The odds in Liar’s Dice come from one simple fact: each die you can’t see shows the face you are thinking about one time in six, or one time in three when ones are wild. From that single number you can work out how many matching dice to expect and how likely any bid is to be true. This guide turns it into tables you can use at the table, walks through four real decisions, and shows how the computer players in our Liar’s Dice game use the same arithmetic.
Every number here was checked with an exact binomial calculation, and every example uses the rules on this site: 2 to 6 players, five dice each by default, bids that only go up, and wild ones unless you switch them off. If you need the rules first, start with the Liar’s Dice rules guide.
The one number behind every bid
A bid such as “six 4s” claims that at least six dice on the whole table show a 4. You can see your own dice, so that part of the claim is certain. Everything else depends on the dice you can’t see, and each of those is an ordinary random die:
- Without wild ones, a hidden die shows a 4 one time in six.
- With wild ones, a hidden die counts if it shows a 4 or a 1: two faces out of six, so one time in three.
That is the whole model. The question at every turn is how many hidden dice have to match for the bid to be true, and how likely that is.
Step 1: count the dice in your hand that count for the bid (the face, plus your 1s when ones are wild). Step 2: subtract that from the bid’s quantity. The answer is what the hidden dice must supply. Step 3: look up the chance in the tables below.
Work out the odds of a bid
44%chance the bid is true
A bid of 6 needs 4 of the 10 dice you can’t see, and about 3.3 of them should count. Calling Liar wins 56% of the time.
How many matching dice to expect
On average, a third of the hidden dice count for any bid face with wild ones, and a sixth without. With five dice each, the hidden dice are everyone else’s:
| Hidden dice | Typical table (five dice each) | Expected matches, ones wild | Expected matches, ones not wild |
|---|---|---|---|
| 5 | 2 players | 1.7 | 0.8 |
| 10 | 3 players | 3.3 | 1.7 |
| 15 | 4 players | 5.0 | 2.5 |
| 20 | 5 players | 6.7 | 3.3 |
| 25 | 6 players | 8.3 | 4.2 |
So a quick estimate of the true count on the table is your matching dice plus a third of the hidden dice (or a sixth without wild ones). A bid near that number is reasonable. Each die above it makes the bid much less likely, as the tables show.
Liar’s dice probability tables
Find the column for the number of hidden dice and the row for the matches the bid needs from them. The cell is the chance the bid is true. The chance that a call of Liar wins is 100% minus that. A dash means the bid needs more matches than there are hidden dice, so it is certainly false.
With wild ones
| Matches needed | 5 hidden | 10 hidden | 15 hidden | 20 hidden | 25 hidden |
|---|---|---|---|---|---|
| 1 | 87% | 98% | over 99% | over 99% | over 99% |
| 2 | 54% | 90% | 98% | over 99% | over 99% |
| 3 | 21% | 70% | 92% | 98% | over 99% |
| 4 | 5% | 44% | 79% | 94% | 99% |
| 5 | under 1% | 21% | 60% | 85% | 95% |
| 6 | – | 8% | 38% | 70% | 89% |
| 7 | – | 2% | 20% | 52% | 78% |
| 8 | – | under 1% | 9% | 34% | 63% |
| 9 | – | under 1% | 3% | 19% | 46% |
| 10 | – | under 1% | 1% | 9% | 30% |
Without wild ones
| Matches needed | 5 hidden | 10 hidden | 15 hidden | 20 hidden | 25 hidden |
|---|---|---|---|---|---|
| 1 | 60% | 84% | 94% | 97% | 99% |
| 2 | 20% | 52% | 74% | 87% | 94% |
| 3 | 4% | 22% | 47% | 67% | 81% |
| 4 | under 1% | 7% | 23% | 43% | 62% |
| 5 | under 1% | 2% | 9% | 23% | 41% |
| 6 | – | under 1% | 3% | 10% | 23% |
| 7 | – | under 1% | 1% | 4% | 11% |
| 8 | – | under 1% | under 1% | 1% | 4% |
Look down any column and you will see the same shape: a bid stays very likely up to about the expected count, then drops off quickly. With 15 hidden dice and wild ones, needing 5 is a little better than even (60%), needing 6 is 38%, and needing 7 is only 20%. One extra die in a bid can change it from a fair claim to a likely lie.
- At least 2
- 98%
- At least 3
- 92%
- At least 4
- 79%
- At least 5
- 60%
- At least 6
- 38%
- At least 7
- 20%
- At least 8
- 9%
- At least 9
- 3%
Fifteen hidden dice, ones wild: the chance stays high up to 5, then falls fast.
The 50% line
The simplest thing to remember is the most a bid can ask of the hidden dice and still be more likely true than false:
| Hidden dice | 5 | 10 | 15 | 20 | 25 |
|---|---|---|---|---|---|
| Ones wild: up to | 2 | 3 | 5 | 7 | 8 |
| Ones not wild: up to | 1 | 2 | 2 | 3 | 4 |
Add your own matching dice to these numbers and you have the highest bid on that face that is more likely true than false, before you take anyone’s bids into account.
