Alberto and Benito practice a game with two dice where each one has the same probability of winning as the other. The dice they use do not have numbers on their faces but colors. Some of the faces are painted red and the rest blue.
The game consists of throwing the dice at the same time. Alberto wins when the two upper faces are the same color and Benito wins when the colors of the faces are different.
One of the dice has five red and one blue faces.
How many red faces does the second die have?
When two dice are rolled, there are 36 equally possible results. For both of them to have the same probability of winning, there have to be 18 ways to get the same color in a roll.
If we call X the number of red faces on the second die, we have to: 18 = 5X + 1 (6 - X) ⇒ X = 3. So the second dice must have 3 red faces and 3 blue faces.