jmc

The only way in which there won't be two dice of the same number is if for each number between 1 and 6 there is exactly one die displaying that number. If we line the dice up, there are a total of $6!$ ways in which we could order 6 dice all displaying different numbers and a total of $6^6$ possible outcomes since each of the 6 dice can have 6 outcomes and all of the rolls are determined independently. That means that the probability of all of the dice showing different numbers is $\frac{6!}{6^6}=\frac{5}{324}$, so the probability that we want is $1-\frac{5}{324}=\boxed{\frac{319}{324}}$.