Variance, swings and bankroll
Chapter 4: The math that matters · Lesson 15 of 21 · 7 min read
Why short sessions are mostly luck, what odds do to your swings, and how much to bring.
The house edge is the average. Real sessions bounce around it, often wildly. That spread is called variance, and its usual measure is the standard deviation (SD): about two sessions in three land within one SD of the average.
For one hour (about 100 rolls) at a $10 table, simulated:
| Strategy | Expected result | SD (typical swing) | Hours you finish ahead |
|---|---|---|---|
| $10 Pass, no odds | −$4 | ±$55 | about 43% |
| $10 Pass + 2× odds | −$4 | ±$155 | about 49% |
| $10 Pass + 3-4-5× odds | −$4 | ±$265 | about 48% |
| $10 Don't Pass + 6× odds | −$4 | ±$265 | about 49% |
Look at that: the expected cost barely moves, but the swing grows five times with full odds. Odds don't make you lose more on average. They make each hour more of a coin flip, with bigger wins and bigger losses.
When does the edge win? With flat Pass bets the average loss only matches one typical swing after about 5,000 bets, roughly 170 hours of play. In one evening you're far more likely to see luck than math. The casino sees millions of bets, so for them the math is certain.
Bankroll. Bring enough that a normal bad run doesn't knock you out. Simulating one hour at a $10 table:
| Strategy | Chance of going broke in an hour, with bankroll of… |
|---|---|
| $10 Pass, no odds | $50: ~38% · $100: ~7% · $150: <1% |
| $10 Pass + 3-4-5× odds | $300: ~29% · $500: ~7% · $800: <1% |
Rules of thumb for one hour: about 10–15 flat bets for flat play, and about 50–80 flat bets if you take full odds. If that's more than you're comfortable with, lower the odds multiple, not the line bet's quality. 2× odds is still a very good 0.61%.
Set a loss limit and a time limit before you start. They don't change the math, but they decide how a bad night ends.
Show the math: Where these numbers come from
SD per Pass decision (flat $1): ≈ 1.00. With 3-4-5x odds: ≈ 4.92 flat units. Don't Pass with 6× lay: ≈ 4.91. About 29.6 decisions per 100 rolls → hourly SD ≈ √29.6 × SD. Flat $10: ≈ $54; full odds: ≈ $268. Edge equals one SD after N decisions when N × 0.01414 = √N → N = 1/0.01414² ≈ 5,000 decisions ≈ 16,900 rolls. The broke-in-an-hour figures come from 20,000 simulated 100-roll sessions, betting only when the bankroll can cover the bet plus full odds. The simulator's multi-session auto-roll reproduces them.
Same cost, bigger swings
Interactive practice table. Try this in the app (free, play money).
Bankroll test
Interactive practice table. Try this in the app (free, play money).
Taking full odds instead of no odds mostly changes...
- Your expected loss per hour
- Your swings
- The chance of a 7
- Nothing
Show answer
B: Your swings. Expected cost stays about $4/hour at $10; the hourly swing goes from about ±$55 to about ±$265.
You played 2 hours and you're up $300. What does that tell you about your strategy?
- It beats the house
- Almost nothing; 2 hours is mostly luck
- You should bet more
- The dice are hot
Show answer
B: Almost nothing; 2 hours is mostly luck. A swing like that is normal with odds. Edges only show up over thousands of bets.
A sensible one-hour bankroll for $10 Pass + 3-4-5× odds is around...
- $50
- $100
- $500
- $5,000
Show answer
C: $500. About a 7% chance of going broke in an hour with $500; under 1% with $800.