Before the Arithmetic
Casino games are games of chance with a built-in house edge. Understanding the probabilities does not make any strategy profitable, and most players lose money over time. Never gamble money you cannot afford to lose, and if gambling is becoming a problem, a local support service can help.
The arithmetic below is useful for a different reason: it lets you check what a game actually offers, rather than relying on how it feels.
The Core Rule
For events where every outcome is equally likely, probability is a fraction:
probability = number of favorable outcomes ÷ total number of outcomes
The result is between 0 and 1. A probability of 0 means impossible, and 1 means certain. It can be shown as a fraction, a decimal or a percentage, so 1/4, 0.25 and 25% are the same thing. The basic probability toolkit comes down to this one fraction and a few ways of combining it.
Single Events
- One die: There are 6 faces. The chance of rolling a 4 is 1/6, about 16.7%.
- One card from a 52-card deck: The chance of an ace is 4/52, which simplifies to 1/13, about 7.7%. The chance of a heart is 13/52, or 1/4.
- Roulette, single zero: There are 37 pockets. The chance of landing on red is 18/37, about 48.65%. The chance of any single number is 1/37, about 2.70%.
- Roulette, double zero: There are 38 pockets. Red is 18/38, about 47.37%.
Two Dice
With two dice there are 6 × 6 = 36 equally likely combinations. The number of combinations that give each total is not equal: a total of 7 can be made in 6 ways, while a total of 2 can be made in only 1 way (both dice showing 1). So the probability of a 7 is 6/36 = 1/6, and the probability of a 2 is 1/36. Counting combinations, rather than just totals, is the key step, and the full set of two-dice outcomes shows why some totals are far more common than others.
The Complement
The probability that an event does not happen is 1 minus the probability that it does. If the chance of a 7 is 1/6, the chance of anything else is 5/6. This is often easier than counting directly. For example, the chance of getting at least one 6 in two rolls of a die is easier to find as 1 minus the chance of no 6 on both rolls: 1 − (5/6 × 5/6) = 1 − 25/36 = 11/36, about 30.6%.
Combining Events
“And” with independent events: multiply. If one event does not affect the other, the chance of both is the product. Two reds in a row on a single-zero wheel is 18/37 × 18/37 = 324/1369, about 23.7%. Each spin is separate, so the second spin is still 18/37 regardless of the first.
“Or” with events that cannot both happen: add. The chance of rolling a 1 or a 2 on one die is 1/6 + 1/6 = 2/6 = 1/3.
Cards drawn without replacement. Here the first draw changes what is left, so the events are dependent. The chance that the first two cards from a full deck are both aces is 4/52 × 3/51 = 12/2652 = 1/221, about 0.45%. After one ace is gone, only 3 remain among 51 cards. Dependent events behave differently from the independent spins of a wheel, and that difference is the starting point for many card-game calculations. Counting hands rather than drawing one card at a time uses card combinations.
Reading Quoted Odds
Odds and probabilities describe the same chances in different words.
- Odds against compare the ways of not winning with the ways of winning. If the probability is 1/6, the odds against are 5 to 1.
- Odds for reverse this: 1 to 5.
- To go from odds against “a to b” to a probability, use b ÷ (a + b). So 5 to 1 against gives 1 ÷ 6.
- To go from probability to odds against, use (1 − p) ÷ p. So a probability of 0.25 gives 0.75 ÷ 0.25 = 3, which is odds of 3 to 1 against.
Converting odds between fractions, decimals and percentages is mostly a matter of keeping track of whether a number includes the returned stake. A quoted “3 to 1” payout and a “4 for 1” payout are the same thing.
A Common Mistake: Confusing Odds With Probability
“Even odds” means 1 to 1, a probability of one half. A bet on red in roulette is often paid at 1 to 1, but its probability of winning is 18/37, less than half, because of the zero. The payout looks like a 50% chance, but the game’s actual chance is a little lower. That small difference is the house edge for the bet.
What Probability Does and Does Not Tell You
Probability describes how often something happens across many repetitions. It does not say what will happen next. A probability of 1/6 does not mean one 6 appears every six rolls. Streaks of the same result are normal, and they do not make the event “due”.
It also does not tell you whether a bet is worth taking. For that, you need to compare the probability with what the bet pays, which is the work done by the payout odds and by expected value. For a casino game, that comparison shows an average cost. Probability is a tool for seeing the game clearly, not for beating it.