CRA 304: Craps Variance

Craps variance is the mathematical spread of possible results around expected value. It explains why a low-house-edge betting pattern can still produce sharp losses, why adding fair odds can make the bankroll swing more, and why one long hand or one fast seven-out sequence says little about the underlying percentages.

Expected value answers “What is the average result over repeated comparable action?” Variance answers “How widely can individual results scatter around that average?” The two measures must be kept separate.

A Pass Line Bet Resolves in Whole Units

A $10 Pass Line wager does not gradually lose about $0.14. It eventually wins $10 or loses $10.

For a standard Pass Line decision:

• the probability of ultimately winning is 244/495, or about 49.293%;
• the probability of ultimately losing is 251/495, or about 50.707%.

The expected result is:

Expected value = ($10 × 244/495) + (−$10 × 251/495)
               = −$70/495
               = −$0.1414 per completed decision

That is the familiar 1.414% house edge on the base wager. The actual decision remains all-or-nothing.

For a result X that is either +$10 or −$10, the variance is:

Variance = E[X²] − (E[X])²
         = $100 − (−$0.1414)²
         ≈ 99.98 dollars²

The standard deviation is the square root of variance:

Standard deviation ≈ $9.999 per decision

The units matter. Variance is expressed in squared dollars; standard deviation converts it back to dollars and is easier to compare with a bankroll.

Free Odds Lower the Blended Edge but Widen the Distribution

Once a point is established, a player with a Pass Line wager may be allowed to add an odds bet. The odds portion is paid at true odds and has zero house edge. That improves the percentage cost of the combined amount placed, but it adds money to every point resolution.

Consider a fixed $10 Pass Line wager with $10 odds whenever a point is established. The possible net decisions include:

The base wager still creates expected loss of about $0.1414 per completed decision. The odds contribution has zero expected value, so the combined expectation remains approximately −$0.1414.

The standard deviation, however, rises to about $18.93 per completed decision under this fixed one-times-odds example. The expected cost barely changes, but the typical size of the swing almost doubles.

This is the central lesson of craps variance: a wager can improve the ratio of expected loss to total money placed while making the dollar path rougher.

The Gambling Regulatory Authority of Singapore’s current craps rules describe Pass Line Odds as an additional wager available after a point is established and tied to the Pass Line result. Offered multiples and table procedures vary.

What 100 Decisions Might Look Like

If completed decisions were repeated under identical conditions and treated as independent, standard deviation grows approximately with the square root of the number of decisions:

Session standard deviation ≈ single-decision SD × √n

For 100 fixed $10 Pass Line decisions:

$9.999 × √100 ≈ $100
Expected loss = 100 × $0.1414 ≈ $14.14

For 100 decisions with the fixed one-times-odds structure above:

$18.93 × √100 ≈ $189.30
Expected loss ≈ $14.14

These are distribution measures, not promises that the result will fall within a particular dollar band. Real sessions may contain changing wager sizes, incomplete decisions, different odds multiples, and several simultaneous bets. The approximation is useful because it exposes the difference between a small average loss and a much larger normal swing.

NIST’s statistical handbook defines variance as the average squared deviation from the mean and standard deviation as its square root. Craps uses the same statistical concepts; only the outcome distribution is game-specific.

Several Bets Can Fail Together

Players sometimes assume that spreading chips across several numbers diversifies risk. That is not necessarily true when the wagers share the same losing event.

Suppose the point is on and one player has:

$10 Pass Line
$30 Pass odds
$18 Place 6
$18 Place 8

A seven-out can remove all $76 at once. The bets are positively correlated around the seven because they lose together. Their combined variance is not calculated by pretending that each wager resolves independently.

By contrast, a Place 6 and a Place 8 do not win on the same ordinary roll, but both lose on 7. The shared loss event dominates the short-term risk of carrying both.

This is why “number of bets” is a poor substitute for exposure analysis. Count the dollars that can disappear on the same roll.

Fast Bets Create a Different Kind of Volatility

One-roll proposition wagers resolve immediately. A rare event can pay a large multiple; every non-winning total loses at once. Their variance comes from a payout distribution concentrated in frequent small losses and occasional larger wins.

The experience can feel very different from a Pass Line cycle:

• more decisions occur per hour;
• each result is known after one roll;
• high payouts create visible spikes;
• repeated misses can drain small wagers quickly;
• the house edge may also be much higher.

Variance alone does not identify a good or bad bet. A high-edge proposition can be volatile and expensive. A zero-edge odds wager can also be volatile. House edge measures price; variance measures spread.

Pressing Makes the Session Non-Stationary

A fixed-wager calculation assumes the same exposure each time. Pressing after wins violates that assumption because later decisions carry more money than earlier ones.

Suppose a player starts with $12 each on Place 6 and Place 8, then adds $6 to the number after every hit. During a long roll, exposure rises. When 7 eventually appears, the ending loss depends on how far the presses progressed.

The betting system has changed the distribution of session outcomes, not the dice probabilities. It tends to produce many modest sequences, some heavily loaded seven-outs, and occasional large wins during unusually long rolls.

A useful session record therefore includes wager size by roll, not just the opening bet.

Bankroll Questions Need More Than House Edge

A player choosing between $10 Pass only and $10 Pass with $50 odds is not choosing merely between two percentage edges. The second structure can lose $60 on a point cycle instead of $10.

Before adding action, ask:

1. What is the maximum amount exposed to one seven-out?
2. How many comparable losses can the session bankroll absorb?
3. Are several bets tied to the same result?
4. Does the betting pattern increase after wins or losses?
5. How many decisions are likely at the current table speed?

The expected-loss calculator estimates average cost. The variance simulator is more appropriate for comparing possible short-term paths.

Why Casino Results Also Swing

A casino can lose on a craps table for an hour, a shift, or longer without the game being mathematically defective. A few long rolls, concentrated odds action, or large winning propositions can push actual table win far from theoretical expectation.

Operational review should separate ordinary variance from control failures. Managers compare drop, decisions, wager mix, unusual payouts, table limits, procedural exceptions, and surveillance evidence. A surprising result is a reason to investigate context, not proof of cheating or dealer error.

The same principle applies to players. A winning session does not prove a betting system has positive expectation, and a losing session does not disprove the published odds. Short results are noisy because the outcome distribution is wide.

The Useful Distinction

Craps variance is not simply “luck.” It is a measurable feature of the wagers and their relationships.

• Expected value identifies the long-run average cost.
• Variance measures squared spread around that average.
• Standard deviation translates the spread into wager or dollar units.
• Correlation explains why several bets may win or lose together.
• Exposure shows how much money can move on one resolving event.

A player can reduce house edge and still increase session risk by adding odds. A player can spread action across the layout and still concentrate risk on the seven. Understanding both dimensions gives a more honest picture than any single house-edge percentage.