Question 1031·Hard·Math

The sample is selected from all positive samples. That is the condition, the fact limiting which samples can be picked. The denominator is the count in the group you pick from. You want the North samples within that group, so $P(\text{North}\mid\text{positive})=\frac{\text{positive North samples}}{\text{all positive samples}}$. The lab’s full sample count is not the denominator here.

The $8.2\%$ is of all $2{,}500$ samples. Type $8.2\%\operatorname{of}2500$ in Desmos. It returns $205$, the positive-sample total needed for the denominator.

The $205$ positive samples came from the three clinics. Remove the $50$ from East and $50$ from West: type $205-50-50$ in Desmos. It returns $105$ positive North samples. The table’s $35\%$ describes North’s share of all samples, not its share of the positive samples.

Divide the North-positive count by the positive-sample total: $P(\text{North}\mid\text{positive})=\frac{105}{205}=\frac{21}{41}$. Type $\frac{105}{205}$ in Desmos. It shows $0.512195\ldots$; tap the fraction button to see $\frac{21}{41}$. So the probability that the selected positive sample came from North is $\frac{21}{41}$. Choice A.