The time between calls to a corporate office is...

The time between calls to a corporate office is exponentially distributed with a mean of 10 minutes. (a) What is the probability that there are more than three calls in one-half hour? (b) What is the probability that there are no calls within onehalf hour? (c) Determine $x$ such that the probability that there are no calls within $x$ hours is 0.01 . (d) What is the probability that there are no calls within a twohour interval? (e) If four nonoverlapping one-half-hour intervals are selected, what is the probability that none of these intervals contains any call? (f) Explain the relationship between the results in part (a) and (b).

The time between calls to a corporate office is exponentially distributed with a mean of 10 minutes.
(a) What is the probability that there are more than three calls in one-half hour?
(b) What is the probability that there are no calls within onehalf hour?
(c) Determine $x$ such that the probability that there are no calls within $x$ hours is 0.01 .
(d) What is the probability that there are no calls within a twohour interval?
(e) If four nonoverlapping one-half-hour intervals are selected, what is the probability that none of these intervals contains any call?
(f) Explain the relationship between the results in part (a) and (b).

Key Concepts

Exponential Distribution

The exponential distribution models the time between events in a continuous process and is defined by its constant rate of occurrence. Its key attribute is the memoryless property, meaning that the likelihood of an event occurring in the future is independent of how much time has already elapsed. This distribution is central to modeling waiting times between random occurrences in many stochastic processes.

Poisson Process

A Poisson process models the number of events occurring within fixed time intervals, assuming that events happen independently and with a constant average rate. The number of events in any interval of time follows a Poisson distribution, linking the continuous-time perspective of the exponential distribution with discrete event counts over specified periods.

Rate Parameter

The rate parameter (commonly denoted as lambda) represents the average number of events per unit time in a Poisson process. It is the reciprocal of the mean inter-arrival time in an exponential distribution. This parameter is essential for determining the probability of a given number of events occurring over a defined time interval.

Complement Rule

The complement rule is a fundamental probability concept stating that the probability of an event not occurring is equal to one minus the probability of the event occurring. This rule is often used in Poisson and exponential contexts to calculate probabilities indirectly, such as determining the chance of more than a certain number of events by first considering the probabilities of the complementary event counts.

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