Question

Suppose that the number of accidents occurring on a highway per hour follows a Poisson distribution with a mean of 1.25.

- What is the probability of exactly three accidents occur in hour?
- What is the probability of less than two accidents in ten minutes?
- What is the probability that the time between two successive accidents is at least ten minutes?
- If ten minutes have gone by without an accident, what is the probability that an accident will occur in the next twenty five minutes?

Answer #1

Here number of accidents will follow a poisson distribution.

p(x) = e^{-}^{x}/x!

= 1.25

(a) p(X = 3) = e^{-1.25}1.25^{3}/3! = 0.0933

(b) Expected number of accident in 10 minutes = 1.25 * 10/60 = 1.25/6 = 0.2083

P(x < 2) = P(x = 0) + P(x = 1)

= e^{-0.2083}0.2083^{0}/0! +
e^{-0.2083}0.2083^{1}/1! = 0.9810

(c) P(at least ten minutes between 2 accidents) =P(no accident in 10 minutes)

= e^{-0.2083}0.2083^{0}/0! = 0.8119

(d) The last 10 minutes will not impact the next 25 minutes.

so,

P(an accident will occur in next 25 minutes) = 1- P(no accidient
in next 25 minutes) = 1 - e^{-25*1.25/60}

= 1 - 0.5940 = 0.4060

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