Gambling Odds Explained: House Edge and Expected Value

Every bet has two numbers attached to it: the chance it wins, and what it pays when it does. Gambling odds are just those two numbers written in one of several formats. Once you can read them, you can see the gap between them, and that gap is the house edge. This guide covers how odds are written, how to work out the house edge and the expected value of any bet, why no betting system closes the gap, and then works through real games with published figures so you can check every step.

1. How gambling odds are written

The same price can be written three ways, depending on where you are and what you are betting on. All three answer the same question: if I stake one unit and win, how much do I get?

Fractional odds (5/1, 1/2) are traditional in the UK and Ireland. They give your profit relative to your stake: 5/1 means you win 5 for every 1 you stake, and get the stake back as well. 1/2 means you win 1 for every 2 staked.
Decimal odds (6.00, 1.50) are standard in Europe, Australia and most online games. They give your total return per unit staked, stake included. 6.00 turns 10 into 60. A game's "1.90×" multiplier is decimal odds under another name.
American odds (+500, −200) are used by US sportsbooks. A plus number is the profit on a 100 stake; a minus number is what you must stake to win 100. +500 wins 500 on 100; −200 needs 200 to win 100.

Every price also carries an implied probability: the chance of winning at which the bet would exactly break even. For decimal odds it is simply 1 ÷ the odds.

Fractional	Decimal	American	Implied probability
5/1	6.00	+500	16.7%
2/1	3.00	+200	33.3%
1/1 (evens)	2.00	+100	50.0%
9/10	1.90	−111	52.6%
1/2	1.50	−200	66.7%

One trap in the wording: casino and racing odds are often quoted against. "3 to 1 against" means three losing outcomes for every winning one, which is a probability of 1 in 4, or 25%, not 1 in 3.

2. True odds vs payout odds

The true odds of a bet come from the game itself: how many ways it can win against how many ways it can lose. The payout odds are what the house actually pays. In a perfectly fair game they would be the same. In any commercial game they are not.

Roulette shows it cleanly. A European wheel has 37 pockets: 1 to 36 and a single zero. Bet on one number and there is one way to win and 36 ways to lose, so the true odds are 36 to 1 against. The table pays 35 to 1. That one unit of difference is where the casino makes its money, and it is there on every spin, whether you win or lose.

3. What is house edge?

The house edge is the share of every stake the operator expects to keep over the long run. It is the gap between true odds and payout odds, expressed as a percentage of the bet. For a bet with a single winning outcome:

House edge = 1 − (probability of winning × decimal payout)

On a European single-number bet: 1 − (1/37 × 36) = 1/37 = 2.70%. An American wheel adds a double zero, so the same bet pays the same 35 to 1 against 38 pockets: 1 − (1/38 × 36) = 5.26%. One extra pocket nearly doubles the price of playing.

For familiar reference points, these are the standard house edges of common casino bets. Several depend on house rules, so treat them as typical figures rather than guarantees:

Bet	House edge	Note
Blackjack, basic strategy	≈0.5%	Rule-dependent; a 6:5 blackjack payout adds about 1.4 points
Baccarat, banker	1.06%	After the usual 5% commission
Baccarat, player	1.24%
Craps, pass line	1.41%	Free odds taken behind it carry no edge
European roulette	2.70%	Every standard bet
American roulette	5.26%	7.89% on the five-number bet
Baccarat, tie at 8 to 1	14.36%	The worst bet on the table

Sports betting hides the edge in the prices instead. A match quoted at −110 on both sides has an implied probability of 52.4% for each team, which adds up to 104.8%. The extra 4.8% is the bookmaker's margin, also called the vig or overround. If the match really is a coin toss, either bet carries a house edge of about 4.5%.

4. Expected value in gambling

Expected value (EV) turns the house edge into a sum of money. Multiply each possible payout by its probability, add them up and subtract the stake:

EV = Σ (probability × payout) − stake

A $10 single-number bet on a European wheel returns $360 (35 × 10 in winnings plus the $10 stake) one time in 37. That is worth 360 ÷ 37 = $9.73, so the EV is −$0.27, which is 2.7% of the stake. A $10 bet on red returns $20 eighteen times in 37: also $9.73, also −$0.27. The two bets feel nothing alike, and they cost exactly the same.

The same method works on a game with several payouts. On Magic Door, five doors are shuffled and you pick one. Behind them are 0×, 0×, 0.5×, 1.25× and 3×. On a 40-Star stake the doors pay 0, 0, 20, 50 and 120 Stars, and each is picked one time in five: (0 + 0 + 20 + 50 + 120) ÷ 5 = 38 Stars. The EV is −2 Stars, a 5% house edge, whichever door you choose.

