🚀 go-pugleaf

I have just been playing drinking games with a friend of mine and we had an
argument about the odds of wining [losing??!!], and I was wondering if
anyone could clarify the situation.

The game was a very simple one (you don't need to be at all sober to play)
and it basically goes like this:-

There are 3 die(dice). You call a number before throwing the die and however
many times that number comes up (out of the three die) you have to 'skull'
that many glasses. My friend reckoned that there was a 1 in 6 chance of
'skulling' but obviously that was completely wrong. No matter how I tried to
point out he was wrong, he wouldn't beleive me and as I was [am] quite drunk
I couldn't come up with a suitable arguement. What I would like to know is
the odds of throwing a single, a double and a triple of the number.

Any written proof in my argument would be much appreciated.
Cheers!!!!!
RJ.