Liar's dice

• Instead of the current player being the only one who can raise the bet, challenge (or "call up") the previously-made bid, any player may raise or challenge a bid at any time. The first challenge made ends the round, and the challenger closest to the current bidder in the direction of play has priority if multiple players challenge at the same time.
• If played with the above variant, the player who made the last bid may count aloud from 1 to 10. If he reaches 10 with no one challenging or increasing the bid, the round ends with that player earning back a die. A player may have more than 5 dice that way, and any player who reaches 10 dice that way wins the game.
• With the above-mentioned variants, some players may stay quiet and win easily. To avoid that, the following rule may be added: Each time a player loses a challenge, he loses a die normally, but the two players sitting to their left and right lose a die as well (unless one of them was the player to win the challenge).
• Another solution to the above-mentioned variants is to force all players to choose a side: Each player holds a two-sided item (preferred a coin or a card), and decides which side means 'true', and which means 'lie'. When a player challenges, all players must join the challenge, placing their items on the table on either 'true' or 'lie', hidden beneath their hands. Once all players have joined, the items are revealed and the table is divided into players who support either side of the challenge. Every player on the losing side loses a die at the end of the challenge.
• With some bidding systems, a player may elect to choose one or more dice of matching value from under their cup, place them outside the cup in view of the other players, re-roll the remaining dice, and make a new bid of any quantity of that face value.
• When a player has no two dice with the same face, he may choose to pass once in a game round. If he does so, the bid will not be raised. The next player can raise the bid using standard rules, or call the bluff. By doing so, he challenges the claim of the passing player having no two dice with the same face. This is commonly used in multi-round games where dice are removed from the game, as it helps players with few dice left to gain more information about the other dice without risk.
• If a bidder is challenged, yet their bid was "spot on", they may win back a die.
• Instead of raising or challenging, the player can claim that the current bid is exactly correct ("Spot On"). If the number is higher or lower, the player loses to the previous bidder, but if they are correct, they win. A "spot-on" claim typically has a lower chance of being correct than a challenge, so a correct "spot on" call sometimes has a greater reward, such as the player regaining a previously lost die or all other players losing a die.

As with any game of chance, probability is highly important. The key element is the "expected quantity": the quantity of any face value that has the highest probability of being present. For six-sided dice, the expected quantity is one-sixth the number of dice in play, rounded down. When wilds are used, the expected quantity is doubled as players can expect as many aces, on average, as any other value. Because each rolled die is independent of all others, any combination of values is possible, but the "expected quantity" has a greater than 50% chance of being correct, and the highest probability of being exactly correct. For example, when 15 dice are in play and wilds are used, the expected quantity is 5. The chances of a bid of 5 being correct are about 59.5%; in contrast, the chances of a bid of 8 being correct are only about 8.8%.

However, a high bid is not necessarily incorrect, because bids incorporate information the player knows. A player who holds several dice of a single value (for instance, four out of the five dice in their hand are threes) may make a bid, with fifteen dice on the table, of "six threes". To an outside observer who sees none of the dice, this has an extremely low probability of being correct (even with wilds), however since the player knows the value of five of those dice, the player is actually betting that there are two additional threes among the ten unknown dice. This is far more likely to be true (about 40%).

Each bid gives others at the table information. Players, through subsequent bids, reveal the players' confidence in the quantity of each face value rolled. A player with two or three of a certain face value under his or her own cup may make a bid favoring that face value. Players can thus use these bids to build a mental picture of the unknown values, which either strengthens or weakens their confidence in a bid they are considering. Others may consider a bid as evidence it is true, and if their own dice support the same conclusion, may increase the bid on that face value, or if their dice refute it may bid on a different face, or challenge the previous bid.

Conversely, bids can also be bluffs. Bluffs in liar's dice can be split into two main categories: early bluffs and late bluffs. An early bluff is likely to be correct by simple probability (depending on the number of players), but other players may believe the bidder made that bid because his or her dice supported it. Thus, the bluff is false information that can lead to incorrect higher bids being made on that face value. Players will thus attempt to trick other players into overbidding by use of early bluffs to inflate a particular face value. A late bluff, on the other hand, is usually less voluntary; the player is often unwilling to challenge a bid, but as a higher bid is even more likely to be incorrect it is even less appealing. A late bluff is thus a critical part of the game; convincing bluffs, as well as reliable detection of bluffs, allow the player to avoid being challenged on an incorrect bid.

Playing Liar's dice involves interpersonal skills similar to other bluffing games such as poker. Being able to reliably detect bluffs through giveaways, or "tells", and analyzing a player's bidding history for patterns that can indicate the likelihood of a bluff, are important skills here just as in poker.

For a given number of unknown dice n, the probability that exactly a certain quantity q of any face value are showing, P(q), is

${\displaystyle \ P(q)=C(n,q)\cdot (1/6)^{q}\cdot (5/6)^{n-q}}$

Where C(n,q) is the number of unique subsets of q dice out of the set of n unknown dice. In other words, the number of dice with any particular face value follows the binomial distribution ${\displaystyle B(n,{\tfrac {1}{6}})}$.

For the same n, the probability P'(q) that at least q dice are showing a given face is the sum of P(x) for all x such that q ≤ x ≤ n, or

${\displaystyle \ P'(q)=\sum _{x=q}^{n}C(n,x)\cdot (1/6)^{x}\cdot (5/6)^{n-x}}$

These equations can be used to calculate and chart the probability of exactly q and at least q for any or multiple n. For most purposes, it is sufficient to know the following facts of dice probability:

• The expected quantity of any face value among a number of unknown dice is one-sixth the total unknown dice.
• A bid of the expected quantity (or twice the expected value when playing with wilds), rounded down, has a greater than 50% chance of being correct and the highest chance of being exactly correct.[1]