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)  $\dfrac{1}{64}$

Step-by-step explanation:

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

$P[N=n]=(\dfrac{1}{2})^{n+1}$, 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. ]

= $=\sum^7_{n=0}(\dfrac{1}{2})^{n+1}(\dfrac{1}{2})^{7-n+1}\\\\=\sum^7_{n=0}(\dfrac{1}{2})^{n+1+7-n+1}=\sum^7_{n=0}(\dfrac{1}{2})^{9}\\\\=\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}\\\\=\dfrac{8}{512}=\dfrac{1}{64}$

Hence, the probability that exactly seven claims will be received during a given two week period = $\dfrac{1}{64}$