Expected value in craps is the average profit or loss attached to a wager after its possible outcomes, probabilities, and payouts are combined. It does not forecast the next roll or one session. It prices the bet.
A $10 wager with an expected value of -$0.14 loses fourteen cents on average each time that decision is completed. It can still win $10 right now. The negative value appears only across a large collection of similarly resolved bets.
The general formula
For outcomes (1) through (n):
[ EV=\sum_{i=1}^{n}P_iR_i ]
where:
- (P_i) is the probability of outcome (i);
- (R_i) is the net result of that outcome, positive for profit and negative for loss;
- probabilities across all included outcomes sum to 1.
For a simple win-or-lose wager:
[ EV=P(\text{win})\times\text{Net Win}-P(\text{lose})\times\text{Stake} ]
House edge expresses the negative player EV as a percentage of the amount initially risked:
[ \text{House Edge}=\frac{-EV}{\text{Initial Stake}} ]
That definition matters in craps because some wagers take several rolls to resolve and some bets combine a house-edge portion with a zero-edge odds portion.
One-roll example: Any Seven
Any Seven wins when the next roll totals 7. Six of the 36 ordered dice combinations make 7, while the other 30 lose.
At the common 4-to-1 payout, a $1 bet has:
[ EV=\left(\frac{6}{36}\times$4\right)-\left(\frac{30}{36}\times$1\right) ]
[ EV=$0.6667-$0.8333=-$0.1667 ]
The average loss is 16.67 cents per $1 decision, so:
[ \text{House Edge}=16.67% ]
The fair payout would be 5 to 1 because the bet loses on five times as many combinations as it wins. Paying 4 to 1 creates the edge.
Multi-roll example: Place 6
A Place 6 wins if 6 appears before 7. Rolls such as 4, 5, 8, or 10 do not settle the wager.
There are:
- five combinations totaling 6;
- six combinations totaling 7.
Conditioning on the 11 combinations that resolve the bet:
[ P(\text{win})=\frac{5}{11},\qquad P(\text{lose})=\frac{6}{11} ]
The standard $6 wager wins $7. Its expected value is:
[ EV=\left(\frac{5}{11}\times$7\right)-\left(\frac{6}{11}\times$6\right) ]
[ EV=\frac{35-36}{11}=-\frac{1}{11}\approx-$0.0909 ]
Relative to the $6 stake:
[ \text{House Edge}=\frac{0.0909}{6}\approx1.515% ]
The harmless rolls before resolution do not change that percentage. They change how long the chips remain on the layout.
Why Pass Line EV takes more work
The Pass Line is a two-stage wager.
On the come-out roll:
- 7 or 11 wins;
- 2, 3, or 12 loses;
- 4, 5, 6, 8, 9, or 10 establishes a point.
If a point is established, the bet wins if that point repeats before 7 and loses if 7 appears first.
Across the complete decision tree, the Pass Line wins with probability:
[ \frac{244}{495} ]
and loses with probability:
[ \frac{251}{495} ]
At even money, the EV of a $1 Pass Line bet is:
[ EV=\frac{244}{495}(+$1)+\frac{251}{495}(-$1) =-\frac{7}{495} ]
[ EV\approx-$0.01414 ]
The house edge is therefore approximately 1.414%.
This result includes immediate come-out decisions and every point. Looking only at the visible 8 winning come-out combinations versus 4 losing combinations would miss the point-cycle outcomes and produce the wrong answer.
Odds bets have zero expected loss—but not zero risk
After a point is established, a player may usually place an odds bet behind the Pass Line. The odds portion pays at the mathematical odds of the point:
| Point | Ways to make point | Ways to make 7 | True-odds payout |
|---|---|---|---|
| 4 or 10 | 3 | 6 | 2 to 1 |
| 5 or 9 | 4 | 6 | 3 to 2 |
| 6 or 8 | 5 | 6 | 6 to 5 |
For $5 odds on point 6, the conditional EV is:
[ EV=\left(\frac{5}{11}\times$6\right)-\left(\frac{6}{11}\times$5\right)=0 ]
The odds wager has no house edge because the payout matches the resolving probabilities.
That does not make the combined Pass Line plus odds position a guaranteed winner. The base Pass Line still has negative EV, and the larger total amount on the table increases variance.
Suppose a player has $10 on the Pass Line and adds $20 odds. The expected loss of the combined position remains tied to the $10 base wager:
[ $10\times1.414%\approx$0.1414 ]
Measured against the total $30 at risk, the average loss rate is lower:
[ \frac{$0.1414}{$30}\approx0.471% ]
The dollar expectation did not improve because of a new player advantage. The same expected loss is spread over a larger combined wager, with more money exposed to short-term swings.
EV per resolved bet and EV per roll are different
A one-roll proposition resolves every roll. A Place bet or Pass Line bet may remain active.
