Bayesian Liar's Dice

You can't see the computer's dice, but its bids give them away. Read them, and call it when it lies.

How to play

You and the computer each roll five dice. You can see only your own.
Take turns bidding on all the dice together: "at least q dice show face f". Each bid must go higher: more dice, or the same number of a higher face. Tap a square in the grid to bid.
Instead of bidding, you can call Liar! on the computer's bid. All dice are shown: if the bid was true, the caller loses a die; if not, the bidder does. Ones are not wild.
Lose all your dice and you lose the game.
The heat map in the grid shows how likely each bid is to be true, counting your dice and what the computer's bids give away. Under about 50%, calling is the better bet.

Playing with a class? Tap Join a class room and enter the code you were given, and your games go on the class leaderboard.

Your bid

Your dice

Computer's dice

Make a bid

Tap a square: "at least this many dice, across both hands, show this face".

The dice odds

P(q) = C(n, q) · (1/6)^q · (5/6)^n−q exactly q P′(q) = Σ_{x ≥ q} P(x) at least q

Bayes' rule on the computer's last bid

Prior Likelihood (rescaled to sum to 1) Posterior

You

Computer

Dice left

Rounds won

Your read on its hand

Expected number of each face among its hidden dice, from your posterior. Before it bids, every face averages one sixth of its dice.

What its bids give away

The computer bids what it thinks is likely to be true, judged from its own dice. Playing solo it treats yours as unknown, so it never reads you; you can read it. In a class room it has also learned how often this class bids a face they hold, and leans on that when deciding to call.

Prior: its dice are fair, so every hand has its usual chance. Likelihood: how probable this exact bid would be from each of the 252 possible hands, under its known way of bidding. Posterior: prior × likelihood, rescaled. A bid on fours is evidence of fours, but it bluffs sometimes, so it is never proof.

The heat map turns that into the chance each possible bid is true, counting your own dice too. Under about 50%, calling liar is the better bet.