I'm trying to understand what kind of probability distribution I need to use in order to calculate a very simple example using a deck of cards. Assume that there is a standard deck of cards (52 cards): Let X be the number of non-hearts until I get 13 hearts (without replacement). What would be the distribution I need to use?

samuelaplc 2022-09-06 Answered
I'm trying to understand what kind of probability distribution I need to use in order to calculate a very simple example using a deck of cards.
Assume that there is a standard deck of cards (52 cards):
Let X be the number of non-hearts until I get 13 hearts (without replacement). What would be the distribution I need to use?
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Answers (1)

procjenomuj
Answered 2022-09-07 Author has 8 answers
Step 1
Place the cards in a row. If we only discern hearts and non-hearts then there are ( 52 13 ) arrangements. Now fix n { 13 , , 52 } and notice that there are ( n 1 12 ) arrangements such that the last heart is placed on spot n. So denoting N as the spot of the last heart we come to:
P ( N = n ) = ( n 1 12 ) ( 52 13 ) 1
Now realize that X = N 13 so that P ( X = k ) = P ( N = k + 13 ) = ( k + 12 12 ) ( 52 13 ) 1
This for k = 0 , 1 , , 39
Step 2
Equality n = 13 52 P ( N = n ) = 1 leads to the observation that n = 12 51 ( n 12 ) = ( 52 13 ) . More generally it can be shown (e.g. by induction) that k = r n ( k r ) = ( n + 1 r + 1 ) .
So a more general setting here would lead to a combinatorial proof of this equality.
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