Question

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 10 cars per hour. What is the probability that 15 or more cars will arrive during any given hour of operation?

Answer 1

Answer:

Therefore, the probability is P=0.161.

Step-by-step explanation:

We know that 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.  We calculate the probability that 15 or more cars will arrive during any given hour of operation.

We know that the mean arrival rate is 10 cars per hour ⇒ λ=10.

We use the Possion formula:

$\boxed{P_r(X=k)=\frac{\lambda^k \cdot e^{-\lambda}}{k!}}$

we get

$P(X\geq 15)=1-P(X=0)-P(X=1)-P(X=2)-...-P(X=14)\\\\P(X\geq 15)=1-\frac{10^0\cdot e^{-10}}{0!}-\frac{10^1\cdot e^{-10}}{1!}-\frac{10^2\cdot e^{-10}}{2!}-...-\frac{10^{14}\cdot e^{-10}}{14!}\\\\P(X\geq 15)=1-(0.00004+0.0004+0.002+0.007+0.01+0.03+0.06+0.09+0.1+0.1+0.12+0.11+0.09+0.07+0.05)\\\\P(X\geq 15)=1-0.839\\\\P(X\geq 15)=0.161$

Therefore, the probability is P=0.161.