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### The First Ace

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Date: 06/12/97 at 15:42:49
From: Saul Klinow
Subject: Probability With Statistical Applications

I am having difficulty understanding the solution of an example
problem provided by my textbook (Subject title by Mosteller, Rourke &
Thomas). The problem is:

"The first ace.  An ordinary bridge deck of 52 cards is thoroughly
shuffled.  The cards are then dealt face up, one at a time, until an
ace appears.  What is the probability that the first ace appears at
the kth card or sooner?"

The solution rendered by the text is as follows:

"Denote by F the event "first ace at the kth card or sooner".  Then
the complementary event Fbar is the event "4 aces after the kth card".
The first k symbols of every of every sample point in Fbar are all N's
(non aces).  Therefore the number of sample points in Fbar is the
number of ways of arranging 4 A's (aces) and 48-k N's in the remaining
52-k places.  This number is

(52-k)!/(48-k)!(4!)

Hence

P(Fbar)=(52-k)!/(48-k)!(4!)  / 52!/(48!)(4!)

and

P(F) = 1 - P(Fbar). "

If the first ace is contained within the group of cards drawn up to
and including the first ace, which describes the event F, how can
Fbar, its complement include 4 aces?  Also, why are the first k
symbols of every point in Fbar non aces?

Thank you for you  great service and wonderful cooperation.

Saul Klinow
```

```
Date: 06/12/97 at 20:10:06
From: Doctor Anthony
Subject: Re: Probability With Statistical Applications

Since F is the event "first ace at the kth card or sooner" then the
converse of this (the complementary event), Fbar, must be that all
four aces come after the kth card. We therefore have 52-k cards left,
which must contain all 4 aces.

The number of ways of arranging 4 A's and 48-k  N's is the number of
ways of arranging 52-k letters, 4 being alike of one kind and 48-k
being alike of a second kind. This number is given by

(52-k)!
-----------  =  (52-k)_C_4
4! (48-k)!

(52-k)_C_4
So prob(Fbar)        =  ------------
52_C_4

and of course the required probability, that the first ace occurs at
the kth card or earlier, is this probability subtracted from 1.

(52-k)_C_4
P(F) =  1 -  -----------
52_C_4

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Probability
High School Statistics

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