Question

During off hours, cars arrive at a tollbooth on the East-West toll road at an average rate of 0.5 cars per minute. The arrivals are distributed according to a Poisson distribution. What is the probability that during the next five minutes, three cars will arrive

Answer 1

Answer:

$P(X=3) = 2.5^3 \frac{e^{-2.5}}{3!}=0.214$

Step-by-step explanation:

Definitions and concepts

The Poisson process is useful when we want to analyze the probability of ocurrence of an event in a time specified. The probability distribution for a random variable X following the Poisson distribution is given by:

$P(X=x) =\lambda^x \frac{e^{-\lambda}}{x!}$

And the parameter $\lambda$ represent the average ocurrence rate per unit of time

For this case we know that $\lambda =0.5 cars/min$

And we want the probability that during the next five minutes, three cars will arrive, so then our rate becomes:

$\lambda = 0.5 \frac{cars}{min}* 5 min = 2.5 cars$

And we want this probability:

$P(X=3)$

And if we use the pmf we got:

$P(X=3) = 2.5^3 \frac{e^{-2.5}}{3!}=0.214$