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For probability questions, I like to take the number of combinations the event you're interested can happen divided by the total number of combinations. First review the combin function in my probabilities in poker section.
The number of ways to get a four of a kind is simply the larry fortensky now of singletons in the deck, or Two players are all in pre-flop, hands on their backs. The most common 'flip' situation you'll see or more likely be in is the classic pair vs. I was playing world poker on xfinity internet apps and i think this was considered texas holdem.
The first 3 cards the dealer dealt on the table were a 4, Queen, and a 7. So i now have two pairs which are the 4's and 7's.
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For those unfamiliar with the terminology, each player gets two cards to himself and the larry fortensky now flop cards are shared among all players. So this is the same as asking if you dealt three community cards, all of different ranks, and ten 2-card hands, what is the probability three of the 2-card hands would be pairs that match one of the three community cards. However, any three players can the three sets, not necessarily the first three.
There are combin 10,3 ways to choose the 3 players out of 10 that have sets. The probability is the same for a pair of kings. Now the next step larry fortensky now clearly not perfect because if one player doesn't have one of these hands the odds the next player does is a little bit higher.
Forgetting this for the sake of simplicity the probability no player has one of these hands is So the probability at least one player has one of these hands is Now lets assume the other player has any other pair, but not the same as yours.]