The odds in No Limit Holdem (NLHE) poker

No Limit Holdem (NLHE) is one of the most common variations of poker, where each player receives 2 hole cards.

Poker is played with a pack of 52 cards divided into 4 suits:

• 13 spade cards
• 13 club cards
• 13 heart cards
• 13 squared cards

NB: We talk about color to talk about the sign of the card and not to designate whether the card is red (diamonds, hearts) or black (spades, clubs).

The number of possible card combinations is 1326. There is therefore a one in 1326 chance of receiving a specific combination.

The probability of receiving a specific card is 1/52.

For example, I want to receive the ace of hearts and the ace of diamonds:

I have a 2 in 52 chance of receiving one of the two when I am dealt the first card, then a 1 in 51 chance of receiving the other card when I am dealt the second card.

The probability of receiving two specific cards is therefore 2/52 * 1/51 = 2/2652 = 1/1326

The probability of receiving a card of a specific suit is 13/52 = ¼

The probability of receiving two cards of the same suit is 13/52*12/51 =156/2704

For your calculations, it is important to remember that when you use the word “and”, the calculation will be a multiplication

Ex: receiving an Ace and a King

8/52*7/51 = 56/2652 = 28/1326 = 14/663 or around 2% chance.

When using the word “or”, the calculation will be an addition

Ex: receiving at least an ace or a king

(4/52+4/52) + (4/51+4/51) = 206/663 or around 30% chance.

Another very useful calculation tool in poker consists of defining the probability that certain cards fall on the turn and/or the river. These cards are called “outs”. For example, if we have two heart cards and two other heart cards are on the board on the turn, we have all the remaining heart cards in the deck as outs to make a flush on the river.

Concretely, given that we have two heart cards, two others are on the board and there are 13 heart cards in total in the deck, we have 13 - 4 = 9 outs.

Once you know how many outs you have, you can find the approximate probability that they will come out by multiplying this number:

• by 2 if there is one card left to come (the river)
• by 4 if there are two cards left to come (turn and river)

In our example, we had 9 outs on the turn, or 9*2 = 18% chance that a heart would complete our hand to give us a flush.