Let's do a multinomial experiment i.e. there are \(m\) no. of possibilities on each trial and we do \(n\) number of trials.
Moreover, the probability of getting a particular possibility at \(i\)th trial also varies, each differently, like \(a_k(i), 1 \le k \le m\)
What is the expected number of times of getting a particular possibility (let's say the one with index 1)?