# Probability of filling M boxes with 2 or more elements when sampling S elements from N total elements

Similar to this question, Frequency of Item in Combination.

I am randomly sampling S objects out of N=99 objects into 9 boxes labeled by a single character, "A-I". Question 1: I want to find the probability of M boxes having 2 or more objects, dependent on how many objects I sample, S out of N. Currently I am only able to do so by simulation (see code below), which is RNG dependent, but accurate. Question 2: Is there a distribution I can follow here? I'm bouncing between binomial and hypergeometric, but I am uncertain how to implement it.

mycountL <- double(9)
names(mycountL) <- LETTERS[1:9]

# Change this for sample drawing size
xTimes = 18

set.seed(12)
for(i in 1:10000){
nL <- names(which(table(sample(rep(LETTERS[1:9],11), xTimes ))>=2))
lL <- length(nL)
mycountL[lL] <- mycountL[lL]+1
}

mycountL/10000 #For probabilities. Drawing 18 times is the lowest sample possible to draw exactly 2 in each LETTER, except that it is highly unlikely.


### Edits: Clarification

• This is not simple. At stats.stackexchange.com/a/410506/919 I show how to solve this problem using the smaller example of M=2 with 99 replaced by 5 and 9 replaced by 6.
– whuber
Commented Mar 26, 2020 at 21:38
• I somewhat understand the mathematics but implementing it is another challenge altogether. I'm afraid I can't wrap my head around your "first" step. Commented Mar 27, 2020 at 19:33
• If it helps, thing of that step as just being two applications of the tabulate function.
– whuber
Commented Mar 27, 2020 at 20:46