How to calculate the probability of finding 3 specific numbers from the range 1-100 within a sample of 100 randomly generated numbers, with repetition, that is also in the 1-100? Order is not important in any context. An example is finding 1,5 and 7 among these 100 numbers.
I know the general formula for a combination with repetitions but my doubt is whether it should be adjusted because the 3 numbers are different, i.e. I am not looking for a triple repeat of one same number.
We are going to compute the complementary probability $q$ that such a triple will not be realized. This will involve an inclusion/exclusion process. One obtains $$q=3\cdot 0.99^{100}-3\cdot 0.98^{100}+0.97^{100}=0.747791\ .$$ Here the first term takes care of all cases where at least one of $1$, $2$, $5$ is missing, the second term takes care of the double counting of the cases where both $1$ and $2$, etc., are missing, and the third term throws in again the cases where all three numbers are missing. The probability $p$ you are after is therefore given by $$p=1-q=0.252209\ .$$