Combined House Edge With Odds

A Pass Line or Come bet with odds is not one wager with one uniform price. It is a house-edge flat bet joined to a true-odds wager. The flat portion supplies the casino’s expected profit. The odds portion adds no expected profit, so including it in the denominator lowers the combined house-edge percentage.

That lower percentage is real. It does not mean the package is safer on the next roll. Larger odds increase the amount that can disappear when seven arrives before the point.

The calculation in one line

For a right-side line bet, the combined edge can be written as:

[ H_{combined}=\frac{EL_{flat}+EL_{odds}}{A_{flat}+A_{odds}} ]

where:

• (H_{combined}) is the house edge on the complete betting package;
• (EL_{flat}) is expected loss from the Pass Line or Come bet;
• (EL_{odds}) is expected loss from the odds wager;
• (A_{flat}) is flat-bet action;
• (A_{odds}) is odds action averaged across comparable decisions.

The odds wager is paid at true odds, so (EL_{odds}=0). The Pass Line’s exact expected loss is (7/495) of the flat stake, or about 1.4141%. The formula therefore becomes:

[ H_{combined}=\frac{0.014141\times B}{B+A_{odds}} ]

where (B) is the flat-bet amount.

The denominator requires care because odds are not posted on every come-out roll. A point is established on 24 of the 36 possible dice combinations, so fixed-multiple odds are present on two-thirds of Pass Line decisions. Treating the odds amount as though it were wagered on every decision understates the real blended edge.

Standard Pass Line combined-edge figures

For equal-multiple odds, average odds action per original Pass Line decision is:

[ A_{odds}=B\times M\times\frac{24}{36} ]

Here, (M) is the permitted odds multiple. This produces the familiar figures below.

These percentages describe expected cost relative to all flat and odds action. They do not say that the player is likely to lose only 0.184% of the chips sitting behind the line in one session. Short-run results are driven by the point-versus-seven race, not by the long-run blended percentage.

Why 3-4-5x odds needs a separate calculation

A 3-4-5x table normally permits:

• 3x odds on points 4 and 10;
• 4x odds on points 5 and 9;
• 5x odds on points 6 and 8.

The limits differ so a maximum-odds win produces the same odds profit: six flat units. A $10 player can therefore place $30, $40, or $50 in odds depending on the point and win $60 from that odds portion.

The average odds action must be weighted by how often each point is established:

[ A_{odds}=B\times\frac{(3\times3)+(3\times3)+(4\times4)+(4\times4)+(5\times5)+(5\times5)}{36} ]

The first number in each product is the odds multiple; the second is the number of dice combinations producing that point. The result is:

[ A_{odds}=2.7778B ]

Total average action is therefore (3.7778B), giving:

[ H_{3-4-5}=\frac{0.014141B}{3.7778B}=0.003743 ]

So full 3-4-5x odds produces a combined Pass Line edge of approximately 0.374%.

A $10 worked decision

Suppose the point is 6. You leave $10 on the Pass Line and place the maximum $50 odds.

• If 6 repeats before 7, the line wins $10 and the odds pay $60 at 6 to 5. Total profit is $70.
• If 7 appears first, the entire $60 package loses.

The odds are mathematically fair because five combinations make 6 and six combinations make 7. A $50 risk paying $60 exactly matches that 5-to-6 chance. The casino’s expected profit remains attached to the $10 flat bet.

This is why “lower edge” and “lower loss on the next decision” are different statements. The price per dollar of action improves; the dollar result becomes more volatile.

Expected loss stays attached to the flat bet

For a $10 Pass Line decision:

[ EL=10\times\frac{7}{495}=$0.1414 ]

That expected loss is approximately the same whether the player takes no odds, 1x odds, or the full permitted amount. Adding fair odds changes the percentage denominator, not the expected loss created by the $10 line wager.

This creates two valid but different ways to report cost:

A player comparing games should ask which denominator a published percentage uses. “House edge per flat bet,” “edge per total action,” and “expected loss per resolved decision” are related measures, but they are not interchangeable.

Come bets use the same pricing, with more moving pieces

A flat Come bet has the same underlying edge as the Pass Line. Once it travels to 4, 5, 6, 8, 9, or 10, the player may add odds at the table’s permitted multiple. The same blended-edge logic applies.

The practical difference is exposure. Several Come bets can occupy different numbers simultaneously. A seven can win a newly placed Come bet on its first roll while losing older Come bets, their odds, the Pass Line, and Pass odds. The settlement is mathematically consistent but can look contradictory if the positions are not tracked separately.

Read the full Come bet procedure before using several traveling bets, and use the craps odds reference for point-specific true-odds payouts.

