5.3 Exercise 59

Use the following information to answer the next three exercises. The average lifetime of a certain new cell phone is three years. The manufacturer will replace any cell phone failing within two years of the date of purchase. The lifetime of these cell phones is known to follow an exponential distribution.

At an urgent care facility, patients arrive at an average rate of one patient every seven minutes. Assume that the duration between arrivals is exponentially distributed.

1. Find the probability that the time between two successive visits to the urgent care facility is less than two minutes.
2. Find the probability that the time between two successive visits to the urgent care facility is more than 15 minutes.
3. If 10 minutes have passed since the last arrival, what is the probability that the next person will arrive within the next five minutes?
4. Find the probability that more than eight patients arrive during a half-hour period.