For a continuously working $6 Place 6, the per-roll outcomes are:
- five combinations win $7;
- six combinations lose $6;
- 25 combinations produce $0 and leave the bet active.
The expected result per roll is:
[ EV_{\text{per roll}}=\frac{5}{36}($7)-\frac{6}{36}($6) =-\frac{$1}{36}\approx-$0.0278 ]
Per completed decision, the expected loss is about 9.09 cents. Per working roll, it is about 2.78 cents because most rolls do not settle the wager.
This distinction becomes important when estimating expected loss per hour. The calculation needs the number of actual resolutions or a correct per-roll model, not a rough guess that every chip on the layout represents a new bet every roll.
Payout language can change the calculation
Expected value must use net profit, not the total amount returned.
If a wager pays “7 to 1,” a winning $1 bet earns $7 and the original $1 is returned. Net result: +$7.
If a promotion says “7 for 1,” the total return is $7 including the original stake. Net profit: +$6.
Confusing those expressions can turn a correct probability calculation into a wrong EV. The payout glossary explains the settlement difference in detail.
Rules can also vary. The Field may pay double or triple on 12, and sometimes pays differently on 2. Those changes alter EV. A correct analysis must use the exact posted payout, not only the bet name.
Comparing several common wagers
The following examples show why payout size alone is a poor guide:
| Wager | Common house edge | Resolution style | Practical reading |
|---|---|---|---|
| Pass Line | 1.414% | Multi-roll | Low base edge; point cycle matters |
| Don’t Pass | About 1.36% | Multi-roll | 12 normally pushes, not wins |
| Place 6 or 8 | About 1.52% | Multi-roll | Wins 7 on 6; efficient place price |
| Field | Varies, often 2.78% or 5.56% | One roll | Depends on 2 and 12 awards |
| Hard 6 or 8 | 9.09% | Multi-roll | One hard pair versus easy ways and 7 |
| Any Seven | 16.67% | One roll | High edge and rapid resolution |
A 30-to-1 proposition payout may still be worse value than an even-money bet because the winning event is much rarer than the sign makes intuitive.
For derivations of the hardway prices, see Hardways House Edge. For the full ranking, see Craps House Edge.
Official rules define the events before math prices them
The expected-value formula is universal, but the wager rules are not. A valid calculation must first establish:
- what wins and loses;
- which rolls push or leave the wager unresolved;
- the posted payout;
- whether a bet is working on the come-out roll;
- when a wager may be made, changed, or removed.
The Massachusetts Gaming Commission’s official rules for craps and mini-craps provide one regulated example of Pass Line, Don’t Pass, Come, Place, Field, hardway, and proposition procedures. Other jurisdictions or approved house rules may differ, so the layout and posted rules control the live wager.
That source is useful not because Massachusetts changes probability, but because it fixes the legal definition of each event before the combinations are counted.
Total action converts percentages into money
Expected loss over repeated wagers is:
[ \text{Expected Loss}=\sum_j \text{Amount Wagered}_j\times\text{House Edge}_j ]
Suppose a session contains:
- $600 of resolved Pass Line action at 1.414%;
- $360 of resolved Place 6/8 action at 1.515%;
- $100 of Any Seven action at 16.67%.
Then:
[ \text{Pass expectation}=$600\times0.01414=$8.48 ]
[ \text{Place expectation}=$360\times0.01515=$5.45 ]
[ \text{Any Seven expectation}=$100\times0.1667=$16.67 ]
Total expected loss:
[ $8.48+$5.45+$16.67=$30.60 ]
Only $100 of the $1,060 resolved action was on Any Seven, yet it contributed more than half of the theoretical loss. This is why occasional proposition bets can dominate the cost of an otherwise low-edge session.
The result says nothing about the actual cash-out. The player may finish ahead or behind by far more than $30.60. EV is the center of the long-run distribution; variance describes how widely sessions can move around it.
Betting systems do not reprice the dice
A progression may change wager size after wins or losses. It can change:
- the frequency of small winning sessions;
- the size of occasional losses;
- bankroll exposure;
- emotional pressure.
It cannot change the EV of the underlying bet unless it changes the wager itself or the conditions under which the wager is made.
If every $10 unit is placed on a 1.414% edge event, the expected loss per $10 of resolved action remains about 14 cents. Doubling after losses changes how many dollars face that edge. It does not remove it.
The page Why No Betting System Changes Probability covers that distinction.
The useful way to apply EV
Expected value helps answer three practical questions:
- Which wager is priced better? Compare the EV or house edge under the exact rules.
- What is the likely long-run cost? Multiply each category of resolved action by its edge.
- How much short-term risk remains? Examine wager size, variance, number of decisions, and bankroll separately.
The lowest-edge choice can still be unaffordable at a large stake. A high-edge bet can still win once. Neither observation changes the price.
For calculations, use the craps odds calculator and expected loss calculator. For interpretation, continue to Craps Probability Basics, Craps Odds, and Craps Variance.