Independent solution

How to solve this Conditional Probability question

Setup

Setup

Let X count affected subjects among the three independent trials. The common affected probability is 0.01.

X∼Bin⁡(3,0.01)X\sim\operatorname{Bin}(3,0.01)

Model

Model

The observed pooled alert is equivalent to X being positive. Because the event X=2 is contained in that alert event, conditional probability reduces to a ratio.

Pr⁡(X=2∣X≥1)=Pr⁡(X=2)Pr⁡(X≥1)\Pr(X=2\mid X\ge 1)=\frac{\Pr(X=2)}{\Pr(X\ge 1)}

Compute

Compute

Use the binomial mass in the numerator and a complement in the denominator.

Pr⁡(X=2)=(32)(0.01)2(0.99)=0.000297\Pr(X=2)=\binom{3}{2}(0.01)^2(0.99)=0.000297
Pr⁡(X≥1)=1−(0.99)3=0.029701\Pr(X\ge 1)=1-(0.99)^3=0.029701
0.0002970.029701=0.0099996633…\frac{0.000297}{0.029701}=0.0099996633\ldots

Answer

Answer

The requested conditional probability rounds to 0.0100.

0.0100(C)\boxed{0.0100\quad\text{(C)}}