Question

Two 6-sided dice are rolled. What is the probability that their results differ by 1?

Answer 1

The probability is `10/36`, which reduces to `5/18`.

Explanation:

It is easy to construct a `6 xx 6` table of all possible pairs of dice rolls. Let's do this, but instead of filling the table with the rolls, let's fill it with the difference when that pair is rolled.

`" | 1 2 3 4 5 6"`
`"————————"`
`"1 | 0 1 2 3 4 5"`
`"2 | 1 0 1 2 3 4"`
`"3 | 2 1 0 1 2 3"`
`"4 | 3 2 1 0 1 2"`
`"5 | 4 3 2 1 0 1"`
`"6 | 5 4 3 2 1 0"`

Assuming the dice are fair and independent, the probability we seek is equal to:

`"number of favoured outcomes"/"number of total outcomes"`

which gives

`"P"("difference is 1")="number of ways the difference is 1"/"number of possible rolls"`

Looking at the table, we notice that, out of the 36 equally likely rolls, 10 of them result in a difference of 1. So we get

`"P"("difference is 1")= 10/36`,

which reduces to `5/18`.