A probability problem on conditional expectation

Question

Consider $n$ multinomial trials, where each trial independently results in outcome $i$ with $p_i$, $\sum_{i=1}^{k}p_i=1$. With $X_i$ equal to the number of trials that result in outcome $i$. How to find $E[X_1 | X_2 =0]$ ?

Answer 1

Use the linearity of (conditional) expectation.

Let $I_i$ be the indicator variable that the $i$th trial shows outcome 1, given that it does't show outcome 2.

Then, $E[I_i] = \frac{p_1} { 1 - p_2}$.

Hence, $E[X_1 | X-2 = 0 ] = E[\sum I_i] = \sum E[I_i] = n \frac{p_1} { 1-p_2} $.