Question

You have three dice: one red (R), one green (G), and one blue (B). When all three dice are rolled at the same time, how do you calculate the probability of the following outcomes: a different number on all dice?

Answer 1

`5/9`

Explanation:

The probability that the number on the green die is different from the number on the red die is `5/6`.

Within the cases that the red and green dice have different numbers, the probability that the blue die has a number different from both of the others is `4/6 = 2/3`.

Hence the probability that all three numbers are different is:

`5/6 * 2/3 = 10/18 = 5/9`.

`color(white)()`
Alternative method

There are a total of `6^3 = 216` different possible raw outcomes of rolling `3` dice.

• There are `6` ways to get all three dice showing the same number.

• There are `6 * 5 = 30` ways for the red and blue dice to show the same number with the green die being different.

• There are `6 * 5 = 30` ways for the red and green dice to show the same number with the blue die being different.

• There are `6 * 5 = 30` ways for the blue and green dice to show the same number with the red die being different.

That makes a total of `6+30+30+30 = 96` ways in which at least two dice show the same number, leaving `216-96=120` ways in which they are all different.

So the probability that they are all different is:

`120/216 = (5 * color(red)(cancel(color(black)(24))))/(9 * color(red)(cancel(color(black)(24)))) = 5/9`