The part people miss is that EV applies to every stake you place, not to the money you arrived with. Take 100 Stars and play 100 rounds at 10 Stars each, recycling your wins. You have put 1,000 Stars through a 5% edge, so your expected loss is 50 Stars: half your starting budget, even though the edge is "only" 5%. The cost of playing is the edge times how much you turn over.

5. RTP is the same number, turned around

Return to player (RTP) is the house edge seen from the other side of the table: RTP = 1 − house edge. A game with a 5% edge has a 95% RTP; European roulette's 2.70% edge is a 97.3% RTP. Online games and slot machines tend to quote RTP, table games and sportsbooks tend to quote the edge, and they are the same measurement.

Two details are worth knowing. First, where a game can return your stake without settling the bet, such as a push in blackjack, published figures do not always use the same base, so small differences between sources are usually a definitional choice rather than a disagreement. Second, a casino "hold" figure is something else again: the share of chips bought that the casino keeps, which grows the longer players recycle their winnings. For the full picture of how RTP is measured and when a promotion can push the return on a single payout above 100%, see when a promotion pushes RTP above 100%.

6. Variance: same edge, different ride

House edge tells you the average. Variance, or volatility, tells you how far a real session wanders from it. Two games can share an edge and feel completely different.

Take two bets with the same 5% edge. Coin Flip pays 1.90× on a 50/50 call. Roll Over set to its narrowest threshold pays 95× on a 1% chance. Both return 95% on average. But a single coin flip swings by 0.95 stakes either side of the average (one standard deviation), and a single 1% roll swings by about 9.45 stakes, ten times as much.

That difference decides how long the edge takes to show. Here is the exact chance of finishing a flat-stake session ahead of where you started:

Rounds played	Coin Flip, 50% at 1.90×	Roll Over, 1% at 95×
10	37.7%	9.6%
100	30.9%	26.4%
1,000	4.7%	41.7%
10,000	About 1 in 15 million	28.6%

The coin flip is the edge made visible: by a thousand flips, fewer than one session in twenty is ahead. The long-shot is the opposite. Over ten rounds you are ahead only if you hit once, which is rare, but over a thousand rounds a handful of 95× hits can carry a whole session, so it stays close to a coin toss for much longer. It still sinks. By 10,000 rounds the chance has dropped to 28.6%, and it keeps falling the longer you play.

A rough rule: the edge starts to outweigh the luck once your expected loss grows larger than one standard deviation of your result. For Coin Flip that happens after about 361 flips. For a 1% roll at 95× it takes about 35,700. High variance does not remove the edge. It only hides it for longer.

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7. Why no betting system beats the edge

A betting system changes how much you stake and when. It cannot change the odds of any single bet. Each bet has a negative EV, and adding up bets with a negative EV always gives a negative total. No ordering, sizing or stopping rule gets around that.

Martingale (double after every loss) wins a small amount most of the time and occasionally loses everything it has built up and more. A long enough losing run always comes, and a table limit or a fixed set of stakes ends the doubling before your bankroll does. On Coin Flip, for example, Stars stakes come in five tiers from 5 to 100. The martingale calculator simulates thousands of sessions with your own bankroll and limit.
Chasing a due win. A losing streak does not make the next win more likely. That is the gambler's fallacy, and raising your stake because of it only puts more money through the same edge.
Picking the "right" stake size. The Kelly criterion, the standard formula for sizing a bet, recommends staking nothing at all when the edge is against you. The Kelly criterion calculator shows it for any odds.
Cashing out at the perfect moment. On a ladder game priced the way the examples below are, every stopping point returns almost exactly the same share of the stake. The choice changes the ride, not the price; when to cash out explains why.

8. Worked examples on real games

Everything above can be checked on a game that publishes both of its numbers, the probability and the payout. Gift Club's games do, so here they are as worked examples. Each multiplier below is decimal odds (it includes your stake), and each figure is the value the game's server runs on.

Bet	Chance to win	Pays	Return	House edge
Coin Flip, heads or tails	50%	1.90×	95%	5%
Color Predict, red or green	40%	2.375×	95%	5%
Color Predict, violet	20%	4.75×	95%	5%
Roll Over, threshold 49	50%	1.90×	95%	5%
Roll Over, threshold 98	1%	95×	95%	5%
Roll Over, threshold 2	97%	0.98×	95%	5%
Magic Door, any door	20% per door	0×, 0×, 0.5×, 1.25× or 3×	95%	5%

Three things are worth reading out of that table.

The payout is the target divided by the probability. A game priced at 95% pays 0.95 ÷ P(win). A 50% chance pays 0.95 ÷ 0.5 = 1.90×; a 1% chance pays 0.95 ÷ 0.01 = 95×. Roll Over does this in the open: you choose a threshold from 2 to 98, a number from 0 to 99 is rolled, and you win if it lands above your threshold. The multiplier is (100 − 5) ÷ (99 − threshold), where the 5 is the house edge in the game's settings. Whatever threshold you choose, the edge stays at 5%.

