Casino games are games of chance with a built-in house edge. No strategy, system or timing changes the odds in the player’s favor over time, and most players lose money over time. Nobody should gamble money they cannot afford to lose. The examples below use hypothetical numbers to explain a calculation; they are not a method for winning.
What expected value means
Expected value (EV) is the average result per bet you would approach if the same bet were repeated a very large number of times. It is calculated by multiplying each possible result by its probability and adding the products.
The word “expected” is misleading. The expected value of a single bet is often a result that can never actually happen on that bet. It is a long-run average, not a forecast.
The method
- List every outcome and the amount you would gain or lose on it.
- Find the probability of each outcome. They must add up to 1.
- Multiply each amount by its probability, then add everything.
Example 1: a fair coin game (hypothetical)
Suppose a game pays you $1 if a fair coin lands heads and takes $1 if it lands tails.
- Heads: +$1 × 0.5 = +0.50
- Tails: −$1 × 0.5 = −0.50
- EV = 0.50 − 0.50 = $0
A game with an EV of zero is called fair. Casino games are not built this way.
Example 2: a die with an unfair payout (hypothetical)
Suppose you stake $1 and win $4 profit if a die shows a six; otherwise you lose the $1.
- Win: +$4 × 1/6 = +0.667
- Lose: −$1 × 5/6 = −0.833
- EV = 0.667 − 0.833 = −$0.167, or about −16.7 cents per $1 bet
Compare this with a $5 profit on a six. That gives +5/6 − 5/6 = $0, a fair game. The pay of 5 to 1 matches the odds against of 5 to 1. When the payout falls below the odds against, the average is negative. This comparison is the link between payout odds and cost.
Example 3: a straight-up roulette bet
These figures follow from standard rules: a single-zero wheel has 37 pockets, and a straight-up bet on one number pays 35 to 1.
- Win: +$35 × 1/37 = +35/37
- Lose: −$1 × 36/37 = −36/37
- EV = (35 − 36)/37 = −1/37, about −$0.027 per $1 bet, or −2.70%
On a double-zero wheel with 38 pockets and the same 35 to 1 payout:
- EV = 35/38 − 37/38 = −2/38, about −$0.053 per $1 bet, or −5.26%
Same payout, extra pocket, roughly double the average cost. The effect of the extra zero is covered in more detail in the explanation of the double zero.
From expected value to house edge and RTP
For a single-unit bet, the player’s EV as a percentage of the amount bet is the negative of the house edge. For the single-zero example above, the house edge is about 2.70%. The related return to player figure is the flip side: it equals 100% minus the edge, here about 97.30%. These two ways of expressing the same number are compared in the article on house edge versus RTP. Be careful with the conclusion, though. A 97.30% return does not mean you will get back 97.30% in any session.
Scaling up
Expected value adds across bets. If the average cost is $0.027 per $1 bet, then 100 bets of $1 have an EV of about −$2.70, and 1,000 bets of $1 have an EV of about −$27. In dollars, the expected loss gets larger the more you wager. This is why the cost per hour of a game depends both on its edge and on how many bets are made, a point developed in the guides to speed of play and to the expected cost of play.
What expected value does not tell you
- It does not predict a session. After 100 bets of $1 on straight-up numbers, one outcome is a profit, and another is a much bigger loss than $2.70. The spread of results around the average is measured by standard deviation, not by EV.
- It does not mean you are “due.” If you have lost more than the average, the shortfall is not paid back by later luck. Each bet starts fresh. Thinking otherwise is the gamblers fallacy.
- It does not account for enjoyment or money you value differently. EV is only a financial average.
- It is not changed by betting patterns. Since every bet has a negative EV, combining many negative bets does not produce a positive one. Fixed combinations such as bet progressions cannot remove the edge, they only change how the losses are distributed.
A short checklist
When you see a payout described, you can estimate the average result per bet by asking:
- What are all the outcomes and their probabilities?
- What does each pay or cost?
- Do the weighted amounts add to less than zero?
If the answer to the third question is yes, the game costs money on average. For casino games, it nearly always does. Reaching this conclusion early makes it easier to treat gambling as paid entertainment with a price, to set a gambling budget of money you can truly spare, and to seek support if the spending stops feeling like a choice.