Independent solution

How to solve this Hypergeometric Sampling question

Setup

Setup

Let the unknown insured count be determined from the ratio of the probabilities of selecting exactly one insured item and selecting none. Taking this ratio cancels the common sample-count denominator.

Pr⁡(K=1)Pr⁡(K=0)=4r24−r=2\frac{\Pr(K=1)}{\Pr(K=0)}=\frac{4r}{24-r}=2

Model

Model

The simplified probability ratio yields a linear equation whose admissible solution is eight insured items, leaving 19 uninsured items.

4r=48−2r,r=84r=48-2r,\quad r=8

Compute

Compute

Count samples with exactly two insured items by choosing two from the eight insured items and two from the 19 uninsured items, then divide by all four-item samples.

Pr⁡(K=2)=(82)(192)(274)=266975=0.27282\Pr(K=2)=\frac{\binom82\binom{19}2}{\binom{27}4}=\frac{266}{975}=0.27282

Answer

Answer

The required sampling probability rounds to 0.27.

0.27(C)\boxed{0.27\quad\text{(C)}}