Expected Value Formula for Gambling: Worked Examples

Expected value (EV) is the average amount a bet wins or loses per wager if you could repeat it many times. You find it by multiplying each possible result by its probability and adding the products. For a $10 bet on a single number at a European roulette wheel, the EV is −$0.27: on average you lose about 27 cents per spin, even though any one spin either wins $350 or loses $10.

That gap between a single result and the long-run average is the heart of gambling math. Expected value never predicts the next bet. It measures what a bet is worth on average, and for almost every standard casino bet that average is negative for the player.

This guide gives the EV formula with each symbol defined, three worked examples with every step shown, and a table comparing the EV of several common bets. It also explains how EV turns into expected loss over time and lists the misreadings that cause the most confusion. All examples are hypothetical, and each number is calculated on the page.

What Expected Value Means

Expected value is a weighted average. Each outcome of a bet has a net result (what you gain or lose) and a probability. Weighting each result by its probability and adding them gives the average net result per bet. The probabilities come from counting outcomes, as covered in the guide to probability in gambling.

EV can be expressed in dollars or as a percentage of the amount wagered. A bet with an EV of −$0.27 on a $10 stake has an EV of −2.70% of the stake. The percentage form is useful because it lets you compare bets of different sizes.

A bet can have positive, zero or negative EV. Negative EV means the bet loses money on average, zero means it breaks even on average, and positive means it gains on average. Almost every standard casino bet is built so the player’s EV is negative; the rare exceptions, such as craps free odds, sit at exactly zero. The difference between the fair payout and the actual payout is how the casino earns its edge.

Expected value is also not a result you will ever see on one bet. A coin that pays $1 for heads and costs $1 for tails has an EV of $0, yet a single flip always ends at +$1 or −$1. EV describes the average of many repetitions, not what any single bet will do.

The Expected Value Formula

The general formula adds up each possible net result multiplied by its probability.

where:

EV is the expected value per bet, in dollars.

Σ means add up over every possible outcome.

probability of each outcome is a number from 0 to 1, and all probabilities add to 1.

net result is the profit (positive) or loss (negative) for that outcome.

For a bet that either wins or loses, the formula simplifies. A bet that pays X to 1 and wins with probability p has this EV per $1 staked.

where:

p is the probability the bet wins.

X is the payout odds, so a bet paying 35 to 1 has X = 35.

(1 − p) is the probability the bet loses, which costs the $1 stake.

From the EV you can read off the house edge, the casino’s average take per dollar wagered, and the expected loss across many bets.

where:

amount wagered is the stake on one bet, or the total across all bets.

expected loss is an average, not a prediction of any session.

When the EV is zero, the break-even probability for a bet paying X to 1 is 1 ÷ (X + 1). A bet that pays 35 to 1 breaks even at 1 ÷ 36, about 2.78%, which is more than the 1 ÷ 37 chance a European wheel provides.

Worked Examples, Step by Step

Example 1: a single number, European wheel. Suppose you bet $10 on one number on a single-zero wheel. The bet pays 35 to 1.

Step 1: p(win) = 1 ÷ 37, winning $350. p(lose) = 36 ÷ 37, losing $10.

Step 2: EV = (1 ÷ 37) × 350 + (36 ÷ 37) × (−10) = (350 − 360) ÷ 37.

Step 3: EV = −10 ÷ 37 ≈ −$0.27, which is −2.70% of the $10 stake.

Example 2: red, American wheel. Suppose you bet $10 on red on a double-zero wheel. The bet pays 1 to 1 and wins with p = 18 ÷ 38.

Step 1: EV = (18 ÷ 38) × 10 + (20 ÷ 38) × (−10) = (180 − 200) ÷ 38.

Step 2: EV = −20 ÷ 38 ≈ −$0.53, or −5.26% of the stake.

Example 3: a −110 bet. A hypothetical −110 line means risking $11 to win $10. If the true win probability were 50%, EV = 0.50 × 10 − 0.50 × 11 = −$0.50, which is −4.55% of the $11 stake. The break-even win rate is 11 ÷ 21 ≈ 52.38%. If someone estimated the true probability at 55%, EV = 0.55 × 10 − 0.45 × 11 = +$0.55, or +5.00% of the stake. That estimate is the weak point, because an estimated probability is uncertain and small errors can flip the sign.

Expected Value Across Common Bets

The same formula gives the EV per dollar for the standard casino bets below. Rules and payouts vary, so each assumption is stated: European single-zero wheel (37 pockets), American double-zero wheel (38 pockets), a craps Field bet that pays 2 to 1 on 2 and 12, Any 7 paying 4 to 1, and baccarat Tie paying 8 to 1 with 8 decks.

Here is how the Field bet works out. Of the 36 dice combinations, 14 win 1 to 1 (totals 3, 4, 9, 10 and 11), 2 win 2 to 1 (totals 2 and 12), and 20 lose (totals 5, 6, 7 and 8). EV per $1 = (14 × 1 + 2 × 2 − 20) ÷ 36 = −2 ÷ 36 ≈ −$0.0556.

EV turns into expected loss over time. Suppose someone makes 60 bets an hour at $10 each on red at an American wheel. Wagered per hour is $600, so the expected loss is 600 × 5.26% ≈ $31.58 per hour. This is an average. Any particular hour can be far better or far worse.

Common Mistakes and Misconceptions

The first mistake is reading EV as a prediction. An EV of −$0.27 does not mean you lose 27 cents on each spin. It means that across a huge number of identical bets, the average per bet approaches −$0.27.

The second is confusing a high win probability with positive EV. Red wins 47.37% of the time at an American wheel, but its EV is still negative because the payout is too small for that probability. Probability alone never tells you whether a bet is worth it.

A third is believing a system changes EV. Raising or lowering stakes after wins or losses changes how results are distributed, but the EV of each bet is fixed by its probability and payout, so the expected loss stays proportional to the amount wagered.

Finally, positive EV is often assumed rather than shown. In sports betting the true probability is an estimate, so a calculated edge is only as good as the estimate behind it.

Quick Reference and How to Check Your Answer

To check an EV, first confirm that the probabilities add to 1: 1 ÷ 37 + 36 ÷ 37 = 1. Second, confirm the sign makes sense: a fair bet comes to exactly zero, so a bet with a payout below fair odds must be negative. Third, check the percentage against the table. A red bet at an American wheel should land at 5.26%, and a single number at a European wheel at 2.70%.

Related Concepts and Next Steps

Once EV is clear, the next question is how a price in sports betting maps to a probability. The guide on converting American odds to decimal shows how to get the break-even probability that the EV calculation above needs.

From here, the house edge is the same idea viewed from the casino’s side, variance explains why real results scatter widely around the average, and betting systems can be tested by asking whether they change the EV. They do not, and later articles show the math.