Question

suppose 4 dice are simultaneously rolled multiple times until exactly 2 of the dice show an even number. what is the probability that strictly more than 2 rolls are required? express your answer as a fraction or other expression as before.

Answer 1

We can approach this problem by finding the probability that exactly 2 rolls are required and subtracting it from 1.

The number of ways to get exactly 2 even numbers in 2 rolls is 4 choose 2, which is 6. And the number of ways to get exactly 2 even numbers in 4 rolls is 4 choose 2 times 2 choose 2, which is 6 * 1 = 6. So the number of successful outcomes is 6 + 6 = 12.

The total number of possible outcomes is 6^4 = 1296.

So, the probability that exactly 2 rolls are required is 12/1296 = 1/108.

Therefore, the probability that strictly more than 2 rolls are required is 1 - 1/108 = 107/108.