Question

Data from the past shows that on average, a ready-mixed concrete plant receives 100 orders for concrete every year. The maximum number of orders that the plant can fullfil each week is 2. (a) What is the probability that in a given week the plant cannot fulfil all the placed orders? (b) Assume the answer to part (a) is 20% (It is not; I just want to make sure that everybody uses the same number for part (b)). Suppose there are 5 of such plants. What is the probability that in a given week 2 of the plants cannot fulfill their orders?

Answer 1

as there are average 100 orders per year so as we know that there are total 52 weeks in a normal year hence average order per week =100/52 =1.92

let X is number of orders per week so X can be assumed to be poisson with lambda=average =1.92 so

a)

since the plant can fulfill maximum 2 orders per week so if there are more than 2 orders in a given week then plant cant fulfill the order hence we have to find P(X>2)

now

b)

Probability of not fulfilling the order in a week =p=0.2

there are n=5 plants and each have an equal probability of not able to fulfill the order in a given week

so let Y is number of plants out of 5 plants that are not able to fulfill the order in a given week hence

Y is a binomial distribution with n=5 p=0.2 so

now we to find P(Y=2)

so

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