Probability / odds of last person drawing their own name from a hat

So there is a 6.5% chance of this happening? That surprises me. I would have thought it was much less probable.

What if I wanted this expressed in Odds? Isn't it 1 out of the number of possible, which if I understand derangments and my own problem correctly (!), in the example of 4 students, out of 24 possible cases, there are only 9 derangements, so the odds would be 1:9... is that right?

What would it be for 12 students? 1:176,214,841? (I took that from the precalculated outcomes for n=12 number at the bottom)

A probability of 0.06509 is about 1 chance in 15 (more precisely, one chance in 15.36).

In the case of 4 students, a probability of 0.1389 is about 1 chance in 7 (7.199).

Thanks everyone, I really appreciate your time.