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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I&#039;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?

procjenomuj · Accepted Answer

Step 1Place 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  ,  …  ,  39Step 2Equality       ∑          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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