How to Play Liar's Poker With Dollar Bills: Bids, Challenges and Odds

In liar's poker with US dollar bills, each player holds one bill and bids on how many times a digit appears across all the serial numbers. Each bid must increase the quantity or keep the quantity and name a higher-ranking digit. The round ends when every other player challenges the last bid. Our calculation: holding three zeros in a three-player game, “5 zeros” is true 48.5% of the time. Under these assumptions, raising the standing bid is worthwhile only if the new bid is true at least 50.7% of the time—about 51%. Information from opponents' moves changes that estimate: player 1's bid pulls it up, while player 2's challenge pulls it down. Depending on the reading, the probability that “5 zeros” is true ranges from 18.7% to 100%.

This guide covers the eight-digit US-dollar game, not the card variant, Liar's Dice or Michael Lewis's book. Other serial-number lengths require different probability tables.

The worked deal comes from Wizard of Odds. We use it to compare challenging with raising, rather than simply judging the final result. Its three-player threshold of 25% applies only when the standing bid is certainly true, not to every decision.

1. Rules: one bill, a hidden number and a shared count

Each player needs one bill of any denomination, and everyone must agree on the stake and house rules before playing. The bill's denomination does not have to match the amount paid at the end of the round.

The version below draws on the Wizard of Odds rules, updated August 3, 2026, and Wikipedia's “Liar's poker,” revision dated March 30, 2024. Their different digit rankings are explained in the next section.

A bid is a claim about digits. A stake is the money used to settle the result. They are not interchangeable.

2. How to raise a bid and when challenges end the round

To raise the bid, either increase the quantity and name any digit, or keep the quantity and name a higher-ranking digit. Instead of raising, you may challenge the bid.

A bid of “q of digit d” means there are at least q occurrences of that digit across all bills, including yours. For example, after “three sixes,” either “three sevens” or “four fives” is legal under both rankings below.

Wikipedia describes 1 as the highest digit, an “ace,” followed by 0 as “10,” with 2 lowest. The Wizard of Odds author says “I prefer” 0 lowest and 9 highest. Neither is treated here as the only correct ranking. The ranking does not change a bid's probability, but it changes which equal-quantity raises are legal.

Play proceeds as follows:

1. A player raises the standing bid or challenges it.
2. If not every other player has challenged the bid, the round continues: the next player may raise it.
3. Once every other player has challenged the last bid, reveal the serial numbers and count the relevant digit.
4. If the count meets or exceeds the bid, the bidder receives one stake from each opponent. Otherwise, the bidder pays one stake to each opponent.

Players who trust one another may simply report their counts. However, the player whose bid was challenged may require everyone to show their bills.

3. How many matching digits to expect

For a specified digit, the expected count across the table is 0.8 × the number of players, assuming independent, equally likely digits. This is a probability model, not an audit of how real serial numbers are printed.

Each position is assigned a 0.1 probability of containing the chosen digit. Its count follows a binomial distribution: the number of successes in independent trials with a fixed probability p. This definition follows the NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.18, accessed October 17, 2026.

Under this model, earlier bills do not change the next bill. The gambler's fallacy simulator explains why a sequence of results does not make a particular outcome “due.”

Repeats on one bill

The most common maximum repeat on one bill is a pair. Here, “three alike” means the largest group of identical digits has size three—not that a previously chosen digit appears three times.

This Wizard of Odds table was reproduced by exact enumeration. Percentages throughout are rounded, so totals may differ from 100%.

A specified digit across the table

Before you inspect your bill, the probability depends on the total number of digits: 8n, where n is the number of players. The table gives the probability that a specified digit appears at least q times.

The highest blind bid quantity with at least a 50% chance of being true is 1 with two players, 2 with three, 3 with four, 4 with five and 5 with six.

The original repeat table and the special-case raise threshold discussed below come from the Wizard of Odds rules and probability tables. The whole-table and known-bill calculations use the independent-digit model described above.

4. How your serial number changes the odds

Subtract the digits you can see from the count required by the bid. If your bill contains h of the relevant digit, a bid of “q of digit d” is true when the other bills contain at least q − h.

With three players, the two opponents hold 16 unknown digits. Without using information from their moves, the probabilities are:

Each additional known occurrence shifts the probabilities one column to the right: the opponents need one fewer occurrence to make the bid true.

Player count also matters. With four players, there are 24 unknown digits:

The accounting resembles blockers and card removal in a bluff-catch: known information reduces the unknown part of the problem. The analogy stops there. Under the independent-digit model, holding three zeros does not make a zero less likely on an opponent's bill.

5. Challenge or raise: which threshold applies?

The raise threshold depends on how likely the standing bid is to be true. The three-player threshold of 25% applies only when that bid is certain; as it becomes less credible, challenging becomes more attractive and the raise threshold rises.

The formulas require both of these conditions:

• Your challenge ends the round because every other opponent has already challenged the standing bid.
• If you raise the bid, every opponent will challenge your new bid.

Let the stake paid to one opponent equal 1, the player count be n, the probability that the standing bid is true be s, and the probability that your raised bid is true be p. The expected value (EV) of each decision, measured in stake units, is:

• Challenge the bid: EV = 1 − 2s.
• Raise the bid and face challenges from everyone: EV = (n − 1)(2p − 1).