When only a few dice are left
Late in a game the hidden dice are few, and every one matters. These are the chances that at least 1, 2 or 3 of the last hidden dice match:
| Hidden dice | 1 match, wild | 2 matches, wild | 3 matches, wild | 1 match, not wild | 2 matches, not wild |
|---|---|---|---|---|---|
| 1 | 33% | – | – | 17% | – |
| 2 | 56% | 11% | – | 31% | 3% |
| 3 | 70% | 26% | 4% | 42% | 7% |
| 4 | 80% | 41% | 11% | 52% | 13% |
Four worked examples
1. Two players: a call that beats every raise
Five dice each, wild ones. You hold 1, 4, 4, 2, 6, so you have three 4s (two 4s and the wild 1). Your opponent bids “five 4s”. You hold three, so the bid needs two 4s or 1s among their five dice: 54% likely. Calling Liar wins only 46% of the time.
Now look at your raises. “Five 6s” or “six 4s” each need three matches from the five hidden dice: 21%. “Five 5s” or “six 2s” need four: 5%. Every raise is far less likely than the current bid, so if your opponent calls you, you will probably lose a die. A call that wins almost half the time is the best move here, even though the bid is slightly more likely true than not.
2. Four players: keep raising on your face
Five dice each, 20 dice in play, 15 of them hidden, wild ones. You hold 6, 6, 1, 3, 5: three 6s. The bid reaches you at “seven 6s”. You need four from the hidden fifteen: 79%. Don’t call. Raise to “eight 6s”, which needs five: 60%. If the bid comes back around at “nine 6s” (38%) or “ten 6s” (20%), the numbers say call. Remember that the other players bidding 6s probably hold some, which pushes the real odds up a little.
3. No wild ones: lower bids, earlier calls
Three players, five dice each, ones not wild, 10 hidden dice. You hold 3, 2, 4, 5, 6, a single 3. A bid of “four 3s” needs three 3s from ten hidden dice: 22%. Calling wins about four times in five. The smaller bid “three 3s” needs two: 52%, a coin flip. Without wild ones, bids that look modest are often already too high.
4. The endgame: one die each
Two players, one die each, wild ones. You hold a 5. Your opponent opens with “one 6”. By pure chance their die is a 6 or a 1 one time in three, so calling wins 67% of the time. Your only raises are at two dice, such as “two 5s”, which needs their die to be a 5 or a 1: 33%. Calling is the better bet, but your opponent chose 6s for a reason, and with one die left a bid is often an honest one. This is where reading the player matters as much as the table.
How to use the numbers at the table
- Count the hidden dice every turn. The total drops by one each round and players drop out. A bid that was safe in the first round can be wild in the fifth.
- Compare the bid with your raises. The question is not only “is this bid true?” but “is any raise I could make more likely to survive?” When every raise is worse, call.
- Treat bids as clues. Probability assumes the hidden dice are random, but a player who bids 4s probably holds a 4 or a 1. Shift your estimate a little toward faces other players have bid.
- Raise on faces you hold. Your own matching dice are certain, so they lower what the hidden dice must supply.
- Raise in small steps. The smallest legal raise keeps your bid as likely as possible and leaves the next player a believable bid to deal with.
The Liar’s Dice strategy tutorial has more on opening bids, bluffing and reading the table.
How the computer players think
The computer players in our game never see your dice. Each one sees exactly what a person in its seat would see: its own dice, how many dice everyone has, and the bids so far. From there the three levels use the same numbers as the tables above, in different ways.
- Easy uses the expected count with some noise. It calls Liar when a bid is above what it expects, give or take a die or two, and otherwise raises, usually on the face it holds most and sometimes on any face. It makes the kinds of mistakes a new player makes.
- Medium works out the exact probability of each bid, like the tables here. It calls a bid that has about a one-in-three chance of being true or less, or a doubtful bid when every raise it could make is riskier, much like the first worked example. Otherwise it makes the smallest raise that is most likely to be true, preferring faces it holds, with an occasional bluff.
- Hard weighs the risk of calling (losing if the bid is true) against the risk of raising (losing if its raise is false and someone calls it). It treats each other player’s latest bid as a weak hint that they hold that face, bluffs on purpose now and then with a plausible bid on a face it doesn’t hold, and sometimes opens on its second-best face so its first bid doesn’t give its hand away.
That makes Medium a good practice partner for the odds: if you can predict its calls from the tables, you are counting correctly. Hard is where reading bids starts to matter.
Where the odds stop helping
The tables assume every hidden die is random, and in a real game they are not quite. Players bid on faces they hold, bluff to steer the bidding and remember what happened last round. Use the odds as your starting point and the bids as your adjustment. Also remember that a bid means “at least”: an exact count means the bid was true and the caller loses a die.
Try it yourself
The fastest way to get a feel for these numbers is to play a few games where the dice are rolled and counted for you. Liar’s Dice runs on one phone, tablet or computer with 2 to 6 players, each a person or a computer player, and a pass screen keeps every hand private between turns. For more dice games, try Yacht Dice, see the scoring chart for the 10,000 dice game, or browse every dice game on the site. If bluffing is your favourite part, Imposter is a word game built on it.
