Bernoulli Trials Probability Question

Question

I've been super stuck on this question for awhile now. Would really appreciate it if someone could break down the solution for me. Thanks!

Question: Consider now the setting of independent Bernoulli trials, each with probability of success $p$. Let $S_i$ be the number of successes in the first $i$ trials. Compute $\mathbb E[S_m \mid S_n]$. (You will need to consider three cases based on whether $m > n$, $m = n$, or $m < n$. Try using your intuition rather than proceeding by calculations.)

Answer 1

Are you familiar with calculating estimated values? Did you get the case $m = n $? That should be the easiest.

Notice that the outcome of each trial does not influence others and you have $P($success $) = p $.

Do they want exact values in function of $S_n$ or comparing $E[S_m | S_n] $ with $S_n $ is enough?

Answer 2

Basic approach. If $m = n$, then $S_m = S_n$, so I expect you can finish that case.

If $m > n$, then there were $S_n$ successes in the first $n$ trials. How many successes do you expect to see in the remaining $m-n$ trials?

If $m < m$, then the proportion of successes in the first $n$ trials was $q = \frac{S_n}{n}$. If that same proportion were to hold in the first $m$ trials, how many successes would you expect?