David vs Goliath Dice

A collectible dice set with a surprise victory for the underdog

This set of dice has a quite curious property. The bigger die (named Goliath) and the smaller die (named David) are both balanced - just like a standard D6, they have pip values adding up to 21. However, when you roll them against each other, something strange happens.

If you roll each die once and compare the results, Goliath rolls a higher number with probability greater than that of David. The same is true if you roll each die twice and add up the results, or if you roll three times, or five, six or more times, Goliath always wins with probability greater than David...

However, if you roll exactly four times, then - and only then - David wins with greater probability. For any other number of rolls, Goliath is more likely to win. How can this be?

Probabilities can be very tricky! To understand a little bit better why Goliath wins, let's look at all the possibilities for 1 roll:

If we roll the dice once each, five of the possible outcomes are a tie. We are left with 31 possible different outcomes - but since 31 cannot be divided by two, one of the dice must have the advantage! In this case, Goliath has the advantage over David - even though they have the same total number of pips on each die.

For two rolls, Goliath still holds the advantage, and also for three rolls... but when you roll each die four times, David wins on 789,540 of the possible outcomes and Goliath wins on only 789,407. This means David has an advantage of 133 over Goliath! Isn't that strange?

Beyond four, Goliath holds the advantage for every number of rolls.

The David and Goliath Dice are designed by Ivo David & Yogev Shpilman. To read more about the maths behind the dice, take a look at the paper The Paradox of Anti-Inductive Dice, by Summer Eldridge, Ivo David de Oliveira and Yogev Shpilman.

You can get hold of your own set of David and Goliath dice right here on Maths Gear!