Question on Poisson processes vehicles cross highway with rate of lamda = 5 per

Question

Question on Poisson processes vehicles cross highway with rate of lamda = 5 per minute.

Suppose instead that the person Is able to escape any car that Is encountered with probability 1/2, and any van that is encountered with probability 3/4. Assume also that any passing vehicle Is a car with probability 3/4 and van with probability 1/4 independently of other passing vehicles. Again, what is the probability that the person will be uninjured if the total amount of time required to cross the road is 15 seconds?

Explanation / Answer

let

C is event that he is ininjured with Car

V is event that he is ininjured with van

and U is event that he is ininjured

so

P(U)=0.5*0.75 +0.75*0.25=0.5625

in 15 second approximately 2 vehicles will pass

so

P( He is uninured) =(0.5625)2=0.3164

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