   # Cars arrive at a car wash randomly and independently; the probability of an arrival is the same for any two time intervals of equal length. The mean arrival rate is 15 cars per hour. What is the probability that 20 or more cars will arrive during any given hour of operation? 13th Edition
Anderson
Publisher: CENGAGE L
ISBN: 9781305881884

#### Solutions

Chapter
Section 13th Edition
Anderson
Publisher: CENGAGE L
ISBN: 9781305881884
Chapter 5, Problem 69SE
Textbook Problem
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## Cars arrive at a car wash randomly and independently; the probability of an arrival is the same for any two time intervals of equal length. The mean arrival rate is 15 cars per hour. What is the probability that 20 or more cars will arrive during any given hour of operation?

To determine

Compute the probability that 20 or more cars will arrive during any given hour of operation.

### Explanation of Solution

Calculation:

In a car wash, cars arrive independently. The probability of an arrival is the same for any two time intervals of equal length. The mean arrival rate is 15 cars per hour.

Define the random variable x as the number cars that will arrive during any given hour of operation. Here, the occurrence of cars is random and the cars arrive independently at the car wash. The mean number of arrival rate is 15 cars per hour. That is, the occurrences (cars) are uniformly distributed for every hour. Thus, the number of cars that will arrive during any given hour of operation has Poisson distribution with parameter, μ=15.

The Poisson probability function is,

f(x)=μxeμx!

Where, x is the number occurrences of the event in a given interval, μ is the mean occurrences of the event, and e is a constant that has value 2.71828.

The required probability is obtained by substituting the values μ as 15 and the corresponding x values in the Poisson probability function till the probability becomes zero. The probability that 20 or more cars will arrive during any given hour of operation is obtained as shown below:

P(20 or more)=P(x20)

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