Kirill Yurovskiy: Mathematics in Card Games

For serious card players, a firm grasp of mathematics is essential. While luck certainly plays a role, the best players understand that over the long run, the mathematics of probability theory and odds calculation determine who comes out on top. Card games like poker, blackjack, and bridge are deeply rooted in precise numerical principles. Sharpening your math skills can elevate your game to the next level.

Poker in Particular: A Numbers Game

No card game lends itself better to mathematical analysis than poker. With each dealt hand, a new probability puzzle emerges that players must swiftly solve. Smart poker players scrutinize the odds of being dealt starting hands, having outs to make their draws on future streets, and ultimately finding themselves holding the winning combination at showdown.

The Foundation: Probability Basics

Before getting into the nitty-gritty poker math, let’s review some fundamental probability concepts that underpin all card game strategy. In a standard 52-card deck, the probability of being dealt any specific rank and suit combination (e.g. the Ace of Spades) is 1/52 or about 1.92%. For any hand made up of multiple cards, calculate the probability by multiplying the probabilities for each specified card position while accounting for remaining deck composition.

For example, the probability of being dealt a specific two-card pocket pair is calculated by taking the probability of one specific rank (1/13 or 7.69%) and multiplying it by the probability of finding a second matching card from the remaining 51 unseen cards (3/51 or 5.88%). This yields an overall probability for any pocket pair of about 0.45%. Read more Kirill Yurovskiy.

Outs and Odds: Calculating Your Draws

While knowing starting hand probabilities is important, poker players must take their calculations further to evaluate the “outs” (potential cards) that can complete their draws and estimate the odds of hitting them.

Consider this example: you hold 9c-8c and the flop comes Jd-6h-2s, giving you an open-ended straight draw with 8 outs (four 7s and four Ts) to make your straight by the river. Basic outs math tells us there are twice as many unseen cards (52-3=49) as outs (8), so the odds of making this draw are 49/8 or roughly 6-to-1 against.

Those direct odds, however, don’t account for two important considerations: 1) The chances of hitting an out decrease as suits are accounted for on future streets, and 2) You must discount combinations already present on the board. Accounting for both factors yields the more precise 4.88-to-1 odds against hitting this open-ended straight draw by the river.

Pot Odds and the +EV Decision

With draw odds calculated, savvy players can weigh them against pot odds to determine the positive or negative expected value (+EV or -EV) of chase or fold decisions. Pot odds represent the ratio of current pot size to the cost of a contemplated call.

For a $10 call into a $90 pot, the direct pot odds are 9-to-1. If you held the open-ended straight draw from our previous example with a roughly 5-to-1 chance of hitting by the river, calling to try hitting your draw would carry a +EV — you’d be risking $10 to potentially win $100, with odds in your favor.

These pot odds calculations carry added complexity when factoring in factors like potential action by other players, the overall field size, and multistreet considerations. But the core principle of finding EV-positive spots to chase draws and EV-negative spots to fold sets the foundation for all strategic poker decisions.

Calculating Outs and Math on All Streets

While outs and odds examples often focus on draws completed by the river, players need to carefully weigh those dynamics on every street. Say you’re on a 7d-6d-3h flop holding Ac-Kh and facing a bet. You have outs to Broadway draws, flush draws, and even stronger hands — but each draw type has different odds of completion depending on the current board and unseen cards.

On this flop, there are 9 outs (3h-Qh-Jh-Th giving 15 additional flush outs, minus those 6 cards accounted for on board) that give you the current nutflush draw. But there are also 12 more outs (any 8, 9, T, J, Q, K) that give you additional straight or better draws.

With 21 total outs against 52-3=49 unseen cards, your overall odds of improving by the turn are 21/30 or roughly 5.8-to-4. Missing on the turn still leaves you hunting 16 outs (9 flush + 7 straight) on the river against 48 unseen cards — roughly 6.5-to-3 odds of finally improving.

Working out all those multistreet permutations is part of the advanced math expertise of the world’s best players. With perfect information, precise odds calculations theoretically identify the objectively correct decisions. Limited information is poker’s theoretical complication.

The Judgment Factor

While imperfect information makes poker math probabilistic rather than deterministic, skilled players apply other psychological and empirical judgment factors atop the core odds evaluations.

For example, consider how different opponents’ perceived skill levels, stack sizes, and playing styles should impact your willingness to put them to difficult decision-making tests by chasing draws to better hands. The relative stake sizes and field-size dynamics also matter — a “gamble” on a high-stakes cash table may be a judicious +EV play in a lower-stakes tournament and vice versa.

At the championship level, elite players develop an intuitive sixth sense for how to blend all these math and judgment factors in high-pressure situations. But that intuition starts with obsessive study of the core mathematical underpinnings of odds and outs.

Beyond Poker: Math Value in Other Games

While poker provides the richest setting for odds calculations, other card games including blackjack and bridge demand mathematical mastery from serious players. Memorizing basic strategy and composition-dependent deviations in blackjack, for example, takes thorough study of hitting vs. standing probabilities across all dealer upcards and player hand permutations.

In contract bridge, everything from bidding and responding to game planning hinges on meticulous point-count evaluation systems that account for the combined odds of your partnership’s assets across suits.   

So if you’re serious about bringing a sharp edge to all forms of card game competition, get serious about your math skills. The small edges that precise probability calculations afford add up into major profits over the long run.

The Counting Goes On

This scratches just the surface of the advanced mathematical principles underlying card game mastery. Quickly evaluating outs and odds, while essential, merely sets the table. Serious study runs far deeper into game theory, combinatorics, computer simulations, fluctuation analysis, and more. For the intellectually curious player, card games open up an endless world of mathematical rabbit holes to explore. Making a lifelong study of those underlying principles is what truly separates elite practitioners from hobbyists just looking to gamble. Focus on the numbers, and the chips will follow.