Perfect strategy I: the lowest possible edge
Chapter 6: Perfect strategy · Lesson 18 of 21 · 9 min read
The pure-math lens. Which play costs the least per dollar bet, and what "perfect" can and can't promise.
"Perfect strategy" comes in three flavours, and this chapter keeps them apart on purpose:
- Pure lowest edge (this lesson): the least possible cost per dollar bet, with no concern for nerves or budget.
- Practical (next lesson): near-minimum edge, built for a real evening.
- Fun per dollar (lesson 20): the most excitement for each dollar you expect to lose.
They give different answers, and that's fine. Start with the math.
Here's the uncomfortable truth: there is no way to win at craps in the long run. Every bet except odds has a house edge, and odds can only be added to bets that have one. "Perfect strategy" therefore means paying the smallest possible edge for every dollar you put in action.
By that pure measure, the ranking at a 3-4-5x table is (these edges agree with the published Wizard of Odds tables):
| Rank | Play | Edge per dollar bet |
|---|---|---|
| 1 | Don't Pass / Don't Come + 6× lay odds | 0.27% |
| 2 | Pass / Come + 3-4-5× odds | 0.37% |
| 3 | Don't Pass / Don't Come, no odds | 1.36% |
| 4 | Pass / Come, no odds | 1.41% |
| 5 | Place 6 or 8 (in $6 units) | 1.52% |
| 6 | Buy 4/10, Lay 4/10 (vig on win, bought for $20+ and laid for $40+) | 1.67% |
Buy and lay bets only reach 1.67% at the right sizes. Casinos round the 5% commission down to whole dollars with a $1 minimum, so a $10 buy on the 4 still pays a $1 vig (10% of the bet) and costs 3.33%. Buy for $20 or more; lay the 4 or 10 for $40 or more.
So the mathematically perfect recipe is:
- Bet the table minimum on Don't Pass.
- When a point is set, lay the maximum odds (6× here).
- For more action, add Don't Come bets and lay full odds on each.
- Never take down a Don't bet once its point is set.
- Bet nothing else.
The right side is a very close second. Pass + max odds is 0.37%, and some players prefer it for reasons covered in Perfect strategy II. At higher-odds tables the gap all but vanishes: with 10× odds it's 0.12% vs 0.18%, and at 100× it's 0.01% vs 0.02%.
Are the two sides mirror images? Only the flat bets are, and only almost: Don't Pass wins when Pass loses, except for the bar-12 push on the come-out. The odds are not mirror images. Taking odds on a 3-4-5x table puts about 2.78 extra units behind each flat unit on average; laying 6× puts about 4.00 behind it. Take odds win more than you risk but less often; lay odds win less than you risk but more often. Both are fair bets, so the only thing that differs is how much 0% money you get to dilute the flat edge with.
Two caveats keep this honest:
- Edge per dollar isn't dollars per hour. Your expected cost still comes from the flat bets: about $4 an hour at $10 ($4.19 for Pass, $4.04 for Don't Pass). The odds just let you bet much more at that same cost.
- Lay odds risk big to win small. You'll put up $60 behind a $10 bet to win $30–50. Make sure your bankroll (lesson 15) can carry it.
Show the math: Comparing the two sides at different odds
Combined edge = flat edge ÷ (1 + expected odds amount per flat unit). Right side, 3-4-5x: (7/495)/(1 + 25/9) = 7/1870 = 0.374%. Don't side, 6× lay: (3/220)/(1 + 4) = 3/1100 = 0.273%. Lay 6× means 6 units laid on every point (wins 3, 4 or 5 units). The Don't side gets more 0% money in action per flat unit (4.00 vs 2.78), as well as a slightly lower flat edge. At N× (taking N×; laying to win N×): right (7/495)/(1 + 2N/3); don't (3/220)/(1 + N).
Try the optimum yourself, with the dice loaded for the first few rolls.
The optimum, hands-on
Interactive practice table. Try this in the app (free, play money).
Now put the two best plays head to head on identical dice.
Optimum vs near-optimum, 10,000 rolls
Interactive practice table. Try this in the app (free, play money).
Which play has the lowest house edge per dollar at a 3-4-5x table?
- Pass + max odds
- Don't Pass + max lay odds
- Place 6 and 8
- Buy 4 and 10
Show answer
B: Don't Pass + max lay odds. 0.27% vs 0.37% for Pass + odds.
Can any strategy give a player a long-run edge at standard craps?
- Yes, with dice control
- Yes, with a progression
- No
- Only on the Don't side
Show answer
C: No. Every combination is a sum of negative or zero EVs.
You're on the Don't Pass with full lay odds. The point is 4. What should you do?
- Take your Don't Pass down
- Leave it: it's now the favourite
- Add a Hard 4 hedge
- Remove the odds
Show answer
B: Leave it: it's now the favourite. Never remove a Don't bet after the point; it now wins 2 times in 3. Removing odds costs nothing but gains nothing.
Which statement is true?
- Pass + odds and Don't Pass + lay odds are exact mirror images
- Only the flat bets are (almost) mirror images
- Lay odds have a house edge
- Take odds have a house edge
Show answer
B: Only the flat bets are (almost) mirror images. The flat bets mirror each other apart from the bar-12 push. The odds bets are different sizes and pay differently, and both are fair.