Question

Please answer all four parts of the question and show all work. Thank you!

Given an arrival rate (lambda and in terms “so many arrivals per time unit) and an X, where X is defined on the same time unit as lambda, that is, if lambda is 20 per hour, then X is not 5 minutes, it is 5/60 of an hour).

Prob (next arrival less than X) = 1- e^-λX

1. Autos arrive at a toll plaza at a rate of 50 per minute during rush hour. If an auto has just arrived,
   1. What is the probability that the next auto will arrive within 3 seconds?
   2. What is the probability that the next auto will arrive within 1 second?
   3. What is the probability that the next auto will arrive after 2 seconds?
   4. What is the probability that the next auto will arrive between 1 and 2.5 seconds?

Answer 1

here λ =50/minute =(50/60) /seconds =0.8333 autos/seconds

a)

P(X<3)=1-exp(-0.833*3)=

0.9179

b)

P(X<1)=1-exp(-0.833*1)=

0.5654

c)

P(X>2)=1-P(X<2)=1-(1-exp(-0.833*2))=

0.1889

d)

P(1<X<2.5)=(1-exp(-0.833*2.5)-(1-exp(-0.833*1))=

0.3101