Bunco Dice Game & Score Odds
Play the party dice game at a full table, with exact odds. It contains no decisions at all — and yet the seat you sit in still changes your chances.
Round 1 of 6 — hunting 1s
Score anything and you roll again; score nothing and the dice pass to the next seat. The first player to 21 takes the round. You are never asked to choose anything, which turns out to be the point.
The table
| Seat | Player | This round | Rounds won | Buncos | Points |
|---|---|---|---|---|---|
| 1 | You | 0 | 0 | 0 | 0 |
| 2 | Dana | 0 | 0 | 0 | 0 |
| 3 | Ravi | 0 | 0 | 0 | 0 |
| 4 | Nina | 0 | 0 | 0 | 0 |
There are no decisions in this game. None at all.
That sounds like a throwaway line about a party game, so here is the argument in full: every moment where a player could conceivably act, and what actually determines it.
| Moment | Who decides |
|---|---|
| Whose turn it is | Fixed by seating order |
| Which number scores | Fixed by the round number |
| Whether to roll | Compulsory |
| Which dice to keep | All three are rerolled, always |
| Whether to roll again | Forced: yes if you scored, no if you did not |
| When the round ends | Fixed at 21 points, by the dice |
Count of moments where the player chooses: 0. You cannot decline to roll, cannot stop while ahead, cannot keep a die, and cannot pick the target — the round number sets it. The player supplies no information to the game at any point, which makes Bunco a Markov chain that people take turns observing.
Worth contrasting with Pig, on this site, which is the exact opposite: Pig is nothing but a decision, and its whole analysis is about when to stop rolling. Bunco deletes that question. It is a social occasion with dice in it, and the dice are not asking you anything.
You have no decisions — but your seat has an edge
Here is the one thing about a Bunco table that is not symmetric. A round ends the moment somebody reaches 21, and the seats roll in a fixed order. If seat one gets there on the sixth cycle, seat four never rolls a sixth time — the round is already over. So the earlier seats get a shot the later seats do not, and the advantage is real even though nobody at the table ever makes a choice.
This is not simulated. A turn that starts on score s can be solved exactly by working down from 20, because every roll that continues the turn moves the score up and never down — so the answer for s only ever depends on answers already computed. Chaining those turns gives the exact distribution of how many turns one player needs to reach 21: a median of 19 and a mean of 19.17. From there, seat i takes the round when it finishes on some cycle, every earlier seat needs longer, and every later seat has not finished sooner.
| Table | Fair share | Seat 1 | Seat 2 | Seat 3 | Seat 4 | Seat 5 | Seat 6 |
|---|---|---|---|---|---|---|---|
| 2 players | 50.00% | 51.77% | 48.23% | — | — | — | — |
| 3 players | 33.33% | 35.17% | 33.28% | 31.55% | — | — | — |
| 4 players | 25.00% | 26.71% | 25.52% | 24.41% | 23.37% | — | — |
| 6 players | 16.67% | 18.10% | 17.49% | 16.90% | 16.35% | 15.83% | 15.33% |
At the four-player table the real game uses, seat one takes 26.71% of rounds against seat four's 23.37%, where an even split would be 25%. Seat one wins 14.3% more often than seat four for no reason other than going first. The edge shrinks as the table grows, because with more seats any particular one is less likely to be the one that ends the round.
Worth being honest about what that does and does not mean. It is a per-round edge with the order held fixed, which is what happens above — every round here restarts at seat one. A real Bunco night rotates the dice and shuffles players between tables precisely so nobody keeps the first chair, which washes the whole thing out. The advantage is genuine and the etiquette that cancels it is older than the arithmetic.
Most turns score nothing at all
All 216 outcomes of three dice, enumerated rather than sampled:
| Roll | Scores | Outcomes | Chance |
|---|---|---|---|
| nothing | 0 | 120 of 216 | 55.56% |
| one target die | 1 | 75 of 216 | 34.72% |
| two target dice | 2 | 15 of 216 | 6.94% |
| a triple of another number | 5 | 5 of 216 | 2.31% |
| Bunco | 21 | 1 of 216 | 0.46% |
A roll scores something 96 times in 216, which is exactly four ninths. Because a turn ends the instant you fail, a turn scores nothing at all five ninths of the time — 55.6%. Picking up three dice, putting them down again and passing them on is the single most common experience the game offers.
