In addition to the application of probabilities in poker some special considerations apply to Texas Hold'em. The calculations are fairly simple, in that prior to the flop the "deck" (that is, the set of unseen cards, including those in other players' hands) has 50 cards. The chances that an ace will turn up in the flop (or any other specific card of which you have one already), are 16.5% or about 5:1 against. The chances of a match for either the ace or the king (or whatever) are double that, or around 33%. This is 2:1 odds.

From there, the calculations can become a little more complex, but still doable within the conventional methodology used in poker. For example, if the hole cards are suited, then the chances of having at least one match in the flop are pretty good, about 63%. The chances of having two matches are lower, 26%, and three is just 5%.

The phase of the game after the flop is where the play becomes dramatic. Fortunately, the math is much simpler, as each card is dealt one at a time. Players will add up the number of "outs" for their hand, that is, the number of different cards in the deck, which, if dealt on turn or river, would complete the hand. It is possible, of course, to make these calculations if you should need two cards for an "out," but then you would have to ask yourself why you are in the game at this stage, since the odds of getting the two necessary cards are probably much worse than the pot odds.

So let us assume that only one card is required for completion of the hand. However, more than one rank might do the trick, as in drawing for a flush, or more than one suit would work, if drawing for a straight or completion of a house or quad. Suppose a player has 4 "outs" for a full house. This means that he or she has two pair, and two of each are still in the deck. The deck has 47 cards in it before the turn card is dealt. The chances of completing this hand on the turn are 4 / 47 or 8.5%. If the hand did not complete on the turn, the chances of completing it on the river card are 4 / 46 or 8.7%. The compound probability of receiving one of the "outs" on either the turn or the river has a little more work to it, but essentially, there are 184 positive scenarios out of a total of 1081 possible one, for an outcome of 17%. Added to this is the remote possibility that one of the pairs will be matched on the board by a pair in the turn and the river. Only two chances out of 1081 would qualify, for a probability of 0.2%. The following table may be helpful in calculating the odds of a winning hand on the turn and river rounds, given the number of "outs." Be sure to include every combination of rank and suit.