Poker probability question [Texas hold'em]: Do chances to get a certain hand with the flop decrease with number of opponents? In a basic Texas hold'em game if I had 2 6's I understand what the probability of getting a 6 on the flop is.  My question is... I say no matter how many players are in the game that the odds are the same.  My buddy says that if there are 12 players ( meaning that 24 cards are delt ) that the chances of getting the 6 decrease since there are less cards compared to 3 players ( 6 cards delt).  I say the odds are identical since they are all unknown. Can someone please help with this.
 A: The probability that the flop allows you to make three sixes is the same, no matter how many opponents there are. However, the probability that some of your opponents has a better hand than yours grows with the number of players. Even if you didn't take a look at your own hand, you could say that your hand is the best of two players with a probability of $0.5$, and is the best hand among ten players with probability $0.1$. Now after you check your hand, you may find that your hand is (or is not) well above average, but the dependence on the number of opponents is still there.
Let's try different rules: Each player gets one card, highest card wins; there is also a flop, but it is ignored. If there are only two players and you have a king, you can be quite confident to have the better hand; if there are $29$ players and you have a king, you know for sure that some player has an ace. But this doesn't change anything about the probabilities what card can be in the flop if you only know your own card (all you know is, it is not your king).
A: Note that the probability that there is a 6 in the flop boils down to the probability that a 6 is in certain position in deck so when dealer flops cards the 6 comes up. When you add more players all this does is change the position the 6's have to be in which does not change probability that you should get a 6 (since 6 is likely to be in any position in the deck anyway). And since you don't know other players cards, this gives you no more information about where 6s may be positioned
