using exponential distribution, we can calculate the probabilities of different arrival intervals for patients at an urgent care facility, which is important for healthcare management and planning.
Step-by-step explanation:
To address the student's question about the likelihood of patient arrival intervals at an urgent care facility, we'll utilize the principles of the exponential distribution. Given the average rate of one patient every seven minutes, the rate (λ) is ⅖ per minute.
- a. To find the probability that the time between two successive patient arrivals is less than two minutes, we use the exponential distribution formula P(X < x) = 1 - e^(-λx). Here, x = 2.
- b. For the probability of an interval of more than 15 minutes, we calculate P(X > 15) using 1 - P(X ≤ 15).
- c. If 10 minutes have already elapsed, we're interested in the probability that the next patient arrives within 5 minutes, which involves the memoryless property of the exponential distribution.
- d. To determine the probability of more than eight patients arriving in a half-hour period, we would use the Poisson distribution, as it is a discrete distribution useful for counting the number of events over a specified interval.
Through these calculations, students will learn how to apply exponential distribution to real-world scenarios, such as patient arrival times in healthcare facilities.