Calculating the Odds When Playing Poker:

There is a huge variety of poker games with unique rules. Nevertheless, the basic principle of all these games is the same: The player with the highest (or the lowest) poker hand wins. Therefore, although each game requires a unique strategy, all these strategies are based on odds of poker hands.

Single Card Odds:

The odds of a single card to be dealt next are the number of cards of the same kind that still are in the deck divided by the total number of cards in the deck. For example, odds of a card of a certain rank to be dealt as the first card are 4/52 (4 cards of the same rank/52 total cards). Note that on practice you should not subtract hole cards of other players from the total number of cards in the deck, because as far as you do not know what they are, they do not affect your calculations, so replace the number of total cards by the number of cards you do not know.

Normally we want to calculate is the odds of a certain combination and not of a single card. Let's say that we want our two first cards to be a pair of Aces. The operation we should use is multiplication. We already know that the odds of the first card to be an ace are 4/52. If we suppose that the first card is an ace, we have three aces left in the deck and 51 total cards, therefore the odds of the second card to be an ace too will be 3/51. The odds of both first cards being aces are 4/52*3/51 = 1/221. If we want our first two cards to be any pair, we do not care what the first card will be. We only want the second card to match it, so we will take the odds of the second card only: 3/51 = 1/17.

Next Card Odds:

If we already have a number of cards and want to calculate the odds of the next card matching one of them, we should add the odds of every card in the hand to match the next card. For example, if we have 4 unmatched cards and want the fifth card to match one of the four, the odds will be: 3/48 + 3/48 + 3/48 + 3/48 = 12/48 = 1/4 (with 4 cards we have 48 cards left in the deck).

This is right if we do not see cards of other players. In stud poker we can take into the account cards on the board. For example, if in five card stud we have the following board: 2, 10, Ace, 8, 10, King, and our hand is 6-10-Jack-Ace, the total number of unknown cards is 42, and we have 2, 6, 8, three 10s, Jack, King and two Aces among known cards, so the odds of the fifth card to match one of the cards in out hand are: 3/42 (for the 6) + 1/42 (for the 10) + 3/42 (for the Jack) + 2/42 (for the Ace) = 1/4.667.

The same way we can calculate the odds of a certain card appearing as on one of several deals. We add the odds of each deal. For example, if we have 10-Jack-King-Ace and want to receive a Queen as one of the three following cards (in a seven card poker game), our odds will be: 4/48+4/47+4/46 = 1/3.92. To simplify the calculation, you can divide the number of required cards by an average number of total cards and to multiply the result by the number of cards to be dealt. For the example above, the simplified formula would be 4/47*4.

Important:

You do not need to calculate odds very precisely. What you want to know is whether the game is worth playing or not, so an approximate calculation is enough. What is important is to take into the account all the information you have. This is particularly right for stud poker, where you can reduce the odds of cards that already are on the board.