Odds limits, chip units, and partial odds

Maximum odds is a ceiling, not a requirement. A player may normally take less, subject to table minimums and clean payout units.

Right-side odds pay:

• 2 to 1 on 4 and 10;
• 3 to 2 on 5 and 9;
• 6 to 5 on 6 and 8.

Those ratios affect sensible chip amounts. On a point of 5 or 9, an even odds amount avoids half-unit payouts. On 6 or 8, multiples of five produce clean 6-to-5 settlements. A dealer may accept other amounts where chip denominations permit, but the posted rule and house procedure control.

Published craps rules also distinguish the flat wager from the supplemental odds wager and specify the true-odds payout ratios. One official example is New York’s rule for supplemental Pass, Come, Don’t Pass, and Don’t Come wagers.

The Don’t side needs its own denominator

Don’t Pass and Don’t Come flat bets have a slightly lower base edge than right-side line bets when 12 pushes: approximately 1.3636%. Lay odds are also paid at true odds, but the player risks more than the possible win:

• lay 2 to win 1 against 4 or 10;
• lay 3 to win 2 against 5 or 9;
• lay 6 to win 5 against 6 or 8.

A statement such as “5x odds” can be ambiguous on the Don’t side. Some tables define the maximum by the amount laid; others describe it by the amount that may be won. The combined percentage cannot be calculated correctly until the actual permitted risk is known.

Use the Don’t Pass guide and the odds-bet house-edge explanation rather than copying a right-side percentage onto a lay-odds package.

Three denominators that are often confused

Craps discussions frequently quote different percentages for the same betting pattern because the denominator changes. Before comparing two numbers, identify what each one measures.

Edge on the flat wager

The Pass Line remains 1.4141% of the flat amount. If a $10 line decision has about $0.1414 expected loss, that figure is divided by $10. The odds wager does not alter the pricing of the underlying contract bet.

Edge on all average action

The combined figures in this article divide the same expected loss by the flat wager plus the odds action averaged across come-out and point outcomes. This is the best measure for comparing the price of complete Pass-plus-odds packages.

Expected loss as a share of the buy-in

Dividing theoretical loss by the money brought to the table is not a game house edge. A $300 bankroll may be wagered repeatedly. If a player cycles $2,000 of total action through the table, the expected cost is based on that $2,000 and the mix of bets, not merely on the original $300.

This distinction matters in practical session planning. Suppose a player resolves 40 separate $10 Pass Line decisions and always takes full 3-4-5x odds when a point is established. Average action is approximately:

[ 40\times$37.78=$1,511.20 ]

At a combined edge of about 0.374%, estimated theoretical loss is:

[ $1,511.20\times0.003743\approx$5.66 ]

The same result can be obtained from the flat bets alone:

[ 40\times$10\times0.014141\approx$5.66 ]

The equality is a useful audit check. If two methods produce materially different expected losses, the odds action, point frequency, or denominator has probably been counted incorrectly. Actual session results may still be hundreds of dollars above or below that expectation.

Casino rating and table-control implications

Casinos generally do not treat every chip behind the line as equivalent theoretical action. The flat bet creates expected revenue; the true-odds portion does not. A player who wagers $10 flat with $50 odds may have $60 exposed, but a rating system may record the $10 base bet, exclude odds, or apply a property-specific discount. That is why a visually large craps stack does not automatically produce comps comparable with a $60 wager carrying a normal house edge.

The separation also matters operationally. Dealers must keep odds physically associated with the correct contract bet, verify the permitted multiple, and pay the point-specific ratio. Floors need a consistent rule for unusual amounts, late additions, odds working on a come-out roll, and player-rating treatment. These controls do not change the mathematics; they prevent the wrong amount or wrong denominator from entering the transaction.

Low percentage does not cap bankroll risk

Suppose a player starts with $300 and makes $10 Pass Line bets with full 3-4-5x odds. After a point of 6, one seven-out costs $60. Five such resolved positions can consume the entire starting bankroll even though the quoted combined edge is only about 0.374%.

That does not contradict the mathematics. House edge measures average cost across a very large amount of comparable action. It does not describe the size or order of individual wins and losses. The craps variance guide explains this separation in more detail.

The useful decision is not “How do I reach the smallest advertised percentage?” It is:

1. choose a low-edge flat bet;
2. understand the odds limit and payout units;
3. decide how much post-point exposure the bankroll can absorb;
4. keep total decisions within a preset session budget.

The craps odds calculator can check point-specific payouts. The expected-loss calculator is better for comparing total action, pace, and session cost.