The craps odds bet: the one wager on the table with no built-in edge

Every wager in a casino is priced so that the house keeps a slice. There is one documented exception, it sits on the craps layout, and it is not advertised by the dealer because it pays nothing to advertise. We derive it here from the dice, and then do the arithmetic that matters: what it does to the cost of an ordinary pass-line session.

What the pass line costs

The pass line wins 244 times in every 495 decisions and loses the other 251. The difference, 7 in 495, is 1.41 per cent — among the cheapest flat bets in the building, and the figure our table of eight bets converted from edge to return gives as 98.59 per cent returned. On a £10 pass line the expected cost of a decision is 14.1 pence.

Where the odds bet comes from

When the come-out roll establishes a point — 4, 5, 6, 8, 9 or 10 — the pass line is now waiting for that number to come before a seven does. At that moment the table allows a second, separate bet behind the first, and that one is paid at the true odds of the point. True odds means the payout exactly matches the probability, which means the expectation is zero.

Three derivations, since the whole claim rests on them. There are six ways to roll a seven with two dice. A point of six can be made five ways, so the race is 5 against 6: the bet pays 6 to 5, and five wins of 1.2 units cancel six losses of one unit exactly. A point of five can be made four ways, so it is 4 against 6: the bet pays 3 to 2, and four wins of 1.5 cancel six losses of one. A point of four can be made three ways, so it is 3 against 6: the bet pays 2 to 1, and three wins of two cancel six losses of one. In every case the sum is nought.

Point	Ways to make it	Ways to roll a seven	Odds paid	Expected result per unit
4 or 10	3	6	2 to 1	nought
5 or 9	4	6	3 to 2	nought
6 or 8	5	6	6 to 5	nought

Free is not the same as profitable

The bet is often called free odds, and the name misleads in one direction. Zero expectation means the bet neither costs nor earns: it is not an edge, it is the absence of one. You cannot win money in the long run by taking odds, and the house is entirely content for you to do it, because the money is still on a table where the flat bet beside it is being charged 1.41 per cent.

What it does to a ten-pound pass line

Here is the figure worth knowing. Most tables cap odds with the 3-4-5 rule: three times the flat bet on a point of 4 or 10, four times on 5 or 9, five times on 6 or 8. Of the 24 ways a point can be established, six give 4 or 10, eight give 5 or 9, and ten give 6 or 8, so the average multiple allowed is (6×3 + 8×4 + 10×5) divided by 24 — that is 100 over 24, or 4.17 times the flat bet.

A point is established on 24 of 36 come-out rolls, which is two times in three. So on a £10 pass line, the average money in action per decision is £10 plus two thirds of £41.67, which is £37.78. The expected cost has not changed — it is still the 14.1 pence the flat bet always cost, because the odds bet contributes nothing to it. But measured against everything staked, 14.1 pence on £37.78 is 0.37 per cent.

That is the entire trick, and it is worth being precise about what it is. Taking maximum odds does not make the house edge smaller. It leaves the expected loss exactly where it was and spreads it over three and a half times as much action, which is the best price per pound staked available on any table in the room.

The cost of that price is variance

Four times the money in action swings four times as hard. The £10 player taking 3-4-5 odds will see sessions a flat-bet player never sees in either direction, and a bankroll sized for £10 decisions is not sized for £37.78 ones. The sensible version is to size the flat bet so that the flat bet plus full odds is the exposure you actually meant: a £5 line with full odds puts about £19 in action, which is nearer a £20 decision than a £5 one. The same trade — a cheap price per pound bought with a wider spread of results — is what makes picking a game a question about tolerance rather than about return, and our page on choosing a game to suit a risk profile works through the choice from that end.

And it remains the only bet of its kind on the layout. The field, the hardways, any seven, the big six and eight are all priced well above the pass line, and no amount of pattern-watching changes any of the numbers above, since the dice have no memory of the last shooter. Our step-by-step introduction to online craps covers the mechanics of getting the odds bet down in the first place — on a physical table it is placed by hand behind the line, and in most software it is a second button that only becomes live once a point exists.