House edge and RTP: the same game, two different numbers

Two numbers describe the same casino game, and they are printed in different places. A table-game rules card calls it a house edge of 2.70 per cent. A slot information screen calls it a return to player of 96.50 per cent. We treat the pair here as one figure written two ways, because for a single resolved bet that is exactly what they are — and then we go through the three cases where they stop agreeing.

Both numbers come out of one calculation

Take a straight-up number on a European wheel. There are 37 pockets, the bet pays 35 to 1, and a winning unit therefore comes back as 36 units including the stake. Cover every pocket in turn and you stake 37 to collect 36. The return is 36/37, which is 97.30 per cent; the shortfall is 1/37, which is 2.70 per cent. One subtraction separates them: return to player is 100 per cent minus the house edge, and the house edge is 100 per cent minus the return.

Nothing in that calculation is about luck. It is a weighted average of every outcome the bet can have, with each outcome multiplied by how often it happens. Which is why the same wheel gives the same answer whichever way the bet is dressed: red or black on the same European wheel pays even money and wins 18 times in 37, so the return is 36/37 again. Different bet, identical edge.

Eight bets, converted

Bet	How it resolves	Return	House edge	Expected loss per £100 staked
European roulette, straight up	1 pocket in 37, pays 35 to 1	97.30%	2.70%	£2.70
European roulette, red or black	18 pockets in 37, pays evens	97.30%	2.70%	£2.70
American roulette, straight up	1 pocket in 38, pays 35 to 1	94.74%	5.26%	£5.26
Craps, the pass line	wins 244 times in 495	98.59%	1.41%	£1.41
Craps, odds behind a point of six	wins 5 times in 11, pays 6 to 5	100.00%	0.00%	nothing
Baccarat, banker at five per cent commission	wins 45.86 times in 100, paid 0.95	98.94%	1.06%	£1.06
Baccarat, the tie at 8 to 1	wins 9.52 times in 100	85.64%	14.36%	£14.36
A slot whose screen states 96.5	declared return for the game	96.50%	3.50%	£3.50

Three of those rows are worth doing by hand, because the arithmetic is the whole argument. The pass line wins 244 times in every 495 decisions, so it loses 251: the difference of 7 over 495 is 1.41 per cent. The odds bet behind a point of six wins 5 times in 11 — five ways to roll a six against six ways to roll a seven — and it pays 6 to 5, so five wins of 1.2 units exactly cancel six losses of one unit. Nothing is left over for the house, which is why that one row reads 100 per cent. The banker bet wins 45.86 times in 100 and is paid 0.95 rather than 1.00, so 0.95 multiplied by 0.4586 falls 0.0106 short of the 0.4462 it loses. The beginner’s guide to craps sets out the shooter sequence those pass-line numbers come from, and the same tables sit behind bankroll play at baccarat.

Where each figure is printed

Table games are described by edge, slots by return, and the reason is practical rather than principled. A table game has a small number of bets, each with its own edge, and a rules card can list them. A slot has one bet, and the figure that matters is cumulative — so the information screen behind the i button states a single return for the whole game, usually to one decimal place. Live-dealer tables carry both: the game rules give the edge per bet type, the lobby tile sometimes gives a return. Either way it is the same subtraction.

Three places where the two numbers part company

Side bets are not in the headline figure. A blackjack table advertising a low edge is advertising the main game. The perfect-pairs box next to it is a different wager with a different, usually far worse, edge, and staking on both means your money is no longer running at the headline number. Baccarat is the clean example in the table above: banker at 1.06 per cent and the tie at 14.36 per cent are the same table, the same shoe and the same shoe arithmetic.

A stated return can include a jackpot you are unlikely to see. On a game with a pooled jackpot, part of every stake feeds the pool and the declared return counts that contribution as returned — to somebody. The return you experience while the jackpot is not hit is lower than the number on the screen, and the gap is exactly the jackpot’s share.

Edge is quoted per initial bet; return is quoted per unit staked. In blackjack you can double and split, so the money at risk by the end of a hand is often more than the money you put up at the start. The conventional edge figure is expressed against the opening bet, which makes it look better than the same game expressed against total action. When you compare two numbers, the question to settle first is always per what — and in baccarat the same question turns 1.06 per cent of money staked into 1.17 per cent of hands actually decided, once ties are taken out as pushes rather than losses.

What neither figure promises

Both numbers are long-run averages, and the long run in this arithmetic is very long indeed. A 3.50 per cent edge means an expected loss of £3.50 for every £100 that passes across the table — not a loss of £3.50 from your pocket, and not anything at all about the next hour. How far a single session can sit from the average is a separate calculation, which we do spin by spin in five hundred spins at a stated return of 96.5 per cent; the shape of that spread is what a game’s volatility band is trying to describe. For the longer view on the same pair of figures, our newsroom piece on how return and edge shape results over time takes it out to the scale where the average starts to mean something.