Setting these equal gives the threshold:

p* = 1/2 + (1 − 2s) / (2(n − 1)).

This is the point where the two actions have equal EV, not a promise of a positive return. Challenging itself breaks even at s = 0.5. Comparing probability with financial consequences is also the principle behind break-even equity from pot odds, although this game uses a different formula.

When the standing bid is certainly true

At s = 1, challenging certainly loses one stake, and the raise threshold simplifies to (n − 2) / (2n − 2). This is the assumption used by Wizard of Odds.

Under these assumptions, raising is no worse than challenging when p meets or exceeds the threshold. Both choices can still have negative EV.

When the standing bid may be false

The lower s is, the stronger your new bid needs to be.

At s = 0.5, raising is at least as good as challenging only when p ≥ 50%. Compare the threshold with the probability of a raise you can legally make.

6. Worked deal: five zeros, but no sixth

From player 3's seat, ignoring information from opponents' moves, challenging “5 zeros” is better than raising to “6 zeros.” Their probabilities are 48.5% and 21.1%, while the raise threshold is 50.7%. The eventual result does not replace an assessment based on information available at the time.

The Wizard of Odds example has three players:

The deal uses the Wizard of Odds ranking, with 0 lowest and 9 highest. Under Wikipedia's ranking, move 2—“2 fives” after “2 zeros”—and move 4—“3 eights” after “3 zeros”—would be illegal.

The moves are:

1. Player 1 bids “2 zeros.”
2. Player 2 bids “2 fives.”
3. Player 3 bids “3 zeros.”
4. Player 1 bids “3 eights.”
5. Player 2 bids “3 nines.”
6. Player 3 bids “4 zeros.”
7. Player 1 bids “5 zeros.”
8. Player 2 challenges the bid.
9. Player 3 raises the bid to “6 zeros.”
10. Players 1 and 2 challenge the bid.

Three eight-digit serial numbers with the zeros highlighted: 2 + 0 + 3 give 5 zeros in total, so a bid of 5 zeros is true and a bid of 6 zeros is false.Teaching diagram of the three-player deal from Wizard of Odds. Observation: the highlighted zeros add up to 5. Calculation from player 3's seat, with 3 zeros in hand and no information from the other players' moves: "5 zeros" is true 48.5% of the time, "6 zeros" 21.1%.

The players reveal the serial numbers and find five zeros: 2 + 0 + 3. That is fewer than six, so player 3 pays each opponent $1.

What player 3 could decide from their bill alone

With only player 3's bill known, “5 zeros” is true 48.5% of the time: at least two zeros are needed among 16 unknown digits. Wizard of Odds gives the same probability as 0.485272 before rounding.

The probability of “6 zeros” is 21.1%. Every legal raise under the chosen ranking, including “5 twos,” has a probability no greater than 21.1%. With s = 48.5%, the raise threshold is 50.7%, well above every available raise.

The rounded EV estimates in stake units are:

• Challenge “5 zeros”: +0.03.
• Raise to “6 zeros”: −1.16.

These calculations use independent, equally likely digits and the round-ending assumptions in the previous section. In the actual deal, “5 zeros” was true, so challenging it would have lost. EV evaluates the decision before the numbers are revealed, not whether every individual outcome succeeds.

How reading the moves changes the decision

Player 1's bid can suggest more zeros, while player 2's challenge can suggest fewer. The following calculations are conditional on particular readings; the moves do not establish those readings as facts.

Raise EV refers to “6 zeros,” and the monetary unit is one stake. The raise threshold follows the section 5 formula at each row's own s: for example 65.7% at s = 18.7%, and 25% only at s = 1, when those assumptions make the standing bid certainly true.

Under these assumptions, challenging beats raising in five of the six rows: the row that uses no information from the moves and every row that takes player 2's challenge into account. Raising has positive EV only in the last row, which trusts player 1's bid by assuming they have at least two zeros and ignores player 2's challenge.

Reading player 2 as having no zeros is an extreme interpretation, not knowledge. It happened to match the revealed bill. The Wizard of Odds author writes:

Had player 2 had a zero then player 3 would have won.

How sensitive the result is to the assumptions

The baseline 48.5% changes with serial-number length, the assumed probability of a zero or the zeros in your hand. Each row changes only the stated condition from the baseline calculation, without using information from moves.

The 7- and 10-digit rows illustrate model sensitivity; they do not describe particular currencies.

7. What these numbers leave out

The tables do not automatically account for information in bids, opponents' behavior or departures from the independent, uniform-digit model. A bid may reflect the bidder's bill or be a bluff—see what a bluff is—so it cannot be treated as certain knowledge.

Keep the calculation's boundaries separate:

• The probability model is an assumption, not a printing audit. Its results are conditional on the digit probabilities used.
• Reads need their own justification. The readings table calculates the consequences of assumptions; it does not prove them.
• House rules define the available actions. Agree on digit ranking, turn order and the stake before playing.
• Serial-number format determines which table applies. Use calculations suited to the number of digits in play.
• The formula covers round-ending decisions only. The threshold formula does not model further bidding after your move.

This is a zero-sum game: what one player wins, the others lose. There is no casino house edge in the calculations described here. This guide does not assess the legality of playing for money in any jurisdiction.

8. Games and works not to confuse with liar's poker

This guide is about bids on digits in bill serial numbers. Similar names do not mean the rules are interchangeable.