What is implied probability?
Every odd encodes a probability. This probability is called the implied probability, because it is the probability that the odd implies would make the bet produce zero expected profit in the long term. In other words, the implied probability is the break-even probability for that odd: if the actual probability of the event were exactly equal to the implied probability, the player would neither gain nor lose money over an infinite number of bets.
The implied probability of a decimal odd is its reciprocal:
d: decimal odd
i: implied probability
i = 1 / d
For example, a decimal odd of 4 implies a probability of 25%:
i = 1 / 4 = .25 = 25%
And a decimal odd of 1.5 implies a probability of approximately 66.67%:
i = 1 / 1.5 ≈ .6667 ≈ 66.67%
Implied probability and actual probability
The implied probability is not the actual probability of an event. It is the probability that the bookmaker's odd assumes, and it is almost always higher than what the bookmaker truly estimates the actual probability to be. The difference exists because the bookmaker adds a margin to the odds, which shifts the implied probabilities upward.
Consider an event with two options. Suppose the bookmaker estimates the actual probabilities to be 50% for each option. If the bookmaker offered fair odds, the decimal odd for each would be 2 (since 1 / .5 = 2). However, the bookmaker might instead offer odds of 1.91 for each option. The implied probabilities would then be:
i₁ = 1 / 1.91 ≈ .5236 ≈ 52.36%
i₂ = 1 / 1.91 ≈ .5236 ≈ 52.36%
The sum of the implied probabilities is approximately 104.72%, which exceeds 100% by 4.72 percentage points. This excess is the bookmaker's margin. Each implied probability has been inflated above the bookmaker's own estimate of the actual probability, ensuring that the bookmaker profits regardless of the outcome, provided the wagers are distributed proportionally.
For the player, the critical task is to estimate the actual probability of an event as accurately as possible, and then compare it with the implied probability. If the player's estimated actual probability exceeds the implied probability, the bet has positive expected value. If the actual probability is lower, the bet has negative expected value.
Removing the margin
Since the implied probabilities of all options in a market sum to more than 100%, a player might want to estimate what the bookmaker's actual probability estimates are, by removing the margin from the implied probabilities. The simplest method is proportional normalization: dividing each implied probability by the total sum of all implied probabilities.
Consider these odds for three options:
Option 1: 1.80 → i₁ = 1 / 1.80 ≈ .5556 ≈ 55.56%
Option 2: 3.60 → i₂ = 1 / 3.60 ≈ .2778 ≈ 27.78%
Option 3: 5.00 → i₃ = 1 / 5.00 = .2000 = 20.00%
Total: 55.56% + 27.78% + 20.00% = 103.34%
The margin is 3.34 percentage points. To remove it proportionally:
p₁ = 55.56% / 103.34% ≈ 53.76%
p₂ = 27.78% / 103.34% ≈ 26.88%
p₃ = 20.00% / 103.34% ≈ 19.36%
These normalized probabilities sum to 100% and represent an approximation of the bookmaker's actual probability estimates, stripped of the margin. However, this method assumes the margin is distributed equally across all options in proportion to their probability, which may not always be the case. The bookmaker may apply different margins to different options, for example charging more margin on the favorite or on the underdog.
Consensus probability
A single bookmaker's implied probabilities reflect that bookmaker's particular estimation, its margin, and its market position. A more robust estimate of the actual probabilities can be obtained by aggregating the odds from multiple bookmakers.
If many independent bookmakers are offering odds on the same event, their individual estimations can be combined through statistical measures such as the mean, the median, or a weighted mean, to produce a consensus probability. This consensus tends to be more accurate than any single bookmaker's estimation, as it benefits from the diversity of information, models, and expertise across different bookmakers.
For instance, suppose four bookmakers offer the following decimal odds for the same option:
Bookmaker A: 2.10 → i = 47.62%
Bookmaker B: 2.15 → i = 46.51%
Bookmaker C: 2.05 → i = 48.78%
Bookmaker D: 2.12 → i = 47.17%
The mean implied probability is:
(47.62% + 46.51% + 48.78% + 47.17%) / 4 ≈ 47.52%
This mean still contains each bookmaker's margin. After removing the margins from each bookmaker's full market and then averaging, the result is a consensus probability that represents the collective estimation of the actual probability of the event. This consensus probability can then be compared against the odds offered by any individual bookmaker to identify opportunities with positive expected value.