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 / iii / Solution

The joint probability of supplier $C$ and a non-defective bulb is $\mathbb{P}(C\cap D^c)=(2/5)(1-\alpha /4)$. Normalize by the positive non-defective probability:This is defined even at $\alpha =0$, where it reduces to the original supplier fraction $2/5$. At $\alpha =1$ it is $3/5$.