True Odds vs Payout Odds in Casino Games Explained

True odds describe how likely a bet really is to win; payout odds describe what the bet pays when it does. When the two match, the bet is fair. When the payout is lower than the true odds, the gap is the house edge. A straight-up bet on a European roulette wheel wins 1 time in 37 (true odds 36 to 1) but pays only 35 to 1, and that one-unit gap is worth 2.70% to the casino on every bet.

This guide explains the difference, gives the formula that turns the gap into a house edge, and works three examples with every step shown. A table then compares roulette and craps bets side by side. All examples are hypothetical, and each states its assumptions, since rules and paytables vary between casinos.

True Odds vs Payout Odds Explained

True odds (also called odds against) compare the chances of losing and winning. If a bet wins with probability p, the true odds are (1 − p) to p. A bet that wins 1 time in 37 loses 36 times for every 1 win, so its true odds are 36 to 1. True odds come from counting outcomes, not from the casino.

Payout odds are the casino’s price: the profit paid per unit staked. A payout of 35 to 1 means a winning $1 bet returns $35 in profit plus the $1 stake. Payout odds are set by the game’s rules, and for standard casino bets they sit at or below the true odds, never above.

A wording note prevents many errors. “35 to 1” pays $35 profit per $1 staked, while “35 for 1” would return $35 in total, which is only $34 of profit. This guide always uses “to 1,” meaning profit.

The house edge lives in the gap. If a casino paid true odds on every bet, the expected result would be zero and the game would be fair. Because standard bets pay less than true odds, every one of them has a negative expected value for the player. The sections below show exactly how large each gap is, building on the ideas in probability in gambling.

The Formula: Turning the Gap Into a House Edge

Start with a bet that wins with probability p and pays Y to 1. The expected value of a $1 bet is the chance of winning times the profit, minus the chance of losing times the stake, which is the same expected value formula covered in the expected value formula guide.

where:

p = probability that the bet wins

Y = payout odds, profit per $1 staked (35 for 35 to 1)

EV = expected value, the long-run average result per $1 staked

Now bring in the true odds X = (1 − p) ÷ p. Rearranging gives p × (X + 1) = 1, so the edge can be written as a direct comparison of the two prices.

where:

X = true odds against the bet, (1 − p) ÷ p

Y = payout odds actually paid

p = probability that the bet wins

In words, the house edge equals the win probability times the shortfall in the payout. The same relationship gives the break-even probability for a payout of Y to 1: 1 ÷ (Y + 1). A bet paying 35 to 1 breaks even at 1 ÷ 36 ≈ 2.78%, but a European straight-up bet wins only 1 ÷ 37 ≈ 2.70% of the time, which is why it loses on average. These formulas assume fixed probabilities and a single bet resolved in one step; bets that depend on a longer sequence, like craps point numbers, need the probability worked out first.

Worked Examples: Roulette and Craps

Example 1: a European straight-up bet. Suppose you bet $10 on one number on a single-zero wheel (37 pockets). The bet pays 35 to 1.

Step 1: p = 1 ÷ 37 ≈ 0.02703. True odds X = 36 ÷ 1 = 36 to 1.

Step 2: Winning profit = 35 × $10 = $350. Losing = −$10.

Step 3: EV = (1/37) × $350 − (36/37) × $10 ≈ $9.46 − $9.73 = −$0.27.

Step 4: House edge = $0.27 ÷ $10 ≈ 2.70%. Check: p × (X − Y) = (1/37) × (36 − 35) = 1/37 ≈ 2.70%.

Example 2: Any 7 in craps. Suppose you bet $5 that the next roll of two dice is a 7. Six of the 36 combinations make a 7, and the bet pays 4 to 1.

Step 1: p = 6 ÷ 36 = 1/6. True odds X = 5 to 1.

Step 2: Winning profit = 4 × $5 = $20. Losing = −$5.

Step 3: EV = (1/6) × $20 − (5/6) × $5 ≈ $3.33 − $4.17 = −$0.83.

Step 4: House edge = $0.83 ÷ $5 ≈ 16.67%. Check: (1/6) × (5 − 4) = 1/6 ≈ 16.67%.

Example 3: a Place 6 bet. Suppose you bet $12 on 6 before 7. Of the 11 combinations that end the bet (5 ways to roll a 6, 6 ways to roll a 7), 5 win, and the bet pays 7 to 6.

Step 1: p = 5 ÷ 11 ≈ 0.4545. True odds X = 6 to 5 = 1.2 to 1.

Step 2: Winning profit = $12 × 7/6 = $14. Losing = −$12.

Step 3: EV = (5/11) × $14 − (6/11) × $12 ≈ $6.36 − $6.55 = −$0.18.

Step 4: House edge = $0.18 ÷ $12 ≈ 1.52%. Check: (5/11) × (1.2 − 1.1667) ≈ 1.52%.

Each result is an average per bet, not a prediction. A single bet either wins or loses; the edge describes the long-run cost per dollar wagered.

Comparing True Odds and Payouts Across Bets

The same formula applies to every bet with a fixed payout. The table lists the win probability, true odds, payout and house edge for several bets. Assumptions: a European wheel has one zero and an American wheel has two (0 and 00); craps bets assume the standard payouts shown, which vary by casino.

Two patterns stand out. First, the same payout can carry very different edges: a 35-to-1 straight-up bet costs 2.70% on a single-zero wheel but 5.26% on a double-zero wheel, because the extra zero lowers the win probability without raising the payout. Second, a bet with a high edge is not necessarily unlikely to win. Any 7 wins 1 time in 6, yet it has the largest edge in the table. The Pass Line figure comes from adding the chance of winning on the come-out roll and on every point. Only the free odds bet, which pays exactly true odds, has a zero edge, and it can be placed only alongside a Pass Line bet that carries its own edge. For roulette-specific derivations, see the roulette house edge guide.

Common Mistakes With True and Payout Odds

Confusing “to 1” with “for 1” shifts the profit by one unit and changes the edge. Treating a high win probability as a low edge is the second error: Any 7 wins often but costs 16.67%.

Comparing odds across different wheels or rule sets is the third. Pay attention to the number of zeros and the exact payout. The fourth is believing that a payout “close to true odds” is nearly free: a small gap on a bet with a high win probability, repeated hundreds of times, still adds up. The fifth is reading the edge as the chance of losing. A 1.41% edge does not mean you lose 1.41% of bets; it means you lose about $1.41 per $100 wagered on average.

None of this makes any standard bet profitable. Because payout odds sit below true odds, the expected value is negative, and gambling should be treated as paid entertainment.

Quick Reference and Check Your Answer

Use this four-step check for any bet. First, find the win probability p by counting outcomes. Second, compute the true odds X = (1 − p) ÷ p. Third, read the payout Y in “to 1” form. Fourth, compute the edge as p × (X − Y).

Verify the result a second way: the edge should also equal 1 − p × (Y + 1). For the example, 1 − (1/37) × 36 = 1/37 ≈ 2.70%. If the two methods disagree, recheck the win probability.

Related Concepts and Next Steps

To read odds in other formats, start with converting American odds to decimal and fractional odds, which shows how the same price appears as American, decimal and fractional odds. For the average result of a bet, work through the expected value formula guide. For a full derivation on one game, see the roulette house edge calculation, and for the counting skills behind every win probability, read probability in gambling.