# Probability theory in sports betting: how to evaluate the outcome and calculating the probability of bets

The theory of probability in betting helps professionals outplay the bookmakers. A correct assessment of the outcome increases the chances of a long-term profit. In this article, we will explore the theory of probability in sports betting and also discuss how to calculate the bookmaker's margin and determine the value of a bet.*This article was published thanks to the "Excelbetting.com" Betting Spreadsheet - Analysis, study sports betting, take tests, earn experience and beat the leaderboard. Outrun the editor-in-chief!*

## Probability theory in sports betting

Probability theory is a branch of mathematics that studies the patterns of random events and quantities, as well as their properties and operations on them. It allows you to assess the likelihood of an event in comparison with another.

There are three approaches to determining the probability:

- A priori. Also called Bayesian method.
- Empirical.
- Subjective.

## Bayesian method

With the a priori approach, the probability of each event is determined in advance. For example, when flipping a perfectly symmetrical coin, the probability of getting heads or tails is 50%.

According to the Bayesian method, each flip will alternate between heads and tails. In practice, this is unlikely. Heads or tails can come up several times in a row, but with multiple tossing of a coin, the ratio evens out. This uneven distribution is called variance.

## An empirical approach

According to the empirical method, the probability of an event is determined by the formula:

P = N / X, where N is the number of suitable outcomes, X is the number of all possible outcomes.

For example, PFC CSKA in the 21st century won 13 out of 24 home games against Spartak. The probability of a victory for the "army" in the next match with the red and white in their field is 54.16%.

## Subjective probability

Determined based on the experience of the observer, public opinion or analysis of a specific situation. When assessing the likelihood, take into account as many factors as possible.

Read about how to beat the bookmaker using mathematics here: The mathematics of betting on football and other sports: how to beat the bookmaker using algorithms

## How to calculate bookmaker's margin and probability of outcome

The odds per outcome reflects the probability of the event, according to the bookmakers, taking into account the margin.

Margin is a bookmaker's commission, which allows you to go in positive territory at a distance.

The probability can be calculated using the formula:

P = 1 / K, where K is the bookmaker's coefficient.

Find the bookmaker's margin using the formula:

M = (S - 1) x 100%, where S is the sum of the probabilities.

For example, in a match between Manchester City and Real Madrid, the bookmakers give odds of 1.60 for the home team, 4.55 for the draw and 5.30 for the away team. We calculate the probability:

● Manchester City win 0.625: 1 / 1.60.

● Draw 0.2198: 1 / 4.55.

● Real Madrid win 0.1887: 1 / 5.30.

In total, we get 1.0335: 0.635 + 0.2198 + 0.1887.

The margin is 3.35%: (1.0335 - 1) x 100.

## How to determine the value of a bet

The odds show the opinion of the bookmakers, so some outcomes are underestimated. Such bets are called value bets.

The bet is valuable if:

K x P> 1, where K is the coefficient, P is your estimate of the probability.

For example, in the PSG - Borussia Dortmund match, the bookmakers give odds of 1.69 for the home win, 4.40 for the draw and 4.82 for the away win.

You analyzed the match and decided that there was a 25% chance of a draw.

We estimate the rate:

4.40 x 0.25 = 1.10> 1, the rate is valuable.

Determine the mathematical expectation of profit at a distance using the formula:

N x V x (K x P - 1), where N is the number of bets, P is your estimate of the probability, K is the coefficient, V is the amount of the bet.

If you estimate the odds correctly and make 20 similar bets for the amount of 500 rubles, then the profit will be 1000 rubles: 20 x 500 x (4.40 x 0.25 - 1).

If you learn to correctly assess the likelihood of all possible outcomes, value bets can be profitable over the long run. There are no methods for objectively assessing a particular outcome.

Read more about value betting here: Valuable bets scare the bookmakers. What is value, how to find it and calculation examples

## What is variance

Variance is the spread of a random variable relative to its mathematical expectation.

Determine the probabilities of a series of defeats using the formula:

D = (1 - 1 / coefficient) to the power of S, where S is the number of losses in a row.

For example, in the hockey match "Ak Bars" - "Siberia" analysts give a coefficient of 1.25 for the home win. The probability of winning for Kazan is 0.80: 1 / 1.25. In theory, Ak Bars should win eight out of ten matches. In reality, Kazan may lose in three meetings in a row, but at a distance the results will approach the mathematical expectation.

The probability of three defeats in a row is 0.008: (1 - 1 / 1.25) ^ 3.

## Monte Carlo method

Monte Carlo method - modeling all possible outcomes for given parameters. The more factors you consider, the more accurately you get the probability for each choice.

When placing bets on sports at bookmakers, consider not only the odds in the line, but also your own estimate of the probability. To be profitable in the distance you need value bets.

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