Dice Probability

Master dice probability step by step

Rolling Two Dice

With two dice, each die can show 1-6, so we have:

Each outcome is an ordered pair (Die 1, Die 2). For example, (3, 4) means die 1 shows 3 and die 2 shows 4.

	Die 2=1	Die 2=2	Die 2=3	Die 2=4	Die 2=5	Die 2=6
Die 1=1	(1,1)	(1,2)	(1,3)	(1,4)	(1,5)	(1,6)
Die 1=2	(2,1)	(2,2)	(2,3)	(2,4)	(2,5)	(2,6)
Die 1=3	(3,1)	(3,2)	(3,3)	(3,4)	(3,5)	(3,6)
Die 1=4	(4,1)	(4,2)	(4,3)	(4,4)	(4,5)	(4,6)
Die 1=5	(5,1)	(5,2)	(5,3)	(5,4)	(5,5)	(5,6)
Die 1=6	(6,1)	(6,2)	(6,3)	(6,4)	(6,5)	(6,6)

Each cell represents one equally likely outcome with probability

Even if the dice look identical, the outcomes are ordered pairs:

(3, 4)

First die shows 3, Second die shows 4

(4, 3)

First die shows 4, Second die shows 3

Both give sum = 7, but they are different outcomes in the sample space.

(1,6)

(2,5)

(3,4)

(4,3)

(5,2)

(6,1)

outcomes

Key Insight: Sum = 7 is the most likely sum when rolling two dice because it has the most ways to occur (6 combinations).