A "win" can pay less than you staked. At threshold 2 you win 97% of rolls, and each of those wins pays 95 ÷ 97 = 0.98×. The screen says win, but your balance goes down. When a win probability is high enough, the fair multiplier falls below 1×, and the house edge takes it lower still.

Rounding is handled without moving the edge. At the smallest 5-Star stake, a 1.90× Coin Flip win is worth 9.5 Stars, and Stars only come in whole numbers. The game pays 9 or 10 on a provably fair draw weighted so the average is exactly 9.5, so the rounding costs you nothing on average. You can check any round with the method in how to verify a provably fair round.

A board with many payouts: Plinko

On an 8-row Plinko board the ball makes eight left-or-right bounces, so the nine slots are reached 1, 8, 28, 56, 70, 56, 28, 8 and 1 times in 256. The Low-risk board pays 3.8×, 1.6×, 1×, 1× and 0.6× from the edge to the centre. Multiply each payout by how often it is hit and add them up: 243.2 ÷ 256 = exactly 95.000%. It is also a good example of how variance hides inside a table. The ball lands in the 0.6× centre slot 27.3% of the time and on a 1× slot 65.6% of the time, and only 7.0% of drops pay more than you staked.

A ladder: Mines Tower

Ladder games price every rung the same way. Mines Tower on Medium hides two mines among five boxes per level, so you survive each level 60% of the time and the multiplier for level k is 0.95 ÷ 0.6^k, rounded down to two decimals:

Level	Multiplier	Chance of reaching it	Return if you cash out here
1	1.58×	60.0%	94.80%
2	2.63×	36.0%	94.68%
3	4.39×	21.6%	94.82%
4	7.33×	13.0%	95.00%
5	12.21×	7.8%	94.94%
6	20.36×	4.7%	94.99%

Multiply any multiplier by its chance and you get the return for stopping there. Because each rung is rounded down rather than to the nearest hundredth, the return is never above 95% and sits a fraction below it on most rungs. That is the whole of the rounding, and it only ever goes in the house's favour by a few hundredths of a percent. The mines calculator does the same sum for a mines grid of any size and any number of mines.

One thing these figures leave out on purpose is promotions. A promotion changes the price or the payout of one specific product for a limited number of claims, and it is never folded into a game's published figure. How that works, and when it can lift a single payout above 100%, is covered in the promotions article. The published figure for every game is in the RTP index.

9. What this means for your budget

A house edge is the price of playing, charged a little at a time. Over one evening luck dominates and you may well finish ahead. Over enough rounds the edge dominates, and the expected cost is simply the edge times everything you stake. That makes a few habits worth keeping:

Set a budget before you start, sized as money you are happy to spend on entertainment, and stop when it is gone.
Count turnover, not deposits. Recycling winnings puts the same Stars through the edge again. A 5% edge on 1,000 Stars of stakes costs 50 Stars on average, however small the starting balance was.
Pick volatility on purpose. Low-variance bets make a budget last longer and end ahead less spectacularly. High-variance bets give you a real shot at a big multiple and more sessions that end with nothing.
Never chase losses. A bigger stake buys the same odds at a bigger size.

10. FAQ

What is house edge in simple terms?

It is the share of each bet the operator keeps on average. With a 5% house edge, every 100 staked returns 95 on average over a very large number of bets. It does not tell you what will happen in one session.

What does 5/1 mean in betting odds?

You win 5 for every 1 you stake, and get your stake back too. In decimal odds that is 6.00, in American odds +500, and it implies a 1 in 6 (16.7%) chance of winning at break-even.

What does −110 mean?

You must stake 110 to win 100. It implies a win probability of 52.4%, so on a genuine 50/50 event the bookmaker keeps about 4.5% of every stake.

How do you calculate expected value in gambling?

Multiply each possible payout by its probability, add the results and subtract your stake. A 100-Star Coin Flip returns 190 Stars half the time: 0.5 × 190 − 100 = −5 Stars per flip.

Can you beat the house edge?

Not with a betting system. You can only choose bets with a smaller edge, play fewer rounds, or stake less. The exceptions are bets where the player really does gain information the game does not price in, such as counting cards in blackjack, and that is not a system but a change in the odds themselves.

Which casino bets have the best odds?

Among standard casino games: blackjack with basic strategy (about 0.5% on good rules), baccarat on banker (1.06%) and craps on the pass line (1.41%), which drops further as you take free odds behind it. Roulette is 2.70% on a single-zero wheel and 5.26% on a double-zero one.

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