All the calculations in the mathematics section of the site rest on a few simple ideas from probability. This article presents them one at a time, each with an example from games of chance: equally likely outcomes, counting combinations, complementary events, independence and expected value.
Equally likely outcomes
The probability of an event is a number between 0 and 1: 0 for an impossible event, 1 for a certain one. Expressed as a percentage, it is the same value multiplied by 100. When all the possible outcomes of an experiment have the same chance, the probability is found by counting:
probability = number of favourable outcomes / number of all possible outcomes
A fair die has 6 faces, each with a chance of one in 6. A European roulette wheel has 37 pockets, so a given number comes up with probability 1/37, that is, 2.70%.
The condition "equally likely" is essential. When two dice are rolled there are 36 equally likely ordered pairs (6 outcomes for the first die × 6 for the second), but the 11 possible totals, from 2 to 12, are not equally likely. A total of 7 can be made in more ways than any other, while 2 and 12 can each be made in only one way:
| Sum | Rolls out of 36 | Probability |
|---|---|---|
| 2 | 1 | 2.78% |
| 3 | 2 | 5.56% |
| 4 | 3 | 8.33% |
| 5 | 4 | 11.11% |
| 6 | 5 | 13.89% |
| 7 | 6 | 16.67% |
| 8 | 5 | 13.89% |
| 9 | 4 | 11.11% |
| 10 | 3 | 8.33% |
| 11 | 2 | 5.56% |
| 12 | 1 | 2.78% |
A common mistake is to treat the pairs (1, 6) and (6, 1) as a single outcome. They are distinct outcomes: the first die shows 1 and the second 6, or the other way round. This is why a total of 7 has a probability of 16.67%, the highest of all the totals. The game of craps is built entirely on this table.
Counting
When there are a great many outcomes, they are not listed one by one but counted with two rules.
The product rule. If one choice can be made in a ways and, for each of them, a second choice can be made in b ways, the two choices together can be made in a × b ways. The 36 outcomes of two dice are an example.
Combinations. The number of ways of choosing k objects out of n, when order does not matter, is:
C(n, k) = n! / (k! × (n − k)!)
where n! is the product 1 × 2 × … × n. A few examples:
- Poker. A pack of 52 cards gives C(52, 5) = 2,598,960 different five-card hands. Of these, only 4 are royal flushes, one for each suit, so the probability of a royal flush is 0.000154%. All the hands are on the Poker page.
- Lottery. A "6 from 49" format has C(49, 6) = 13,983,816 combinations. A single ticket matches all 6 numbers with a probability of 1 in 13,983,816.
- Texas hold'em. Two cards out of 52 can be dealt in 1,326 ways. Counting the pairs (13 ranks, each in C(4, 2) ways according to the suits of the two cards) and dividing by the total, the chance of being dealt a pair as the first two cards is 5.88%.
Complementary events
The complement of an event A is the event "A does not happen". Together the two cover every possibility, so:
probability that A does not happen = 1 − probability of A
In European roulette, an even-money bet wins with probability 48.65% and loses with probability 51.35%. The gap between the two comes from the zero, on which the bet loses. In poker, a five-card hand with no combination at all has a probability of 50.12%; its complement, a hand that makes at least a pair or a better combination, has the rest.
The complement is especially useful for "at least once" questions. The probability that an event with probability p happens at least once in n independent trials is 1 − (1 − p)ⁿ, that is, one minus the probability that it never happens. Thus a run of ten losses in a row on an even-money bet has a probability of 0.13%, but the chance that it happens at least once in 100 series is 11.98%. The example is discussed in Betting systems.
Independence
Two events are independent when the occurrence of one does not change the probability of the other. For independent events, the probability that both happen is the product of their probabilities:
probability that A and B both happen = probability of A × probability of B
Roulette spins are independent. The probability of three losses in a row on an even-money bet is 13.54%, and the probability of ten reds in a row is 0.0743%: the same probability, multiplied by itself ten times.
Cards drawn from a pack without being put back are not independent. The first card is an ace with probability 4/52; if it was an ace, the second card is an ace with probability 3/51, because 3 aces are left among 51 cards. The probability that the first two cards are both aces is the product (4/52) × (3/51), that is, 0.45% (1 in 221). Here the second probability depends on the result of the first draw. Confusing the two situations lies at the root of the gambler's fallacy.
Expected value
The expected value (the average) of a bet is the sum of its possible results, each multiplied by its probability:
expected value = Σ probability of the result × the result
A bet on a single number in European roulette wins 35 units with probability 1/37 and loses one unit with probability 36/37. The expected value is (1/37) × 35 + (36/37) × (−1), an average loss of 2.70% of the stake. This is the house edge, explained in House edge and RTP.
The expected value is a long-run average, not the most likely result of a bet. A bet on a single number ends either with a win of 35 units or with the loss of one unit; never with the average. How far actual results stray from the average is described by variance, explained in Variance and the law of large numbers.
See also
- The gambler's fallacy
- Betting systems
- House edge, RTP and variance in the glossary