Question

On Sundays, cars arrive at Sami Schmitt's Scrub and Shine Car Wash at the rate of 8 cars per twenty minute interval. Using the Poisson distribution, the probability that five cars will arrive during the next thirty minute interval is_________.

Answer 1

Answer:

The probability that five cars will arrive during the next thirty minute interval is 0.0127.

Step-by-step explanation:

Let X = number of cars arriving at Sami Schmitt's Scrub and Shine Car Wash.

The average number of cars arriving in 20 minutes is, 8.

The average number of cars arriving in 1 minute is, $\frac{8}{20}=\frac{2}{5}$.

The average number of cars arriving in 30 minutes is, $\frac{2}{5}\times 30=12$.

The random variable X follows a Poisson distribution with parameter λ = 12.

The probability mass function of X is:

$P(X=x)=\frac{e^{-12}12^{x}}{x!};\ x=0,1,2,3...$

Compute the probability that 5 cars will arrive in 30 minutes as follows:

$P(X=5)=\frac{e^{-12}12^{5}}{5!}$

                $=\frac{6.1442\times10^{-6}\times 48832}{120}$

                $=0.0127$

Thus, the probability that five cars will arrive during the next thirty minute interval is 0.0127.