What Is The Math Behind Poker Pot Odds

We’ve looked previously at real-world applications of math; from the arithmetic required to compose a weekly household budget to the advanced algorithms used in computer programming languages. A branch of math that we use constantly, and perhaps unknowingly, is probability. If the weather app says 80% chance of rain, we’ll likely take an umbrella or rain jacket outside with us. At 10%, we might rationalize a trip to the supermarket won’t involve much time outdoors and take our chance. At 50%, we’ve got a brave call to make!

A poker table is a place where probability is paramount. Far from being a game of chance, the top professionals are constantly calculating. In the most popular variant, Texas Hold ‘Em, players are dealt two cards of their own and (up to) another five are turned over one by one as community cards for each player to make the best possible hand with. Thus, the player must ascertain what they have, what hands may be likely to be made as cards are overturned, and what their opponents may be holding. 

Probabilities 

Before a card is dealt, a player has 1 in 2 chance of ending up with a pair. The odds of hitting the royal flush – the best hand in the game – are 1 in 649,740. To put that into statistical perspective, you’re almost 50 times more likely to be struck by lightning! Each poker hand starts with a ‘big blind’ and a ‘small blind’ – two players will put in a stake before any cards are dealt. Let’s say that pot size is $3. Upon seeing their two personal cards, an opponent forwards $2. To continue playing, you must match (or raise) their wager. With the pot now at $5, you must put in one-third of that value ($1.66) to have a chance at winning the whole pot. This lets you calculate poker pot odds at 3 to 1. For it to be worthwhile to continue, a player would calculate that their hand will be the winner more than one-third of the time. If a player’s initial two cards are a Jack and 3 of different suits, they have few potential combinations to make a winning hand and may fold at that point. If they’re dealt an Ace and King of the same suit, you’d expect to see them back their cards fairly heavily. 

Progression 

As the example hand moves forward – let’s say we had two hearts for our personal cards and then another two come out – we have the potential to finish with a flush, which is five cards of the same suit. If we have equity of 3:1, per the pot odds, then with a circa 20% chance of getting another heart, the math doesn’t work in our favor. However this doesn’t take into account our opponent’s actions. Let’s say they’ve been very aggressive in the lead-up to the final (river) card and we suspect they got a high pair in their initial deal. A flush with even a 7 as the highest card would beat three of a kind, even if they were higher value cards like Kings or Aces. Therefore we can use implied odds to work out if we can finish on a flush then we would likely have a better hand than our opponent and it may be worth the risk. And now we move on to psychology!

Of course, a poker player can only guess at what their opponent may have. The next time you watch a televised tournament – where the cameras under the table show you each player’s hand – keep an eye out for the displays that show you each player’s percentage chance of winning the hand and how it adjusts as each card is dealt. The most fascinating aspect is math in action.