Fundamental concepts of probability
Probability is the mathematical framework for quantifying uncertainty. In the context of betting, probability is the foundation upon which all analysis rests: odds encode probabilities, expected value is computed from probabilities, and every decision a player makes depends on estimating probabilities as accurately as possible. This article covers the fundamental concepts necessary to work with probabilities rigorously.
Definition and notation
The probability of an event is a number between 0 and 1 (or equivalently, between 0% and 100%) that represents the likelihood of that event occurring. A probability of 0 means the event is impossible; a probability of 1 means it is certain. All events in between have some degree of uncertainty.
P(A): the probability of event A occurring.
For any event A:
0 ≤ P(A) ≤ 1
The probability of event A not occurring is the complement:
P(¬A) = 1 − P(A)
For example, if the probability of a team winning is .60, the probability of that team not winning is 1 − .60 = .40. In a two-option market, this is the probability of the other option. In a three-option market (such as a soccer match with local, visit, and draw), the complement of one option is the sum of the other two.
Mutually exclusive events
Two events are mutually exclusive if they cannot both occur simultaneously. In a head to head market, the options are mutually exclusive: exactly one of them will occur. The probabilities of all mutually exclusive and exhaustive options must sum to 1:
P(A₁) + P(A₂) + ... + P(Aₙ) = 1
This is a fundamental constraint. If a player estimates probabilities for all options in a market, those estimates must sum to 1. If they do not, the estimates are internally inconsistent.
The probability of one or another mutually exclusive event occurring is the sum of their individual probabilities:
P(A or B) = P(A) + P(B) (when A and B are mutually exclusive)
Independent events
Two events are independent if the occurrence of one does not affect the probability of the other. For example, the outcome of a soccer match in England and the outcome of a tennis match in Australia are generally independent events: knowing the result of one provides no information about the result of the other.
The probability of two independent events both occurring is the product of their individual probabilities:
P(A and B) = P(A) · P(B) (when A and B are independent)
This multiplication rule is the mathematical basis of parlay calculations. If a player combines two independent selections with probabilities .60 and .50, the probability of both winning is:
P(A and B) = .60 · .50 = .30 = 30%
The extension to n independent events is:
P(A₁ and A₂ and ... and Aₙ) = P(A₁) · P(A₂) · ... · P(Aₙ)
It is critical to verify that events are truly independent before applying this multiplication. Events from the same match, the same competition, or involving the same participants are often not independent.
Conditional probability
The conditional probability of event A given that event B has occurred is denoted P(A|B) and is defined as:
P(A|B) = P(A and B) / P(B)
If A and B are independent, then P(A|B) = P(A) — knowing that B occurred does not change the probability of A. If they are not independent, then P(A|B) ≠ P(A), and the occurrence of B provides information about the likelihood of A.
In betting, conditional probability arises frequently. The probability that a team wins a match given that a key player is injured is different from the unconditional probability of the team winning. The probability of over 2.5 goals given that a goal has been scored in the first 10 minutes is different from the pre-match probability of over 2.5 goals.
For events that are not independent, the correct formula for the probability of both occurring is:
P(A and B) = P(A) · P(B|A) = P(B) · P(A|B)
Law of large numbers
The law of large numbers is a theorem that states that as the number of independent repetitions of an experiment increases, the average of the observed results converges to the expected value. In the context of betting, this means that if a player places a large number of bets with a known expected value, the average profit per bet will approach that expected value as the number of bets grows.
This theorem is what makes expected value a meaningful concept for practical decision-making. A single bet is uncertain, but the average over many bets is predictable. The law does not guarantee that any particular sequence of bets will be profitable — it guarantees that the average will converge.
The rate of convergence depends on the variance. Higher variance requires more repetitions before the average stabilizes near the expected value. This is why bets at high odds (which have high variance) require a much larger sample to demonstrate their expected value than bets at low odds.
Frequentist and subjective probability
There are two principal interpretations of probability relevant to betting.
The frequentist interpretation defines probability as the long-run relative frequency of an event. If a coin is flipped an infinite number of times and lands heads 50% of the time, the probability of heads is .50. This interpretation applies naturally to repeatable experiments.
The subjective interpretation defines probability as a degree of belief, informed by available evidence. When a player estimates that a team has a 65% probability of winning a specific match, he is expressing a subjective probability — the match will occur only once, and there is no long-run frequency to observe. The estimate is based on analysis of available data, models, and judgment.
In sports betting, both interpretations are relevant. Frequentist probability underpins the statistical models and historical analyses used to estimate probabilities. Subjective probability is what the player ultimately assigns to a specific event, combining all available information into a single number. The quality of a player's subjective probability estimates — measured by their calibration against observed outcomes over many events — determines his long-term profitability.