Independence and correlation explained for betting

In one sentence

Two outcomes are independent when knowing one tells you nothing about the other, and correlated when they tend to happen together or apart.

How it works

If two events are independent, the chance of both happening is simply one chance multiplied by the other. That is how accumulators are priced: leg one times leg two times leg three.

Many betting outcomes are not independent. A favourite winning and over 2.5 goals tend to go together, because a dominant team often scores several. A striker scoring and his team winning are linked too. Multiply those as if they were separate and you get the wrong joint probability.

Correlation also matters for your portfolio. Five bets on one match, or several scorelines in one Correct Score market, are really one big position. Treating them as separate bets understates your risk.

The maths

P(A \text{ and } B) = P(A) \times P(B) \quad \text{only if independent}

\phi = \frac{P(A \text{ and } B) - P(A)P(B)}{\sqrt{P(A)\,(1-P(A))\,P(B)\,(1-P(B))}}

P(A), P(B): the chance of each outcome on its own.
P(A and B): the chance both happen.
φ (phi): the correlation between two yes-or-no outcomes, from −1 (never together) through 0 (independent) to +1 (always together).

In plain English: if the real chance of both is higher than the product of the two chances, the outcomes are positively linked.

Worked betting example

Football, same match. Suppose fair probabilities are:

Home win: 50%.
Over 2.5 goals: 55%.
From your model, both together: 32% (an illustrative model output).

If independent, the joint chance would be 0.50 × 0.55 = 0.275, so 27.5%.
Odds priced as independent: 1 ÷ 0.275 = 3.64.
Fair odds using your joint estimate: 1 ÷ 0.32 = 3.13.
EV per £1 at 3.64 if your 32% is right: 0.32 × 3.636 − 1 = 0.164, so 16.4%.
Correlation: φ = (0.32 − 0.275) ÷ √(0.5 × 0.5 × 0.55 × 0.45) = 0.045 ÷ 0.249 = 0.18.

A modest correlation of 0.18 moves the fair price from 3.64 to 3.13. In practice most bookmakers now adjust bet builder prices for this, and usually add a larger margin as well, so a free 16% is rare.

Where it's good

Pricing same-game multiples and bet builders.
Spotting when a bookmaker multiplies prices that should not be multiplied.
Portfolio risk: measuring how much your open positions really overlap.
Correct score and goal-line modelling, where home and away goals are linked.
Staking several bets at once without over-betting a single underlying outcome.

Limitations and pitfalls

Estimating joint probabilities needs lots of data. You need enough matches where both events could be observed.
Correlation is not causation. Two outcomes can move together because of a shared cause, such as weather or team news.
Correlations change with context. Favourite-and-overs is stronger in some leagues and match-ups than others.
The phi measure only captures simple linear linkage between yes-or-no events; more complex links need a joint model.
Bookmakers know about the obvious ones and price or restrict them.
Hidden correlation in your own betting (same league, same weekend, same model error) is easy to miss and inflates risk.

How to build it

pandas for counting joint outcomes; numpy.corrcoef for correlations; penaltyblog or a bivariate Poisson model for football score dependence.
Data: outcome-level records for each market leg in the same matches.
Practical tip: rather than estimating correlation directly, simulate full scorelines from a goals model and read every joint probability off the same simulation.

Conditional probability: dependence means P(A given B) differs from P(A).
Combinatorics: counting multiples, which assumes independence.
Simultaneous Kelly: staking several correlated bets together.
Bivariate Poisson: modelling linked home and away goals.
Monte Carlo simulation: getting joint probabilities by simulation.