10,000 Trials Independent software studio

The only bet in the casino with no house edge, and what it does for you.

Taking maximum odds is the one piece of casino advice almost everyone gets right and nobody checks. It lowers a percentage without lowering a loss.

undefined shoes dealt undefined hands undefined patterns searched undefined null replicates

The short version, in dollars

At $10 on the line and about 100 rolls an hour, you get roughly 30 decisions in that hour. Here is what the advice everyone repeats is actually worth.

$4.19 an hour: what the pass line costs you. This number does not change when you take odds. Not a little. At all.
$1,119 an hour: what you are putting at risk once you back it with 3-4-5× odds, against $296 without. Same expected loss, 3.8× the money on the table.
0.374%: the famous number. It is your expected loss divided by that larger pile — a smaller fraction of a bigger denominator, which is not the same as losing less.
$49.37 an hour: what the same money would cost on any seven, the worst bet on the layout. That is where to spend your attention, not on the odds behind the line.

Words we could not avoid

the line: The pass-line bet: the main bet in craps, made before the first roll of a sequence. It wins on 7 or 11, loses on 2, 3 or 12, and otherwise waits.
the point: If the first roll is 4, 5, 6, 8, 9 or 10 that number becomes the point, and the bet then waits to see whether the point or a 7 arrives first.
the odds bet: An extra bet allowed behind the line once a point is set. It is the only bet in a casino paid at true odds, meaning the payout exactly matches the chance of winning and the house keeps nothing.
true odds: A payout that exactly matches the real chance of the thing happening. Bet on a point of 4, which comes up half as often as a 7, and you are paid double. Nothing is skimmed, so the average outcome is exactly zero.
house edge: The share of every dollar staked that the casino keeps on average. Watch the word staked: it is a fraction, and a fraction can be shrunk by making the bottom of it bigger.
expected value: What a bet is worth on average if you could make it thousands of times. Zero expected value means the average outcome is neither a gain nor a loss — which is what makes the odds bet unique, and also what makes it unable to rescue anything.
bankroll: The money you have set aside to play with — not the size of one bet, but everything you are prepared to put at risk before you stop. Odds bets eat it far faster than the line does, which is the practical catch.
action: The total amount you put at risk over a session, counting every bet separately. Two players can lose the same money with wildly different action.

What this was computed on

Two dice have 36 outcomes, so craps is a closed-form game. Nothing below is simulated, sampled or fitted — every figure is exact, computed in rational arithmetic because the headline compares two numbers that differ by a tenth of a percentage point.

The game

Two fair six-sided dice, 36 equally likely outcomes
Pass line, don’t pass with the 12 barred — the standard rule
Odds behind the line at true odds, up to the house maximum
3-4-5× odds: 3× on 4 and 10, 4× on 5 and 9, 5× on 6 and 8

The bets priced

Line bets, odds, come and don’t come
Place and buy bets on every point
The field, both the 2:1 and the 3:1 version of the twelve
Hardways and the one-roll propositions in the centre

What is not modelled

Dice influence — the dice are assumed fair, which is the whole question for some players
Comps, which max odds genuinely does earn faster
Table minimums, bankroll and ruin
Come/don’t come odds, priced the same as the line’s and left off the layout figure

The pace

100 rolls an hour, a busy table
3.3758 rolls per pass-line decision, computed not assumed
29.6 decisions an hour
$10 on the line

Hourly figures move with the pace assumption, which is why it is stated above rather than buried. The edges do not — those are properties of the dice.

A few layouts pay the field 2:1 on the 12 instead of 3:1. Both are priced below, and the difference is larger than the entire pass-line edge.

Comps are ignored. A casino rates you on action, so max odds does buy you more of them — which is the one honest argument for it, and it is an argument about marketing spend rather than about odds.

We assume the dice are fair. Every figure here rests on that and none of it tests it. Whether a practised shooter can influence a throw is a real question, it is not this one, and it would need thousands of recorded throws from a named shooter to answer — which we do not have and have not measured.

