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Chapter 4: Q82E (page 248)

Waiting for a car wash. An automatic car wash takes exactly 5 minutes to wash a car. On average, 10 cars per hour arrive at the car wash. Suppose that 30 minutes before closing time, 5 cars are in line. If the car wash is in continuous use until closing time, will anyone likely be in line at closing time?

Short Answer

Expert verified

The probability of no car will be a washis likely that someone will be in line.

Step by step solution

01

Given information

x denotes the number of cars that arrive at the car wash in 30-minute intervals.

x follows a Poisson distribution with a mean of 5.

02

Calculate the probability

Let x denote the number of cars that arrive at the car wash in 30-minute intervals. On average 10 cars per hour arrive at the car wash.

Therandom variable x follows a Poisson distribution with parameter,

位=10鈥�鈥�per鈥�鈥�houre=10脳3060鈥�鈥�per鈥�鈥�30鈥�鈥�minutes=5鈥�鈥�per鈥�鈥�30鈥�鈥�minutes

The probability mass function of x is given by,

PX=x=e-位位xx!,鈥�x=0,1,2,...=e-55xx!,鈥�x=0,1,2,...

The probability of x is less than one is,

Px<1=Px=0=e-5500!=0.006738鈮�0.0067

Px<1=0.0067

Therefore, the probability that no car will be a wash is likely someone will be in line.

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Buy-Side Analysts

Sell-Side Analysts

Mean

0.85

-0.05

Standard Deviation

1.93

0.85
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