I get a few percent off from simulation results when I verify these probability calculations, so I am hoping someone spots it.
When there is a suited pair on the flop in texas holdem and nine seats with two cards each. Tree cards are known, the suited pair on the table, and the last card in the flow, so there is 49 cards left in the stack, and 11 of those wear the same suit as the pair.
The probability of any or more players on the table having two cards wearing the same suit as the pair on the table would then be (11/49.0) * (10/48.0) * 9.
Except simulation reveals I'm off by just a few tiny percent on that claim, so what am I missing here? Why is my calculated probability just a notch higher then reality? What am i missing?
You have correctly written down the probability that one person has the flush, and then multiplied by $9$ because you have $9$ players. However multiplying by $9$ over-estimates the total probability. The reason why is that more than one person could have a flush, so your 9 cases can overlap, thus the overall probability is not the sum of those $9$ probabilities. The probability that 2 particular people have a flush is $(11/49)(10/48)(9/47)(8/46)$. There are ${9 \choose 2} = 36$ possible pairs of players. So you have to subtract $36(11/49)(10/48)(9/47)(8/46)$. But then this under-estimates the overall probability because your subtracted cases overlap when $3$ people have the flush. So you have to add the probability that any 3 people have the flush. And so forth, subtract the probability that any 4 players have the flush, then add the probability that any 5 players have the flush. This is the inclusion-exclusion principle, you can look it up. If you apply it all the way out to the maximum number of players that can have the flush (5), you will get the exact answer.