What is Expected Value?

Understanding Expected Value (EV) is foundational for anyone serious about sports betting. It is the measure used to determine whether a bet is likely to make you money in the long run or lose money.

Expected value, when related to betting, is the average amount you can expect to win (or lose) for every wager over the long run. A bet is said to have Positive Expected Value (+EV) if it is a profitable wager over time. If over a large sample size, a wager loses money in the long run, it is said to have Negative Expected Value (-EV).

How to Determine Expected Value

The basic formula for determining the expected value of a single bet:

Looking at the formula above, we can see that accurately calculating the EV of a single wager means we need to know the following variables

1. Probability of winning
2. Profit when you win
3. Probability of losing
4. Loss when you lose

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We can simplify these variables, helping us calculate the EV of a wager

1. Probability of winning
2. Profit when you win = the odds the sportsbook/casino has listed, such as 1.95 to 1
3. Probability of losing = 1 - probability of winning
4. Loss when you lose = bet amount, often called stake

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Now we can see that when placing a wager, the only thing we need to calculate the EV of that wager accurately is the probability of winning. In many scenarios, this can be difficult to determine, but in some scenarios, we know this information. Let's look at a few real-world examples.

Example 1: Coin Toss

Betting on a coin toss is a scenario where we know the true probability of winning is 50%. Let's use this information to determine the EV of that bet.

1. Probability of winning = 50%
2. Profit when you win = ?
3. Probability of losing = ?
4. Loss when you lose = ?

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Now we can determine the probability of losing.

1 - Probability of Winning = Probability of Losing. In our coin toss scenario that is 1 - 0.5 = 0.5.

1. Probability of winning = 50%
2. Profit when you win = ?
3. Probability of losing = 50%
4. Loss when you lose = ?

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Now we know two of the four required variables, the probability of winning (50%) and the probability of losing (50%). The last two variables are tied together; profit when you win and loss when you lose. You determine the loss when you lose by choosing how much to bet. We will use $100 in this example. The profit when you win will be set by whoever is taking the bet, such as a casino or sportsbook. In this example, let's use 1.95 to 1. For every $100 wagered, the profit is $95 on a winning bet.

1. Probability of winning = 50%
2. Profit when you win = $95
3. Probability of losing = 50%
4. Loss when you lose = $100

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Now we have everything needed to determine the expected value of this bet.

The expected value of betting $100 on a coin toss to win $95 is -$2.50. An expected value of -$2.50 does not mean that placing this bet guarantees you will lose $2.50 every time. You could place one $100 bet, win $95, and then walk away $95 richer. However, if you were to place this bet a large number of times, it will average out to an expected value of -$2.50 for each time you place the bet. More on sample size/number of bets later. For now, know the larger the sample size, i.e., the more bets you place, the closer your actual result will be to the expected outcome.

Casinos and sportsbooks make money when bettors consistently place -EV bets. They try to set the payouts (odds) such that each wager is -EV. This does not mean bettors lose every wager or that every time a bettor goes to a casino, they lose money. It simply means that in the long run, over a massive sample size bettors are expected to lose money and casinos are expected to make money.

Example 2: Coin Toss With a Weighted Coin

Now, let's look at a scenario where the coin is weighted so the probability is not 50/50 like a regular coin, but instead, there is a 60% chance it lands on heads. However, the casino does not realize this leaves the payout the same. Below are our new inputs to determine the expected value of this wager.

1. Probability of winning = 60%
2. Profit when you win = $95
3. Probability of losing = 40%
4. Loss when you lose = $100

Because we knew this was a weighted coin and the true probability of it landing on heads was 60% and not 50% we can gain an "edge" over the house. Just like in the first example, placing this bet one time does not guarantee that we will win. We need a large sample size, think thousands of bets, to realize the $17 of expected value.

Why is EV Important in Betting?

Successful sports betting relies on consistently making +EV bets. Even if individual bets might lose, repeatedly placing +EV bets theoretically, over many wagers, leads to profit.

Now that we know how to determine the expected value of a bet, we need a way to identify which sports bets have positive expected value.