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Casino games are games of chance with a built-in house edge. No strategy, system or choice of timing changes the odds in the player’s favor over time, and most players lose money over time. Nobody should gamble money they cannot afford to lose. If gambling is causing you or someone you know difficulty, a local support service can help. This article explains the math; it is not a way to win.
Outcomes and the sample space
Probability starts with a simple question: what can happen? Each distinct result is an outcome, and the complete list of possible outcomes is the sample space.
- A coin flip has two outcomes: heads and tails.
- A standard six-sided die has six: 1, 2, 3, 4, 5 and 6.
- A 52-card deck has 52 outcomes for a single draw.
- A single-zero roulette wheel has 37 pockets (0 through 36). A double-zero wheel has 38.
Before you can calculate anything, you need the sample space written down correctly. Most mistakes in gambling math come from miscounting it, for example forgetting the zero pocket on a wheel.
Events
An event is any set of outcomes you care about. “The die shows an even number” is an event containing three outcomes: 2, 4 and 6. “The card is an ace” contains four outcomes. “The ball lands on red” contains 18 pockets on a standard roulette wheel.
The basic formula
When every outcome is equally likely, the probability of an event is:
probability = number of outcomes in the event ÷ total number of outcomes
Some worked examples:
- Even number on one die: 3 ÷ 6 = 1/2, or 50%.
- An ace from a full deck: 4 ÷ 52 = 1/13, about 7.69%.
- Red on a single-zero wheel: 18 ÷ 37, about 48.65%.
- Red on a double-zero wheel: 18 ÷ 38, about 47.37%.
- A single number on a single-zero wheel: 1 ÷ 37, about 2.70%.
Notice that red is not a 50% bet on either wheel. The zero pockets belong to neither color, and that gap is where the house edge comes from. The idea that equally likely outcomes can still produce a negative result for the player is explored further in the guide to the house edge.
The “equally likely” condition matters. If a die were weighted, counting faces would no longer give the right answer. For well-built, fair equipment and properly tested software, equal likelihood is the design assumption, which is why fairness testing exists.
Probabilities live between 0 and 1
A probability of 0 means impossible and 1 means certain. Percentages are the same numbers multiplied by 100. Converting between fractions, decimals and percentages is a skill of its own, covered in the article on odds conversion.
The complement rule
The probability that an event does not happen is 1 minus the probability that it does. If the chance of an ace is 4/52, the chance of a non-ace is 1 − 4/52 = 48/52 = 12/13.
The complement is a shortcut for “at least one” questions. What is the chance of at least one six in two rolls of a die? Counting directly is awkward. Instead, find the chance of no six at all: (5/6) × (5/6) = 25/36. Then the answer is 1 − 25/36 = 11/36, about 30.56%. Notice that it is not 2/6, which is the tempting but wrong answer from simply adding.
Adding and multiplying
Two rules cover most simple problems.
Either/or (addition). If two events cannot both happen on the same trial, add their probabilities. A die showing 1 or 2: 1/6 + 1/6 = 2/6 = 1/3. If the events can overlap, subtract the overlap so you do not count it twice. A card that is a heart or an ace: 13 hearts + 4 aces − 1 (the ace of hearts is in both) = 16 cards, so 16/52 = 4/13, about 30.77%.
Both/and (multiplication). If events do not influence each other, multiply. Two sixes on two separate rolls: 1/6 × 1/6 = 1/36, about 2.78%. This works because dice rolls and roulette spins are independent events: one result does not change the next. Card draws without replacement are different, because removing a card changes what is left.
Counting when the sample space is large
For two dice there are 6 × 6 = 36 equally likely ordered results, and counting those is the basis of the two-dice outcomes that craps relies on. For five-card hands the count is far larger, and shortcuts for counting such hands are needed. Those methods are covered in the article on card combinations.
Probability does not predict a single result
A probability describes what happens across many repetitions, not what the next trial will do. A 1-in-37 chance does not mean a number “comes up once every 37 spins.” It can appear twice in a row or not appear for 100 spins, and neither fact changes the next spin. Believing otherwise is the gamblers fallacy, one of the most persistent errors in gambling.
Putting it together
With a sample space, an event, and the choice of adding, multiplying or taking the complement, you can answer a large share of the basic questions about casino games. The next steps are to attach money to those chances, which is what expected value does, and to understand how results spread around the average, which is the role of standard deviation. Together they explain why a game can be exciting in the short term and still cost the player money in the long run.