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Chapter 9: Q. 9.3 (page 412)

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

Expert verified

The probability of succesfully crossing the bridge in s seconds is

Ps=e-s/20+se-s/2020

Also the probability for different values of s is

P5=1.25e-0.25P10=1.5e-0.5P20=2e-1P30=2.5e-1.5

Step by step solution

01

Given Information

We have given that cars arrive on highway at the rate of𝜆=3permin.

Al takes s seconds to cross the highway.

02

Simplifying

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

⇒Ps=P(X<2|𝜆=s20)⇒Ps=P(X=0|𝜆=s20)+P(X=1|𝜆=s20)⇒Ps=e-s/20+se-s/2020

03

Calculations

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

P5=1.25e-0.25P10=1.5e-0.5P20=2e-1P30=2.5e-1.5

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

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