Two dice a tossed find the probability that the numbers are the same given that at least one is 3?

Answer 1

There are two different ways to look at this that give two different answers. The first way is this. We already know that one of the dice is a three. So the question of whether the two dice show the same number becomes "Is the second die a 3?" And the probability of the second die being a 3 is 1/6. Without really making it clear, this method ASSUMES that it is the FIRST die that is constrained to be a 3. The other way is this. The question constrains AT LEAST ONE of the two die to be a 3, but we don't know which one. So, what you have to do is draw a chart, listing all the possibilities (1-6) for die #1 on the left and all the possibilities for die #2 (also 1-6)across the top. There are 36 possible outcomes. But in only 11 of them does EITHER of the two dice show a 3. And, among those 11, in only 1 does BOTH dice show a 3. Therefore, the probability is 1/11.