# Help with this combinatorial computation

I am going to run a test with N subjects. There are M=50 stimuli to be evaluated by the subjects but, for time and concentration reasons, a subject cannot evaluate all the stimuli. Therefore, I am going to provide only P < M stimuli (randomly chosen) to each subject.

• How can I compute the expected number of times a given stimulus will be evaluated, knowing N and P?
• Similarly, how can I choose N in order to have an expected number of evaluation greater than X for each stimulus?

If all the stimuli are assigned randomly (uniform) and independently, then subject $n$ would have probability $\frac{P}{M}$ to get a specific stimulus $p_i$ in his set of stimuli $P_n$:

$\mathbb{P}(p_i \in P_n) = \frac{P}{M}$

To get the expected number of times that stimulus is given in total, simply multiply by $N$:

$\mathbb{E}(\sum_n 1_{\{p_i \in P_1, \dots, P_N\}} = N\frac{P}{M}$

Similarly, if you want $N\frac{P}{M} > x$, you can rearrange this to find:

$N > x\frac{M}{P}$

This is just a binomial distribution.

A given stimulus has a probability P/M to be evaluated by a given subjects. Therefore, the number of times an stimulus is evaluated in a N subjects sample follows a binomial distribution $B(N,P/M)$ and its expectation is $N·P/M$. To have an expected number greater than X, you just need to choose $N>X·M/P$.