Poisson Distribution practice problem

Question

I have just started to learn Poisson Distribution and I have no idea how to deal with the following practice from my textbook: Suppose the average amount of cars passing on a street per minute is Poisson (8.6). It takes a lady 5 seconds to cross a street and she waits for two cars to pass. What is the probability that she is safe to cross the street ?

Answer 1

THe first step is to calculate the law of the number of car that pass on the street in 5s.

It can be shown that it follow a Poisson law with $\lambda = 5\times\frac{8.6}{60} \simeq 0.7166$

Then the number of car that pass on the street in 5s is

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

So the probability that no car pass in the street in the next 5s is

$$P(X=0) \simeq e^{-0.7166} \simeq 48.8 \%$$