Independent solution

How to solve this Conditional Probability question

Setup

Setup

Let L and H identify the two risk classes, and let Z be the observed zero-count event. A Poisson variable with mean λ is zero with probability exp(-λ).

Pr⁡(Z∣L)=e−0.05\Pr(Z\mid L)=e^{-0.05}
Pr⁡(Z∣H)=e−0.30\Pr(Z\mid H)=e^{-0.30}

Model

Model

Multiply each zero-count likelihood by its class prior and normalize the low-risk weight.

Pr⁡(L∣Z)=0.70e−0.050.70e−0.05+0.30e−0.30\Pr(L\mid Z)=\frac{0.70e^{-0.05}}{0.70e^{-0.05}+0.30e^{-0.30}}

Compute

Compute

Evaluate the two joint weights and their ratio.

wL=0.70e−0.05=0.665860598w_L=0.70e^{-0.05}=0.665860598
wH=0.30e−0.30=0.222245466w_H=0.30e^{-0.30}=0.222245466
Pr⁡(L∣Z)=0.6658605980.888106064=0.749753464\Pr(L\mid Z)=\frac{0.665860598}{0.888106064}=0.749753464

Answer

Answer

The posterior low-risk probability rounds to 0.750.

0.750(D)\boxed{0.750\quad\text{(D)}}