5 cards are dealt from a perfectly shuffled deck

Date: 03/16/98 at 00:43:08
From: Nicholas Chu
Subject: Card problems...

Hello,

I have a question from my teacher and I don't understand how to do it, 
can you help me?

   "5 cards are dealt from a perfectly shuffled deck. What is the 
   probability of being dealt exactly 2 aces, given that you were  
   dealt the ace of spades? What is the probability of being dealt   
   exactly 2 aces, given that you were dealt at least 1 ace? Do the 
   results agree with your intuition?"

Since my teacher was talking about the theorem:

   P(A|B) = P(A and B) / P(B)

I was thinking,

   P(exactly 2 aces and ace of spade) / P (ace of spades), and

   P(exactly 2 aces and at least 1 ace) / P (at least 1 ace)

   P(exactly 2 aces) = 1/C(52,5) * c(5,2)
   P(ace of spades)  = 1/C(52,5) * ?
   P(at least 1 ace) = ?

Thank you very much!

Nic

Date: 03/16/98 at 08:59:32
From: Doctor Mitteldorf
Subject: Re: Card problems ......

Dear Nic,

The point of this problem is that your intuition says it doesn't 
matter whether the one ace in your hand is the ace of spades or not; 
the answer to the two questions should be the same.  But the 
calculations along the way, using Baye's Theorem for the two different 
cases, will be very different.

In probability analysis even more than in other areas of math, it's 
especially important to think in detail and model the situation in 
your mind, not just apply a formula. It's so easy to be fooled by 
problems that sound as if they ought to be the same but when you think 
them through clearly... Let me get you started with some of the 
calculations you'll need to do:

Easiest first:  

   What's the probability of the ace of spades appearing in your 
   5-card hand? I'm going to calculate what the probability is of 
   NOT having the ace of spades in the hand, then subtract from 1.  

   The probability that the first card is not the ace of spades is 
   51/52. Given that first draw, there are 51 cards left, and 50 of 
   them are not the ace of spades.  

   Continuing for all 5 cards, we find the probability that there is 
   NOT an ace of spades in your hand is

   51/52 * 50/51 * 49/50 * 48/49 * 47/48

   You can cancel a lot of numerators with denominators here, and   
   reduce to 47/52. So the probability of having the ace of spades 
   in your hand is 1-47/52 = 5/52.

   Well, that result is kind of expected. After all, you've got 5 
   cards in your hand, 47 on the table. The chance that the ace of 
   spades is in your hand ought to be just 5/52, and that it's on the  
   table ought to be 47/52. This is the correct answer, and there's 
   nothing wrong with the shortcut.

   But now ask, what's the probability that your hand has at least 
   one ace? Well, let's just take the shortcut here. The probability 
   of having the ace of spades in your hand is 5/52. The probability 
   of getting the ace of diamonds is 5/52. For each of the four aces,  
   the probability is the same, so 4*5/52 or 20/52, right? Let's just 
   check with the long method:

   The probability that the first card you draw is not an ace is 
   48/52. Now there are 51 cards left, and 47 of them are not aces, 
   so the probability of the second card not being an ace is 47/51.  

   Continuing in this way, we find the probability of NO aces in the 
   hand is

   48/52 * 47/51 * 46/50 * 45/49 * 44/48
   (We can do a little canceling here, but not much.)  

   The probability of having at least 1 ace must be one minus the 
   probability of no aces, or

   1 - 48/52 * 47/51 * 46/50 * 45/49 * 44/48.

   This is different from 20/52.  Which is right?  The second 
   calculation is right. Where did our reasoning go wrong in the 
   other one?  I'll leave that one to you.


Next calculation:  

   What's the probability of having 2 or more aces in the hand?  
   It's easier to calculate the probability of 1 ace and no aces.  

   The probability of having 2 or more aces is just 1 minus the  
   probability of having no aces minus the probability of having  
   exactly 1 ace.  The probability of having no aces we calculated 
   above: 

   48/52 * 47/51 * 46/50 * 45/49 * 44/48

   For the probability of having exactly 1 ace, we can do this: 
 
   Say the ace is the first draw. The probability of that is 4/52.  
   Then the second draw must NOT be an ace. Since there are 51 cards
   left in the deck and 48 of them are not aces, the probability for 
   the second draw is 48/51. Continue in that way and you find the 
   probability of getting the ace on the first draw and no aces on the 
   four subsequent draws is: 

   4/52 * 48/51 * 47/50 * 46/49 * 45/48.

   If you apply the same technique to the probability of getting 
   the ace on the second or the fifth draw, you'll find the same  
   expression.  So the probability of having exactly 1 ace in your  
   hand is 5 times the result above:

   20/52 * 48/51 * 47/50 *46/49 * 45/48

   And the probability of having two or more aces is 

   1 - (20/52 * 48/51 * 47/50 *46/49 * 45/48) 
     - (4/52 * 48/51 * 47/50 * 46/49 * 45/48).


There's a lot more to do here, but I think you're off to a good 
start now, and I'll ask you to take the next steps yourself.  

-Doctor Mitteldorf,  The Math Forum
Check out our Web site http://mathforum.org/dr.math/   

Date: 03/16/98 at 11:31:02
From: Doctor Anthony
Subject: Re: Card problems ......

I am using the the notation C(n,r) to mean (n choose r).  It is 
a binomial coefficient and can be calculated from:

              n!                           10!
   C(n,r) = -------      e.g.  C(10,4) = -------  =  210
            r!(n-r)!                      4! 6!

                                          C(3,1) x C(48,3)
   Prob(2 aces|you have ace of spades) =  -----------------
                                               C(51,4)

                                           51888
                                       =  --------  = 0.207635
                                           249900 


Probability of being dealt at least one ace could mean 1, 2, 3, or 4 
aces. This is best calculated from 1 - Prob(no aces)

                        C(48,5)
   Prob(no aces)   =    -------    =  0.658842
                        C(52,5) 

   Probability of at least one ace =  0.341158


                                      C(4,2) x C(48,3)
   Prob(2 aces|at least one ace)   = ------------------
                                     0.341158 x C(52,5) 

                                   =  0.11704

Do these results agree with your intuition?

Yes! Because given that you have the ace of spades, it is a lower 
probability than the probability of at least one ace.  So the sample 
space is less and the probability of 2 aces is proportionally greater.

-Doctor Anthony,  The Math Forum
Check out our Web site http://mathforum.org/dr.math/