Q. 9.2 - A First Course in Probability

Question

Cars cross a certain point in the highway in accordance with a Poisson process with rate λ = 3 per minute. If Al runs blindly across the highway, what is the probability that he will be uninjured if the amount of time that it takes him to cross the road is s seconds? (Assume that if he is on the highway when a car passes by, then he will be injured.) Do this exercise for s = 2, 5, 10, 20.

Step-by-Step Solution

Answer

The probability that AI will be uninjured is $\left(\frac{s}{20}\right)$ . Probability for $s=2$ is $0.9048$, Probability for  $s=5$ is $0.7788$, Probability for  $s=10$ is $0.6065$ and Probability for   $s=20$ is $0.3679$.

Step 1: Given Information

The Poisson process is given with rate $\lambda =3$ per minute.

Step 2: simplify

The number of cars passed by on the highway during the time when Al crosses the read blindly can be modeled as the Poisson process with rate$\lambda =3$. As if he needs $s$ seconds to cross the road, the number of cars that pass by is $N(\frac{s}{60}·3)=N\left(\frac{s}{20}\right)$. The necessary and sufficient condition that he will uninjured is$N\left(\frac{s}{20}\right)=0$.

So,

$P\left(N\right(\frac{s}{20})=0)=e^{{}^{-\frac{s}{20}}}$

Hence, for$s=2,5,10,20$ 

$$\begin{gathered} P\left(N\right(\frac{2}{20})=0)=e^{-0.1}=0.9048 \\ P\left(N\right(\frac{5}{20})=0)=e^{-0.25}=0.7788 \\ P\left(N\right(\frac{10}{20})=0)=e^{-0.5}=0.6065 \\ P\left(N\right(\frac{20}{20})=0)=e^{-0.1}=0.3679 \end{gathered}$$