I have my solution here with a different question:
What is the probability that the number on the red die is smaller than the number on the white die?
5/12
There are 36 possible ways of throwing a dice twice. If the first throw is 1, then there are five possibilities for a greater throw in the second; if the first throw is 2, then there are four, and so on.
Thus the total number where the second throw is greater than the first is 5 + 4 + 3 + 2 + 1 + 0 = 15, so the probability is 15 / 36 = 5 / 12
There is a 5/6 chance that the dice are different.
There is a 1/2 chance that the first will be bigger for two different dice.
The probability is 5/6 x 1/2 = 5/12
I hope the method and technique in my solution might help you solve your problem.