Within an experiment, an observation can only end up in one bin.

If the observations are independent before you bin them, you would look at a multinomial distribution for each experiment.

That is considering experiment $i$, then if the count in bin $j$ is $X_{i,j}$, then $(X_{i,1},X_{i,2},X_{i,3},...,X_{i,n})$ would be multinomial with parameters $n_i$ (the total number of values in experiment $i$) and the set of population proportions for that experiment.

If you focus on a single bin (compared to all others), that would be binomial.

If you consider the set of counts in the bins for all experiments, then if the counts are independent across experiments you'd have an independent collection of multinomials.

Within an experiment, an observation can only end up in one bin.

If the observations are independent before you bin them, you would look at a multinomial distribution for each experiment.

That is considering experiment $i$, then if the count in bin $j$ is $X_{i,j}$, then $(X_{i,1},X_{i,2},X_{i,3},...,X_{i,n})$ would be multinomial with parameters $n_i$ (the total number of values in experiment $i$) and the set of population proportions for that experiment.

If you focus on a single bin (compared to all others), that would be binomial.

If you consider the set of counts in the bins for all experiments, then if the counts are independent across experiments you'd have an independent collection of multinomials.

If you consider the set of counts in one bin over all experiments, that would be a collection of independent binomial observations.