Expected value (EV) is a mathematical way of estimating the average result of a bet if the same situation could be repeated a very large number of times.
It does not predict what will happen on the next bet. Instead, it combines the possible outcomes and their probabilities to calculate the average result over the long run.
Expected value is an important concept because it helps explain how casino games can have a mathematical advantage for the house.
A Simple Example
Imagine a simple game where you have:
- A 50% chance of winning $10
- A 50% chance of losing $6
The expected value can be calculated as:
(50% × $10) + (50% × −$6)
That gives:
$5 − $3 = +$2
So the expected value is +$2 per play.
This does not mean you will win $2 every time. You could win $10 or lose $6. The $2 figure represents the mathematical average if the same bet were repeated many times.
Expected Value in Casino Games
Casino games are generally designed so that the expected value favors the casino.
Suppose a particular wager has a theoretical house edge of 5%.
If you repeatedly wager $100 on that game, the casino’s long-term expected result would be approximately $5 per $100 wagered.
From the player’s perspective, the expected value would be approximately:
−$5 per $100 wagered
Again, this is not a prediction of an individual result.
A player could win significantly more than $100, lose the entire $100 or experience many different outcomes.
How Probability Affects Expected Value
Probability is at the heart of expected value.
The basic formula is:
Expected Value = (Probability of Outcome 1 × Value of Outcome 1) + (Probability of Outcome 2 × Value of Outcome 2) + …
Every possible outcome is given a value and multiplied by its probability.
The results are then added together.
This allows mathematicians to calculate the average theoretical result of a game.
Roulette as an Example
Consider a simple European roulette bet on one number.
There are 37 pockets, so the chance of hitting a particular number is:
1 ÷ 37 ≈ 2.70%
A successful straight-up bet normally pays 35-to-1, while losing bets receive nothing.
The expected value can be represented as:
(1/37 × $35) + (36/37 × −$1)
The result is approximately:
−$0.027
So for every $1 wagered, the theoretical expected loss is about 2.7 cents.
That is another way of expressing the standard house edge for this type of European roulette bet.
Expected Value Does Not Predict Individual Results
This is perhaps the most important point to understand.
If a game has a negative expected value, it does not mean a player must lose the next bet.
For example, if the expected value is −$5 per $100 wagered, an individual player might win $500, lose $100 or finish somewhere in between.
Expected value only describes what the average mathematical result would approach over a very large number of similar wagers.
Expected Value and House Edge
The two concepts are closely connected.
If a game has a 4% house edge, the player’s expected value is generally negative by approximately 4% of the amount wagered, assuming the relevant bet is evaluated on the same basis.
For a $10 wager:
Expected loss = $10 × 4% = $0.40
This means the mathematical expectation is a loss of 40 cents per $10 wager over a large number of repetitions.
It does not mean the player loses exactly 40 cents each time.
Expected Value in Games With Decisions
Expected value can also be used to study games where players make decisions.
Blackjack is a good example. Different decisions can have different mathematical outcomes because the player’s choice changes the possible future results.
Researchers can calculate the expected value of different decisions by considering the probabilities of the possible outcomes.
This is one reason why blackjack mathematics can be more complicated than the mathematics of a simple roulette bet.
Why Expected Value Matters
Expected value provides a simple way to connect probability, payouts and long-term results.
It explains why a casino can offer many winning bets while still maintaining a mathematical advantage overall. Some players will win, some will lose, and results can vary widely in the short term.
Over a large number of repeated outcomes, however, the expected value provides a useful mathematical picture of the game’s underlying structure.
In casino mathematics, EV is therefore best understood as a long-term average calculation, not a prediction of what will happen on the next spin, roll, hand or bet.
