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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.

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1 Answer 1

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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

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