### Instruction: Phil rolls 6 fair 6-sided dice. What is the probability that at least two dice show the same number?### Response: To solve a probability problem, I usually find it helpful to think of the opposite event and then subtract its probability from 1.
In this case, the opposite event is that all 6 dice show different numbers, which means they form a permutation of the numbers 1 to 6.
There are 6! ways to arrange 6 distinct numbers in order, and there are 6^6 ways to roll 6 dice, so the probability of getting a permutation is 6! / 6^6.
Therefore, the probability of not getting a permutation, which is the same as getting at least two dice that show the same number, is 1 - 6! / 6^6.
I can simplify this fraction by canceling out some factors of 6.
I get 1 - (6 * 5 * 4 * 3 * 2 * 1) / (6 * 6 * 6 * 6 * 6 * 6) = 1 - (5 * 4 * 3 * 2) / (6 * 6 * 6 * 6 * 6) = 1 - 120 / 7776.
I can use a calculator or long division to get the decimal value of this fraction.
I get 1 - 120 / 7776 = 0.9846 approximately.
I can express this probability as a percent by multiplying by 100.
I get 0.9846 * 100 = 98.46%.