What are the odds of rolling 6 of a kind on a 6 sided die?

Answer 1

A 1 in 36 chance

The above is not true, though I can understand why this mistake might be made, for example,

if you have 1 die, then your chances of rolling any given number are 1/6, multiply this by six die and you get 1/36 RIGHT?

What actually happens is more complicated.

Imagine you are holding 1 die and want to roll a 5, you have a 1/6 chance to do this.

Now take a second die, assuming you rolled a 5 on the first, you have a 1/6 chance of rolling a second 5, so the total odds that you can roll a pair of fives is actually 1/6*1/6 = 1/36. If you add another die, you have to then multiply this by 1/6 and so on.

Extend this to having six die and it continues:
5 on 1 die =1/6
5 on 2 die =1/6*1/6=1/36
5 on 3 die =1/6*1/6*1/6=1/216
5 on 4 die =1/6*1/6*1/6*1/6=1/1296
5 on 5 die =1/6*1/6*1/6*1/6*1/6=1/7776
5 on 6 die =1/6*1/6*1/6*1/6*1/6*1/6=1/46656

Now to complicate matters, if you want to roll six 5's your chances are 1/46656

HOWEVER, if you don't care what your six of a kind is, then essentially the "first" die doesn't matter in the calculations, it's just there to determine what number we are after, leaving us with a total of five "more" die the we need to get that number on.

So if you just want to roll "SIX OF A KIND" and don't care what number it is, your chances are 1/7776