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How to calculate expected value in betting

Whether you're betting with a bookmaker or on an exchange, calculating the expected value of a trade is fundamental. This article explains how to calculate and measure expected value, and shows how it can be used to find value bets.

Expected value is a predicted value of a variable, calculated as the sum of all possible values each multiplied by the probability of its occurrence.

In betting, the expected value (EV) is the measure of what a bettor can expect to win or lose per bet placed on the same odds time and time again. Positive expected value (+EV) implies profit over time, while a negative value (-EV) implies a loss over time. All bettors should be aiming to identify betting value with every bet they make.

Expected value calculation

The formula to calculate expected value for betting is fairly simple:

(Amount won per bet * probability of winning) – (Amount lost per bet * probability of losing)

Let’s use a coin toss as an example of calculating expected value. Assuming the coin and the toss are fair, each outcome (heads or tails) has an equal probability of 50% - therefore the odds offered on a fair market would be 2.0.

This would result in an EV of 0 for either a Head or Tail - because the probability of the two outcomes is the same, so if you tossed a coin infinitely it would theoretically end up all square.

If however you were offered odds of 2.15 for the coin to land on heads, this is a value bet.

If you placed £10 on the coin landing on heads at 2.15, the EV is calculated likewise:

(11.50 X 0.5) – (10 X 0.5) = 0.75

This shows an EV of 0.75. Therefore you would expect to make an average profit of 75p for each £10 bet, because the odds received are better than the implied odds of the coin toss.

How expected value helps sports bettors

If you constantly find value bets, you will be a profitable bettor in the long run. However, bookmakers build a margin into their odds, while betting exchanges charge commission.

If a bookmaker was to offer odds on a coin toss the price would be priced around 1.90 for both heads and tails. By running the calculation the expected value reduces accordingly:

(9 * 0.50) - (10 * 0.50) = - 0.5

Over time this means you would lose on average 50p for every £10 staked and is, therefore, a bad value bet.

The same basic calculation can be applied to sporting markets on a bookmaker or exchange.

Let’s use the Smarkets 1X2 odds for the Premier League game between Arsenal and Manchester United.

Note: These odds have Smarkets’s industry-low 2% built into the price. Learn how to convert betting odds.

Outcome Odds Implied probability
Arsenal 2.76 36.23%
Draw 3.40 29.41%
Manchester United 2.78 35.97%

To work out the expected value of a £50 bet on Manchester United to win is calculated as:

(89 * 0.36) - (50 * 0.66) = - 0.6

The EV is negative for this bet, suggesting that over time you would lose on average 60p for every £50 staked.

How to identify value bets

However, a negative EV doesn’t mean you’re going to lose money. Unlike a coin toss, betting odds are subjective, and therefore if you accurately predict an outcome compared to the bookmaker or another user on the exchange, you’re likely to make a profit.

If you calculate your own probability for an outcome that differs from the implied probability of the odds, you can identify a positive EV.

Using the same game between Arsenal and Manchester United as our example. The odds give United a 35.97% chance of winning.

Let’s say you calculate (using, for instance, a Poisson Distribution model) United have a 40% chance of winning.

(89 * 0.40) - (50 * 0.60) = 5.6

By running the calculation again, the EV for a United win becomes + £5.60 for every £50 staked.

Traders should aim to build their own handicap models, generating their own implied probabilities of a betting market. When the odds in the model differentiate widely enough compared with a bookmaker or other user on an exchange, this is perceived EV.

The best place to find value is to specialise on niche markets, where the playing field is more level between bookmakers and bettors on an exchange. Once you understand the market well, you will be able to identify odds that are skewed from your implied probability, giving you a positive EV over time.

Through experience of trading these markets, you will gain an accurate sense of how the match may play out, but more importantly, a sense of how the market will react.

Arbitrage a positive expected value

Another way to find a positive expected value is with an arbitrage strategy, which exploits odds from separate bookmakers and exchanges to form a positive EV.

Apply this to betting

Now you understand how to calculate expected value on a market, you have the grounding to become a successful trader.

Remember, identifying value does not guarantee a profit, it is theoretical. A fair coin toss can land on heads 20 plus times in a row, and not be biased. What it does mean is, if calculated correctly, the odds would be in your favour.

The next step is to consistently make value bets. You may not win every one, but you will give yourself a better chance of being profitable over time.

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