Simple probability question. Involves drawing name for gift.

We had a dinner with my family where we draw names to give gift this christmas. The only rule was to not pick our name.

We did it a couple of time because we kept picking our name.

Finally we succeded and i started to think about what was the probability of someone picking his own name thus making this fail.

I built an excel simulation program which simulate drawing and got a consistent 50 % failure chance for 2 persons, about 66 % for 3 persons...and so on...but it's like there is a somekind of a maximum of about 68 % as i add people to my simulation.

For 6 persons i get 68 % failure and for 100 persons, it's about the same. I don't understand why that is.

So i started thinking...and the figures that i think it should be is about :

success rate should be 1/number of person, so 1/6 for 6 persons so a failure rate of 5/6....

Can anyone explain me what i should be getting ?