Finding one

The odds bet has no house edge, and that is not the same as it helping.#

odds bet expected value: exactly 0
expected loss, no odds: 0.014141
expected loss, 3-4-5x: 0.014141
quoted edge: 1.414% to 0.374%

The odds bet is paid at true odds. A point of 4 arrives half as often as a seven and pays double; a point of 6 arrives five times for every six sevens and pays six to five. Work each one out and the expected value is not close to zero, it is zero — the only bet on any table in the building of which that is true.

A bet worth zero cannot change what you expect to lose. Adding it to the line leaves your expected loss at 0.014141 of a line bet, which is the same number you started with. What it changes is the amount staked. Divide an unchanged loss by a larger stake and the quotient falls: 1.414% becomes 0.374%, and at 100× odds it becomes 0.021%.

That is the whole of the famous advice. It is a true statement about a fraction, repeated by people who have not noticed which half of it moved.

Both series are exact. The blue line is what the advice is about; the gold line is what you pay.

Finding two

What it does buy is variance, and a longer night.#

action, flat line: $296/hr
action, 3-4-5x: $1,119/hr
expected loss, either: $4.19/hr
decisions an hour: 29.6

At $10 on the line and 100 rolls an hour you get about 30 decisions, and the house expects $4.19 of them. Back every point with 3-4-5× odds and the house still expects $4.19. What changed is that you now have $1,119 moving across the table each hour instead of $296.

That is a real thing to want. A bet with no edge is the cheapest volatility available anywhere in the building, so if the point of the evening is a big swing rather than a small loss, odds are exactly how to buy one. It is simply not what the advice says it is.

There is one other honest argument for it. Casinos rate players on action, so maximum odds earns comps far faster than the flat line does. That is a claim about a marketing budget, not about dice, and it is the only version of the advice that survives contact with the arithmetic.

Finding three

The cheapest bet in the game is the one nobody makes.#

pass line: 1.414%
dont pass: 1.364%
per resolved bet: 1.403%
why it is not zero: the 12 is barred

The don’t pass wins whenever the pass line loses, so on a fair layout it would be the better side outright. It is not fair: a rolled 12 on the come-out pushes instead of paying, and that single barred number is the entire house edge on the bet. It comes to 1.364% against the line’s 1.414%.

Quoted per bet actually resolved, ignoring the pushes, it is 1.403% — still under the line. Either way it is the cheapest thing on the felt.

Almost nobody bets it. The reason is social rather than mathematical: the don’t pass wins in silence at a table that has just groaned, and loses to a cheer. That is worth roughly half a tenth of a percent, which is a small price for not being the person everyone is glaring at, and a real one.

Finding four

The money is in the middle of the layout, and it is not yours.#

hard 4 or 10: 11.111%
any seven: 16.667%
versus the line: 12x
field, 2:1 vs 3:1 twelve: 5.556% / 2.778%

Everything in the centre of the table runs between 9.091% and 16.667%. Any seven, the bet the stickman calls for most insistently, is 12 times the price of the pass line sitting six inches away.

The field is the instructive one, because it looks generous — seven of the eleven numbers win. It costs 2.778% where the 12 pays 3:1 and 5.556% where it pays 2:1. One number, printed in small type on the felt, and the gap between the two versions is larger than the entire pass-line edge.

This is where a craps player actually loses money. A table full of people arguing about whether to take 3× or 5× odds, while the stickman quietly sells them a bet that costs twelve times the one they are arguing about.

The same prices, laid out where they sit on the table. The cheap bets are the ones under your hands; the row along the bottom is the middle of the felt.

Every bet on the layout, priced exactly

By enumeration over the 36 dice outcomes, not simulation. The list runs from the one bet that keeps nothing to the one that keeps a sixth of everything placed on it.