The rest follows in closed form rather than needing simulation. A roll is worth 0.699 points on average; a turn runs 1.8 rolls, exactly nine fifths, and is worth 1.258 points. A turn reaching 21 or more happens 0.83% of the time, and an actual Bunco lands once in 216 rolls.
| Turn score | How often |
|---|---|
| 0 | 55.56% |
| 1 | 19.29% |
| 2 | 10.56% |
| 3 | 5.00% |
| 5 | 2.49% |
| 4 | 2.47% |
A triple of the wrong number beats two of the right one
One oddity falls out of that table. Two target dice come up 15 times in 216 and score 2. A triple of some other number comes up 5 times — three times rarer — and scores 5. Rarity and reward agree, which is defensible. But it does mean the number you are supposedly hunting all round is worth less than an accident involving any of the other five, and a player chasing the round number has no way to prefer one over the other anyway. Every figure on this page is exact, from enumerating all 216 outcomes, and each is cross-checked against a simulation of hundreds of thousands of turns.
How to use
- Choose how many players sit at the table and which seat is yours.
- Press Roll to throw three dice at the current round number.
- Each die matching the round number scores a point.
- Three of the round number is a Bunco, worth 21.
- Three of any other number is worth 5.
- Scoring lets you roll again; scoring nothing passes the dice to the next seat.
- The first player to 21 takes the round, and six rounds decide the table.
Frequently asked questions
What are the rules of Bunco?
Six rounds, numbered one to six. In each round you roll three dice and every die showing that round number scores a point. Three of the round number is a Bunco and scores 21. Three of any other number scores 5. If you score anything you roll again, and if you score nothing your turn ends and the dice pass on.
Is there any skill in Bunco?
None whatsoever, and that is worth stating plainly rather than hedging. We listed every moment in the game where a player might act: whose turn it is, which number scores, whether to roll, which dice to keep, whether to roll again, and when the round ends. Every one is fixed by the rules or the dice. The player supplies no information to the game at any point.
How often does a Bunco turn score nothing?
More than half the time. Of the 216 outcomes of three dice, 120 contain no die showing the target and are not a triple, so a roll scores something exactly 96 times in 216, which is four ninths. Since a turn ends the moment you fail, a turn scores nothing at all five ninths of the time — 55.6%.
How many rolls does a turn last?
Exactly nine fifths on average, which is 1.8. Because you continue precisely when you score, and that happens with probability four ninths, the number of rolls in a turn is geometric and its mean has a closed form rather than needing simulation.
How many points is a Bunco turn worth on average?
About 1.26 points. Each roll is worth 151/216, or 0.699 points on average, and a turn contains 1.8 rolls, so the expected total per turn is 1.258. A turn reaching 21 or more happens only 0.83% of the time.
What are the odds of rolling an actual Bunco?
One in 216 per roll, since all three dice must show one specific number. That is the same chance as any other named triple, which is why three of a wrong number is a common consolation and three of the right one is the event the game is named after.
Why is three of the wrong number worth more than two of the right one?
Because the scoring ladder tracks rarity rather than relevance. Two target dice come up 15 times in 216 and score 2, while a triple of some other number comes up 5 times and scores 5. It does mean the number you are supposedly hunting is worth less than an accident, which is one of the odder corners of the rules.
How many players do you need for Bunco?
Four to a table is the standard, which is why the table here starts at four, and a full Bunco night runs several tables of four at once. You can set the table to two, three, four or six players above. The rules do not change with the count — only how long you wait between turns, and how likely you are to be the one who reaches 21 first.
Does it matter which seat you sit in?
Yes, and this is the one asymmetry in a game with no decisions in it. A round ends the instant somebody reaches 21, so if an earlier seat gets there first, the later seats never take their turn that cycle. Solving it exactly rather than simulating it, seat one takes 26.71% of rounds at a four-player table against seat four’s 23.37%, where an even split would be 25% — about 14% more often, purely for going first. A real Bunco night rotates the dice and moves players between tables, which is exactly what cancels it out.
How long does a round of Bunco take?
One player needs a median of 19 turns to get from nothing to 21, and 19.17 on average. That is longer than the 1.26 points a turn is worth would suggest, because the turn that finishes the round usually overshoots 21 rather than landing on it — a Bunco alone is worth 21 from a standing start. With four players a round is over once the fastest of them gets there.
Does this send anything anywhere?
No. The dice are rolled and the scores kept entirely in your browser, and nothing is uploaded.
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