Why do two dice make seven more often than two?

Two fair dice can each land on any number from one to six. Yet adding their numbers does not make every total equally likely. Seven has many ways to appear. Two has only one. The surprise comes from counting the number pairs hiding behind each total.

Imagine one blue die and one orange die, rolled independently so neither result affects the other. If the blue die shows one, the orange die can show any of its six numbers. The same is true when blue shows two, three, four, five or six. There are six groups of six pairs: 36 equally likely pairs under this fair-dice model.

To make two, both dice must show one. To make seven, the pairs can be 1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, or 6 and 1. Blue one with orange six is a different outcome from blue six with orange one. Seven therefore has six of the 36 possible pairs; two has just one.

That makes seven six times as likely as two on each roll in this model. It does not promise that seven will appear six times in the next 36 rolls. Short runs can be uneven. A missing seven is not owed to the players: a fresh independent roll has the same chances as before.

This is why a board game that moves a piece by the sum of two dice behaves differently from one that chooses all totals equally. The dice do not favor the middle numbers printed on their faces. There are simply more pairs that add up to a middle total. A game’s small rules can create a surprisingly rich pattern.