The number of automobiles entering a mountain tunnel per 2-minute period has a P

Question

The number of automobiles entering a mountain tunnel per 2-minute period has a Poisson distribution with mean one. An excessive number of cars entering the tunnel during a brief period of time produces a hazardous situation.

(a) Find the probability that the number of autos entering the tunnel during a two-minute period exceeds three.

(b)What is the probability that the number of autos entering the tunnel during a six-minute period is less than four

(c) Assume that the tunnel is observed during ten 2-minute intervals, thus giving independent observations

Y1, Y2,

Explanation / Answer

mean = lambda = 1

P(Y= k) = e^(-1) / k!

a ) P(Y >3) = 1 - P(Y=0) - P(Y=1) - P(Y=2) - P(Y=3) = 0.018988

now mean =3 because of 3 minute period, lambda = 3

P(Y= k) = e^(-3) *(3)^k / k!

b ) P(Y <4) = P(Y = 0) +P(Y = 1) + P(Y = 2) + P(Y = 3) =0.647232

Let X be Binomial with

n = 10 , p = P(Y >3) =0.018988

c) ans = P(X >=1) = 1 - P(X = 0) = 1 - (0.018988)^10 = 1

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