Six 6-sided dice are rolled. What is the probability that the number of dice showing even numbers and the number of dice showing odd numbers is equal?
There are 6 choose 3 ways for 3 of the dice to show even numbers and 3 of them to show odd numbers. Each roll is even with probability 1/2 and odd with probability 1/2, so each arrangement of 3 odd numbers and 3 even number occurs with probability (1/2)^6. Thus, the probability that 3 dice out of 6 show even numbers is 5/16
From the AOPS webstite..