Independent solution

How to solve this Hypergeometric Distribution question

Setup

Setup

Conditioning on the absence of a spade reduces the relevant population to seven cards.

N=7,K=2,n=2N=7,\qquad K=2,\qquad n=2
D∣{S=0}∼Hypergeometric⁡(N=7,K=2,n=2)D\mid\{S=0\}\sim\operatorname{Hypergeometric}(N=7,K=2,n=2)

Model

Model

Use the hypergeometric variance with its finite-population correction.

Var⁡(D∣S=0)=nKN(1−KN)N−nN−1\operatorname{Var}(D\mid S=0)=n\frac{K}{N}\left(1-\frac{K}{N}\right)\frac{N-n}{N-1}

Compute

Compute

Substitute the conditioned population size, diamond count, and sample size.

Var⁡(D∣S=0)=2(27)(57)(56)\operatorname{Var}(D\mid S=0)=2\left(\frac{2}{7}\right)\left(\frac{5}{7}\right)\left(\frac{5}{6}\right)
Var⁡(D∣S=0)=50147=0.3401361\operatorname{Var}(D\mid S=0)=\frac{50}{147}=0.3401361

Answer

Answer

The conditional variance rounds to 0.34.

0.34(D)\boxed{0.34\quad\text{(D)}}