# HyperGeometric distribution : Inutition for symmetry

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.

## 1 Answer

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