Odds behind the line	0.000%
Don’t pass (12 barred)	1.364%
Pass line	1.414%
Place 6 or 8	1.515%
Field (12 pays 3:1)	2.778%
Place 5 or 9	4.000%
Field (12 pays 2:1)	5.556%
Place 4 or 10	6.667%
Hard 6 or 8	9.091%
Big 6 or Big 8	9.091%
Hard 4 or 10	11.111%
Any craps	11.111%
Eleven	11.111%
Two or Twelve	13.889%
Any seven	16.667%

What to do at the table

Craps has more bets than any other game in the room and almost all of them are expensive. The whole of correct play is two bets and a refusal.

research/craps/engine.py · exact enumeration over 36 dice outcomes

The only bets worth making

Two of them, and they are next to each other on the felt.

this is the play, not a mistake

Every sequence, before the first roll

Most people — Spread money around the layout so something hits on every roll.

Correct — The pass line, flat, at 1.414%. Or the don’t pass at 1.364%.

Those two are the cheapest bets in the game and they sit at your elbow. Everything else on the felt is between one and twelve times more expensive.

$1.41per $100 on the line

this is the play, not a mistake

The bet nobody at the table makes

Most people — Betting against the shooter is rude, and it feels like rooting for a crash.

Correct — It is cheaper. 1.364% against 1.414%.

The gap is small and it is real. The reason it is unpopular is social, not mathematical — it wins in silence and loses to a cheer.

$1.36per $100, the cheapest bet in craps

What the odds bet actually does

It is free, which is not the same as it being useful. Know which one you want.

cost of the mistake

Backing the point with odds

Most people — It lowers the house edge to 0.37%, so take the maximum.

Correct — True, and it does not lower what you lose by a cent.

Expected loss is 0.014141 of a line bet with no odds and 0.014141 with 3-4-5×. The percentage falls because you divided the same loss by 3.78× the stake.

$0.00saved by taking maximum odds

not a cost — nothing to compare by length

What the odds bet is actually for

Most people — It is free money, so more of it must be better.

Correct — It is a free way to raise variance. Take it if you want a bigger swing, not a smaller bill.

At $10 a line it turns $296 an hour of action into $1,119. The same $4.19 goes to the house either way; the rest is a longer, wilder ride.

3.8×the action, same loss

The rest of the layout

The middle of the table is where the money is, and it is not yours.

cost of the mistake

Place bets on the 6 and 8

Most people — The most common numbers, so the safest place to put money.

Correct — Acceptable at 1.515%. The 5, 9, 4 and 10 are not.

Place 5 or 9 costs 4.000% and place 4 or 10 costs 6.667% — nearly five times the line.

$1.52per $100

cost of the mistake

The hardways and the propositions

Most people — Small bet, big payout, and the stickman keeps offering.

Correct — Never. This is what the stickman is for.

Hard 4 or 10 costs 11.111%. Any seven costs 16.667% — the worst bet on the table and the one shouted for most often.

$16.67per $100 on any seven

cost of the mistake

The field, because it covers seven numbers

Most people — Seven of the eleven numbers win. Those look like good odds.

Correct — Check what the 12 pays before you decide, then usually no.

At 3:1 on the 12 it costs 2.778%. At 2:1 it costs 5.556%. One number on the felt, and the difference between the two versions is bigger than the whole pass-line edge.

$5.56per $100, the 2:1 version

The odds bet cannot lower your expected loss, because its own expected value is exactly zero. At every multiple from none to 100×, the loss is the same 0.014141 of a line bet.

The don’t pass is cheaper than the pass line — 1.364% against 1.414% — and almost nobody bets it, because it wins when the table groans.

The proposition bets in the middle of the layout run from 9% to 16.667%. That is where a craps player actually loses money, and it has nothing to do with odds.

Max odds is not bad advice. It is a true sentence about a ratio, repeated by people who have not noticed it is silent about the money. If you want a smaller number to quote, take the odds. If you want to lose less, bet less.