Question

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

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

Step-by-step solution

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.

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,

$\begin{array}{rcl}\lambda & =& 10 per houre\\ & =& 10\times \frac{30}{60} per 30 \min utes\\ & =& 5 per 30 \min utes\\ & & \end{array}$

The probability mass function of x is given by,

$\begin{array}{rcl}P\left(X=x\right)& =& \frac{e^{-\lambda }{\lambda }^x}{x!}, x=0,1,2,...\\ & =& \frac{e^{-5}{5}^x}{x!}, x=0,1,2,...\\ & & \end{array}$

The probability of x is less than one is,

$\begin{array}{rcl}P\left(x<1\right)& =& P\left(x=0\right)\\ & =& \frac{e^{-5}{5}^0}{0!}\\ & =& 0.006738\\ & \approx & 0.0067\\ & & \end{array}$

$P\left(x<1\right)=0.0067$

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