Five-Card Straight

Date: 12/10/2002 at 13:23:01
From: David Stadler
Subject: Poker probability

From a standard 52-card deck you pull out a 5-6-7-8 (suit is 
irrelevent for the problem). You shuffle the remaining 48 cards and 
deal 3 more cards. What percentage of the time will you complete the 
5-card straight (by receiving a 4 or 9 with any one of the three cards 
dealt). My friends and I disagree on the answer. Here is my math:

1st draw = 16.6667% 
2nd draw = 17.0213%
3rd draw = 17.3913%

adding the three chances together I get 51.08% you will get a 4 or 9.

My reasoning:

Ignoring the slight increase in percentage when a non-winning card is 
dealt, your odds are 8/48 or 1/6, similar to dice. If I select one 
number from a die and roll 3 times (3 cards), my odds are 50% to roll 
my number. 

Please help us settle this dispute - it is amazing how stubborn we all 
are!

Date: 12/10/2002 at 20:20:07
From: Doctor Ian
Subject: Re: Poker probability

Hi David,

The problem with these 'at least one' situations is that you have to
consider _all_ of the possibilities:

    p(4,-,-)            '-' = not 4 or 9
  + p(-,4,-)
  + p(-,-,4)
  + p(9,-,-)
  + p(-,9,-)
  + p(-,-,9)
  + p(4,9,-)
  + p(9,4,-)
  + ...

It's a headache, to say the least. Fortunately, there is a trick you
can use that works very nicely:  

  p(at least once) = 1 - p(never)

Does that make sense? 

There are four 4's and four 9's remaining in the deck, which has 48
cards. So the probability of drawing something other than a 4 or a 9
would be 

  (48 - 8)/48

The probability of doing it again would be 

  (47 - 8)/47

And again,

  (46 - 8)/46

So the probability of missing on all three draws would be 

     40   39   38
     -- * -- * --
     48   47   46

      5   39   19
  =  -- * -- * --
      6   47   23

      5   13   19
  =  -- * -- * --
      2   47   23

  
  = 1235/2162

  = 0.57

So the probability of hitting at least one would be 

  p = 1 - 0.57
   
    = 0.43

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 12/12/2002 at 10:16:39
From: David Stadler
Subject: Thank you (Poker probability)

Dr. Ian -

Thank you very much for explaining the poker probability question to 
me. I made a program to simulate and came up with the same answer 
before I sent the question, but still couldn't see what I was missing. 
Thanks again.

D. Stadler