Wikipedia page on HyperGeometric distribution says
Swapping the roles of green and drawn marbles:
$$ f ( k ; N , K , n ) = f ( k ; N , n , K ) $$
where in LHS,
N = Total number of marbles
n = number of draws
K = number of green marbles(others are red)
k = number of green marbles in n draws
I understand how the equality holds mathematically, but I can't understand why this equality holds intuitively.
Consider slightly different situations, both involving double selections:
Suppose you have $N$ marbles of which you randomly select $K$ of the $N$ without replacement (to count as green), then put those back and independently select $n$ without replacement from your $N$ of which $K$ are green (to count as drawn in your hypergeometric distribution). You have $f ( k ; N , K , n )$ as the probability mass function for the number $k$ which are selected both times
Suppose you have $N$ marbles of which you randomly select $n$ of the $N$ without replacement (to count as green), then put those back and independently select $K$ without replacement from your $N$ of which $n$ are green (to count as drawn in your hypergeometric distribution). You have $f ( k ; N , n , K )$ as the probability mass function for the number $k$ which are selected both times
It seems intuitively obvious to me that the probability of selecting $k$ marbles both times is the same in the each of these situations