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?

1 Answer
Aug 14, 2017

#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#