Suppose 4 dice are rolled, what is the probability that 1 number appears at least twice?
1 Answer
The probability is
Explanation:
Let's number the dice with 1,2,3, and 4. We first count the number of ways a roll of the four dice does not have a number that appears at least twice. Whatever is on top of the first die, there are 5 ways to have a different number on die 2.
Then, assuming that we have one of those 5 outcomes, there are 4 ways to have a number on die 3 that is not the same as on dice 1 and 2. So, 20 ways for dice 1, 2, and 3 to have all different values.
Assuming we have one of these 20 outcomes, there are 3 ways for die 4 to have a different number than dice 1, 2, or 3. So, 60 ways altogether.
So, the probability of NOT having two numbers the same is
The probability of the opposite, i.e. having at least two, equals 1 minus the above probability, so it is
