So let’s do the math for an example where we’re rolling five dice. Intuitively, you may expect that the five rolls should come up different a lot of the time. (At least this is what people often mention in emails.) So what’s the chance of all the five rolls being different? The first die is trivial. Any of the six possible values is fine (none will result in duplicates), giving a probability of 6⁄6 = 1. After you’ve rolled the first die, the chance of the second coming up different from the first is 5⁄6, because there is now one less value you haven’t seen before. The third is 4⁄6, and so on. Hence the total probability of all your five rolls turning out different is:

6⁄6 × 5⁄6 × 4⁄6 × 3⁄6 × 2⁄6 ≈ 9.26%

Hence, if you roll five dice repeatedly, you should expect over 90% of the rolls to contain duplicates. If you roll six dice, you can multiply the value above by a further 1⁄6 and you’ll get approximately 1.54%. Hence, if you roll six dice repeatedly, you can expect to get six different values only about once in every 65 rolls.