Assume that the tunnel in Exercise 3.132 is observed

Step 1 of 2

Suppose the number of automobiles entering the tunnel is observed during ten 2-minute intervals, thus observing ten independent observation \(Y_{1}, Y_{2}, Y_{3, \ldots \ldots, \ldots,} Y_{10}\), on the Poisson random variable.

We need to find the probability that the number of cars entering the tunnel \((Y>3) during at least one of the ten 2-minute intervals.

Let the random variable \(X\) be the number of 2-minute intervals with more than three cars.

Here we need to find \(P(X \geq 1)\).

First, let's find the probability that the number of cars entering the tunnel during the 2-minute period exceeds three.

Let the random variable Y be the number of cars entering the tunnel in a 2-minute period.

A random variable \(Y\) is said to have a  Poisson probability distribution if and only if

\(p(y)=\frac{\lambda^{y} \times e^{-\lambda}}{y !}, \quad y=0,1,2, \ldots \ldots, \lambda>0\)         

(1)

we can write \(P(Y>3) like this,

\(\begin{array}{l} P(Y>3)=1-P(Y \leq 3) \\ P(Y>3)=1-[P(Y=0)+P(Y=1)+P(Y=2)+P(Y=3)] \end{array}\)