The number of cars washed in a certain car wash station is a Poisson random variable with a mean of 6 cars per hour

Question

(a) What is the probability that next car arriving to this station will wait less than 45 minutes?

(b) What is the probability that at least 1 cars will be washed in 30 minutes period?

Answer 1

The Poisson random variable with mean waiting time of 6 cars per hour, means the mean waiting time between car washes is 10 minutes.

a. When the random variable represents waiting time between events, it is an exponential random variable.

P(X<45) =1-exp(- x/λ) = 1 - exp(-45/10) = 0.9889

b. the poisson mean is 6 cars per hour, which can also be expressed as 3 cars per 30 minutes

P(X≥1) = 1 - P(X=0) = 1 - (λ^xe^-λ)/x! = 1 - ((3⁰e^-3)/0!) = 0.9502

Please let me know if you have any questions.

Dian