Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 221 / 4 / iv / Solution

The stop records estimatebut they cannot estimate the population encounter probabilities $\mathbb{P}(D=1)$ and $\mathbb{P}(D=0)$ because encounters without a stop are absent. Equivalently, they do not determine the race-specific stop-probability ratio.

The scientist needs a representative denominator for all police-civilian encounters, including those with $M=0$. Suitable sources could include a carefully designed population or travel survey, systematic street and traffic observation, dispatch or body-camera sampling that records non-stop encounters, or an external administrative source measuring exposure to police by race. Combining its estimate of the population race odds with the estimable stop-data terms identifies the displayed causal risk ratio, provided the external sample targets the same city, period, and encounter population.