This is a basic question which comes up often, but obtaining the exact answer is rather advanced. It involves the inclusion-exclusion principle again. You can do a search for my explanations and examples of that principle. The most recent one was from yesterday for the
odds that some player has a flush draw.
For your problem, we start by saying that the probability of one particular player having a pair is 78/1326 = 1/17. For the probability that at least 1 of 8 players has a pair, a quick and dirty first approximation is 8/17 = 47%. This is the first step in the inclusion-exclusion method, and this is higher than the exact answer because it double counts all the times exactly 2 players have it, triple counts the times 3 players have it, etc. In most problems, this will actually be within 1%, but not in this case, because the probability of a pair is fairly high. To make it more accurate, we subtract off the probability that 2 people have pairs, and we approximate that probability the same way as C(8,2)* 78/C(52,2) * 73/C(50,2) = 9.8%. C(8,2) is the number of ways to pick the 2 players. There are 73 pairs remaining after the first player chooses a pair. Subtracting this result from 47% gives 37.2%. We can continue in this manner until the answer is as accurate as we like. If we carry it out to 8 terms, the answer will be exact. Note that you alternate adding and subtracting, and each successive term will be smaller than the last, so you know when your answer is sufficiently accurate. Normally this only requires 2 or 3 terms, but in this case it will converge more slowly. Also, the next term will be a little more complicated, because the number of pairs remaining will depend on which pair the second guy chose. This is not as clean as the flush example, because in that case we could consider all combinations of 2n cards dealt to n players, so the answer was very clean and easy to carry out to all the terms. Be sure to understand that example first. I'll leave this one as an exercise to the student, as they say. [img]/images/graemlins/grin.gif[/img] I'll be glad to help you if you run into trouble.
PS - I computed the next term, and it is 1.2%, bringing the answer to 38.4%.