From Feller -- An introduction to probability theory and its applications (p.31):
"In sampling without replacement the probability for any fixed element of the population to be included in a random sample of size r is:
1 - [ ( (n-1)! / (n-1-r)! ) / (n! / (n-r)!) ] = r/n
I understand how we get r/n as a result (in fact, that would be my intuitive answer to the problem), and I also understand the formula of n! / (n-r)! as the number of possible outcomes when one draws without replacement. But where does (n-1)! / (n-1-r)! come from in the numerator? That just makes no intuitive sense to me.
Thank you for any hints!
There are $$ \binom{n}{r}=\frac{n!}{(n-r)!r!} $$ unordered samples of $r$ people from the population of $n$ people without replacement. Fix a particular individual of the population. There are $$ \binom{n-1}{r}=\frac{(n-1)!}{(n-r-1)!r!} $$ unordered samples of $r$ people from the population of $n$ people that do not include this individual. Hence the probability that the individual is not included in the random sample of size $r$ is given by $$ \binom{n-1}{r}\bigg/\binom{n}{r}. $$ So the probability that the individual is included in the random sample of size r (the complementary event) is given by $$ 1-\binom{n-1}{r}\bigg/\binom{n}{r}. $$