Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 18 / b / i / Solution

The law of total probability givesFor $\alpha >0$, Bayes' theorem therefore yieldsAt the allowed endpoint $\alpha =0$, no bulb is defective: $\mathbb{P}(D)=0$. This conditional probability is then undefined, rather than $2/5$. The value $2/5$ is its limit as $\alpha \to 0^{+}$, but does not define conditioning on the impossible event within the stated discrete model.