Question

Suppose that in Problem 9.2, Al is agile enough to escape from a single car, but if he encounters two or more cars while attempting to cross the road, then he is injured. What is the probability that he will be unhurt if it takes him s seconds to cross? Do this exercise for s = 5, 10, 20, 30.

Short answer

The probability of succesfully crossing the bridge in s seconds is

$P_s=e^{-s/20}+\frac{s{e}^{-s/20}}{20}$

Also the probability for different values of s is

$$\begin{gathered} P_5=1.25e^{-0.25} \\ {P}_{10}=1.5e^{-0.5} \\ {P}_{20}=2e^{-1} \\ {P}_{30}=2.5e^{-1.5} \end{gathered}$$

Step-by-step solution

Given Information

We have given that cars arrive on highway at the rate of$𝜆=3\mathrm{per}\min$.

Al takes s seconds to cross the highway.

Simplifying

Let $P_s$ denote the probability that he will return unhurt after s seconds.

$$\begin{gathered} \Rightarrow P_s=P(X<2|𝜆=\raisebox{1ex}{$s$}\!\left/ \!\raisebox{-1ex}{$20$}\right.) \\ \Rightarrow P_s=P(X=0|𝜆=\raisebox{1ex}{$s$}\!\left/ \!\raisebox{-1ex}{$20$}\right.)+P(X=1|𝜆=\raisebox{1ex}{$s$}\!\left/ \!\raisebox{-1ex}{$20$}\right.) \\ \Rightarrow P_s=e^{-s/20}+\frac{s{e}^{-s/20}}{20} \end{gathered}$$

Calculations

Now putting the values of $s=5,10,20,30$in above equation we get,

$$\begin{gathered} P_5=1.25e^{-0.25} \\ {P}_{10}=1.5e^{-0.5} \\ {P}_{20}=2e^{-1} \\ {P}_{30}=2.5e^{-1.5} \end{gathered}$$

which is the required probability for the various values of s.