Independent solution

How to solve this Complement Rule question

Setup

Setup

For each of the seven days, the no-claim probability is r, and the daily claim indicators are mutually independent.

Pr⁡(no claim on one day)=r\Pr(\text{no claim on one day})=r

Model

Model

The complement of at least one claim is a period with no claim on every day.

Pr⁡(at least one claim)=1−Pr⁡(no claims in seven days)\Pr(\text{at least one claim})=1-\Pr(\text{no claims in seven days})

Compute

Compute

Independence allows the seven daily no-claim probabilities to be multiplied.

Pr⁡(no claims in seven days)=∏j=17r=r7\Pr(\text{no claims in seven days})=\prod_{j=1}^{7}r=r^7
Pr⁡(at least one claim)=1−r7\Pr(\text{at least one claim})=1-r^7

Answer

Answer

The required probability is the complement of seven claim-free days.

1−r7(D)\boxed{1-r^7\quad\text{(D)}}