| Move | Normal Prob. | Adjusted Prob. |
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📈 How the Math Works
Learn about the probability calculations that power the Liar's Dice Calculator.
Normal Probability
The normal probability is the chance a claim is true based on all unknown dice in the game, treating them as a single random pool. It doesn't consider the claimant's hand or bluffing.
1️⃣ Count Your Contribution
First, we count how many dice in your hand contribute to the claim. Ones are wild and count as any number.
2️⃣ Unknown Dice Pool
Next, we calculate the total number of dice that are unknown to you. This includes the dice held by all other players. We also determine the remaining needed dice to meet the claim.
3️⃣ Binomial Probability
We use the binomial formula to find the probability of getting exactly the required number of dice (or more) from the unknown pool. What's the probability the unknown dice provide what you need?
| k | Calculation | Value |
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4️⃣ Final Probability
Finally, we sum the probabilities of getting exactly the number of matches we need and all higher possibilities.
Adjusted Probability (Bayesian)
The adjusted probability is a more sophisticated calculation that factors in the claimant's dice and a bluffing model using Bayesian inference. It's the probability the claim is true, given that the claimant made it.
1️⃣ Count Your Dice Contribution
Look at your dice and count how many show the claimed number. Ones are wild and count as any number. The remaining needed dice must come from other players.
2️⃣ Calculate Prior Probabilities (Binomial)
We calculate the probability each possible number of claimant dice contribute to the claim using the binomial formula. The example table below shows the probabilities for to 5 matches from the claimant's dice. What are the likely distributions of the claimant's dice?
| k | Calculation | Value |
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3️⃣ Account for Bluffing with Bayes' Theorem
The claim might be exaggerated. We estimate the probability of the claim being made given a certain number of matches a player holds, considering a fixed bluff rate. When would the claimant make this claim given their hand?
4️⃣ Calculate Posterior Probabilities
We use Bayes' Theorem to adjust the prior probabilities based on the fact that a claim was made. This gives us a new probability distribution for the number of matches the claimant holds, called the posterior probability.
| k | Calculation | Value |
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5️⃣ Calculate Probability Claim is True
For each possible number of matches the claimant holds (\(K\)), we calculate the probability that the remaining unknown dice satisfy the claim. We use a binomial tail sum for this.
Generic Formula:
| \(k\) | Calculation | Value |
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6️⃣ Final Adjusted Probability
Finally, we multiply the posterior probabilities by the probability the claim is true for each and sum them up. This is the final adjusted probability the claim is likely true, considering both dice probability and bluffing.
| k | Posterior | P(true|k) | Weighted Product |
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