Question

Two symmetric dice have had two of their sides painted red, two painted black, one painted yellow, and the other painted white. When this pair of dice is rolled, what is the probability both dice land on different colors?

Answer 1

First let us calculate the probability that the two dice land on the same color

Imagine that only one die was rolled, then we would have
P (red) = 2/6
P (black) = 2/6
P (white) = 1/6
P (yellow) = 1/6

Since two dice are rolled and the outcomes on the two dice are independent, we have, for two-dice rolls

P (red) = (2/6)(2/6) = 1/9

P (black) = (2/6)(2/6) = 1/9

P (white) = (1/6)(1/6) = 1/36

P (yellow) = (1/6)(1/6) = 1/36

So, probability that the two dice land on the same color = 1/9 + 1/9 + 1/36 + 1/36 = 10/36 = 5/18

So, probability that they land on different colors = 1 – (5/18) = 13/18