Bank dice game odds
In the Bank dice game, every roll after the first three has a 1-in-6 (about 16.7%) chance of being a 7 and busting the pot. Here is the math behind every Bank decision: how often a pot survives, how big it gets, and why the expected pot always grows even though the risk never goes away.
What are the odds of rolling a 7 with two dice?
Two six-sided dice have 36 equally likely outcomes, and 6 of them add up to 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). That makes 7 the most common total: 6 in 36, or 1 in 6. Doubles are just as likely, also 6 in 36. The other 24 outcomes add their total to the pot, and those totals average exactly 7.
| Total | Ways to roll it | Chance |
|---|---|---|
| 2 or 12 | 1 each | 2.8% each |
| 3 or 11 | 2 each | 5.6% each |
| 4 or 10 | 3 each | 8.3% each |
| 5 or 9 | 4 each | 11.1% each |
| 6 or 8 | 5 each | 13.9% each |
| 7 | 6 | 16.7% |
How big is the pot after the three safe rolls?
During the first three rolls of a round, a 7 adds 70 points instead of busting, and any other roll adds its total. One safe roll is worth 17.5 points on average, so the pot averages 52.5 points when the danger phase begins. A single safe 7 is worth more than the average of all three rolls together, which is why you should never bank during the safe phase.
How likely is the pot to survive more rolls?
Each roll after the safe phase busts with probability 1/6, independently of what came before. The chance that the pot survives k more rolls is (5/6)k. Survival drops below 50% on the fourth danger roll.
| Danger rolls survived | Chance the pot is still alive | Chance it has busted |
|---|---|---|
| 1 | 83.3% | 16.7% |
| 2 | 69.4% | 30.6% |
| 3 | 57.9% | 42.1% |
| 4 | 48.2% | 51.8% |
| 5 | 40.2% | 59.8% |
| 6 | 33.5% | 66.5% |
| 8 | 23.3% | 76.7% |
| 10 | 16.2% | 83.8% |
Is one more roll worth the risk?
On average, yes. With a pot of P points, the next danger roll busts 1 time in 6, doubles the pot 1 time in 6, and adds an average of 7 points the other 4 times in 6. The expected pot after that roll is:
The doubles exactly cancel out the busts, so every extra roll adds about 4.67 points to the expected pot, no matter how big it already is. The catch is variance: the average keeps creeping up only because a few huge pots make up for the many that go to zero. Bank is won by finishing ahead of the other players, not by maximizing an average, so the scoreboard should drive your decision.
When should you bank?
We simulated 400,000 rounds where a player banks after a fixed number of danger rolls. Waiting longer raises the average a little, but makes the outcome much less certain.
| Bank after | Chance to bank | Avg. pot if banked | Avg. per round |
|---|---|---|---|
| Safe rolls only | 100% | 52 | 52 |
| 1 danger roll | 83% | 69 | 57 |
| 2 danger rolls | 69% | 88 | 61 |
| 3 danger rolls | 58% | 111 | 64 |
| 4 danger rolls | 48% | 140 | 67 |
| 5 danger rolls | 40% | 172 | 69 |
| 6 danger rolls | 34% | 213 | 71 |
| 8 danger rolls | 23% | 318 | 74 |
- Leading? Bank early. A sure 70 to 90 points keeps you ahead while everyone chasing you takes the risk.
- Trailing? You need the long tail. Staying in for big pots is the only way to catch up, and the expected value is on your side.
- Last one in? Bank as soon as the pot covers the gap you needed to close.
Does a long streak make a 7 more likely?
No. Dice have no memory: the chance of a 7 is 1 in 6 on every roll, whether the last ten rolls were 7-free or not. What a long streak does change is the size of the pot you stand to lose.
New to the game? Start with how to play Bank, or keep score at the table with the free printable Bank score sheet.
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