Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability resit / 8 / a / ii / Solution

Using the indicator random variable, sum over the values of the discrete random variable $X$:Dividing by the positive probability $\mathbb{P}(B)$ proves $\mathbb{E}[X\mid B]=\mathbb{E}[X{1}_B]/\mathbb{P}(B)$. On $B^c$ the product vanishes, so no additional contribution is omitted.