Question

An insurance company determines that N, the number of claims received in a week, is a random variable with P[N = n] = 1/2n+1, where n > 0 . The company also determines that the number of claims received in a given week is independent of the number of claims received in any other week. Determine the probability that exactly seven claims will be received during a given two week period.

(A) 1/256
(B) 1/128
(C) 7/512
(D) 1/64
(E) 1/32

Answer 1

Answer: (D)
`(1)/(64)`

Explanation:

Given : An insurance company determines that N, the number of claims received in a week, is a random variable with

`P[N=n]=((1)/(2))^(n+1)`, where n > 0 .

To determine the probability that exactly seven claims will be received during a given two week period.

Let n is the number of claims during first week (
`N_1` ) the 7-n is the number of claims during second week (
`N_2` ).

Then , we have

`P(N_1+N_2)=\sum^7_(n=0)P(N_1)P(N_2)`

[ ∵ The number of claims received in a given week is independent of the number of claims received in any other week. ]

=
`=\sum^7_(n=0)((1)/(2))^(n+1)((1)/(2))^(7-n+1)\\\\=\sum^7_(n=0)((1)/(2))^(n+1+7-n+1)=\sum^7_(n=0)((1)/(2))^(9)\\\\=(1)/(2)+(1)/(2)+(1)/(2)+(1)/(2)+(1)/(2)+(1)/(2)+(1)/(2)+(1)/(2)\\\\=(8)/(512)=(1)/(64)`

Hence, the probability that exactly seven claims will be received during a given two week period =
`(1)/(64)`