The mathematics of parlays
A parlay, also called an accumulator, is a single bet that combines multiple individual selections. The parlay wins only if every individual selection wins. If any one selection loses, the entire parlay loses. The mathematical properties of parlays have important consequences for the expected value, the margin, and the risk involved.
Combined odds
The combined odd of a parlay is the product of the decimal odds of all individual selections:
d₁, d₂, ..., dₙ: decimal odds of each selection
Combined odd = d₁ · d₂ · ... · dₙ
For example, a parlay of two selections at odds 1.50 and 2.40:
Combined odd = 1.50 · 2.40 = 3.60
A ¤100 wager on this parlay would return ¤360 if both selections win: the original ¤100 plus a prize of ¤260.
Combined probability
If the selections are independent events — meaning the outcome of one does not affect the outcome of the other — the actual probability of the parlay winning is the product of the actual probabilities of each individual selection:
w₁, w₂, ..., wₙ: actual probabilities of each selection
Combined probability = w₁ · w₂ · ... · wₙ
If a player estimates the actual probabilities of the two selections above to be 70% and 45%, then:
Combined probability = .70 · .45 = .315 = 31.5%
The implied probability of the combined odd 3.60 is:
Combined implied probability = 1 / 3.60 ≈ .2778 ≈ 27.78%
Since 31.5% > 27.78%, this parlay has positive expected value. The expected value can be calculated as:
e = .315 · (3.60 − 1) − .685 = .315 · 2.60 − .685 = .819 − .685 = .134 = 13.4%
Margin compounding
A critical property of parlays is that the bookmaker's margin compounds with each additional selection. This means the effective margin of a parlay is always larger than the margin of any individual selection.
Consider two independent markets, each with two options and a margin of 5%. For each market, the sum of the implied probabilities is 105%. Now consider a parlay that combines one selection from each market.
The fair probability of both selections winning might be, for example:
w₁ = .50, w₂ = .50
Fair combined probability = .50 · .50 = .25 = 25%
Fair combined odd = 1 / .25 = 4.00
But the bookmaker's implied probabilities for each selection are inflated by the margin. Suppose the implied probabilities are 52.5% each (reflecting the 5% margin distributed proportionally):
Combined implied probability = .525 · .525 = .275625 ≈ 27.56%
Combined implied odd = 1 / .275625 ≈ 3.628
The effective margin on the parlay is:
27.56% − 25% = 2.56 percentage points on the combined probability, which corresponds to a combined implied probability sum that would be approximately 110.25% (since 1.05 · 1.05 = 1.1025) if expressed as a total overround for the parlay.
In general, if each market has a margin factor of (1 + m), the combined margin factor for an n-selection parlay is:
(1 + m)ⁿ
For a 5% margin and 4 selections: (1.05)⁴ ≈ 1.2155, meaning the effective overround is approximately 21.55%. This compounding effect is the primary reason why parlays are, on average, less favorable for the player than individual bets. The more selections in a parlay, the more margin the bookmaker collects.
Expected value of a parlay
The expected value of a parlay depends entirely on the expected values of its individual components. A parlay of selections that all have positive expected value will itself have positive expected value. Conversely, a parlay that includes any selection with negative expected value will tend to have a more negative expected value than the individual bets would separately, because the negative expected value is amplified by the multiplication.
However, even a parlay composed entirely of positive expected value selections is not necessarily a good bet. The Kelly criterion for a parlay will typically indicate a very small fraction of the bankroll, because the high combined odd means the prize is large relative to the probability of winning. The variance is substantially higher than placing the same selections as individual bets.
For this reason, from a pure bankroll growth perspective, placing individual bets is almost always superior to placing parlays of the same selections. The parlay concentrates all the risk into a single outcome — all must win or all is lost — while individual bets allow the player to profit from any subset of correct predictions.
Correlated parlays
The multiplication of probabilities is only valid when the selections are independent events. If the outcomes are correlated — meaning the occurrence of one affects the probability of the other — the combined probability must account for this correlation.
For example, betting on a team to win and the match to have over 2.5 goals may be positively correlated: if the team wins, it is more likely that multiple goals were scored. In this case, the combined probability of both events occurring is higher than the product of their individual probabilities would suggest.
Conversely, some combinations may be negatively correlated. The bookmaker may not always adjust the odds to fully account for these correlations, which can occasionally create opportunities for informed players who understand the relationships between the selections.
Most bookmakers restrict or prohibit parlays of selections from the same event precisely because of these correlations, as mispriced correlated parlays can produce systematic losses for the bookmaker.