



In
the game of draw poker, the idea is to get the best 5 card hand
possible.



The
players bet that they have a better 5 card hand 


than any
other player remaining in the game. 


If you
don't think you have the best hand, 


and don't
think that you can trick the other players 


into
thinking that you do (called bluffing), 


you
can quit for that hand (called folding). 


The
rules of what beats what are a bit complex, 


but
basically more cards of the same number 


or
more cards of the same suit 


or
having all cards in numerical order, is a good thing. 





Poker
is more a game of human nature and Psychology, 


but
it helps to know the odds. 


Leaving
out the betting, 


the
way the game is played is that each player is dealt 5 cards. 


The
player can keep all of the cards or trade in some in 


for
replacement cards from those left in the deck. 


You
don't get to choose what replacement cards you get. 


You
basically get them at random. 





Say
you are in the game and get these 5 cards ... 






10 of Hearts 

10 of Spades 

6 of Hearts 

Jack of Hearts 

Queen of
Hearts 






You
now have to decide if you want to keep these cards 


or
trade some of them in for cards that might be better. 





Here
are some strategies you might consider ... 





1) Keep the four hearts. Trade in the ten of spades



and hope you get another heart so you have cards that are all one
suit. 





2) Keep the two tens. Trade in the other cards 


to try to get more tens. 





3) Keep the ten, Jack, and Queen of Hearts. 


Trade in the 6 of Hearts and the10 of Spades and try to get a Royal Flush. 


That is, a ten, Jack, Queen, King, and Ace all in the same suit. 





What
are the chances of each of these options working? 





There
are 52 cards in a deck. You have 5 of them. 


That means there are 47 cards from the deck
that
you don't have. 





Other players have some of these cards,



but you don't know which ones they have. 


That
means we treat all other cards the same way. 





Say you try for the flush (choice
1) 


You
trade in the 10 of Spades. 


You
hope to get another Heart card. 


There
are 13 Hearts in the deck. 


You
have 4 of them.



That
means there are 9 other Heart cards in the other 47 cards in the
deck. 


The
chance of getting one of them is ... 











OK,
Now let's try for more tens (choice 2)



What
is the chance that you will get more tens? 


One
way to figure this out, 


is
to find the chance of not getting any more 10's 


and
then subtract that from 100% 





There are
two more tens in the 47 cards that are not in your hand. 


That
makes 45 of those cards something other than 10's. 


The
chance of getting one of those other cards next is ... 











If
that next card you get is not a ten, 


there
will still be 2 tens out there and 46 cards total. 


That
means 44 of them will be cards that are not tens. 


The
chance of getting one of the "not a ten" cards next is ... 











so
the chance that neither of those two cards 


is
a ten is ... 











Now
there are 45 cards left, and 2 of them are tens. 


That
means 43 of them are not tens. 


The
chance of getting one of the "not tens" now is ... 











and
the chance that none of the three cards will be tens is ... 











There
is a 90% chance that you will not get any more tens, 


so
there is a ... 





100%
 90% = 10%






10%
chance of getting 1 or 2 more tens. 





How
much of that 10% is the chance of getting exactly 1 more ten? 


How
much of that 10% is the chance of getting exactly 2 more tens? 





To
answer this, 


figure
out the chance of getting 2 more tens. 


The
rest of the 10%, will be the chance of getting 


only
one more ten. 





Remember
the tree diagrams? 


We
can use something like the idea of one here. 


There
are 45 cards of the 47 that are not tens. 


Any
path that includes one of the remaining tens 


for
each of two choices, 


can
have any one of the 45 other cards for the third card. 





So,
for example, there are 45 different paths 


where the first selection is the ten of clubs, 


and
the second selection is the ten of diamonds, 


and
the third selection is one the other 45 cards. 





The
path types through the two other tens are ... 





10
of Clubs 
10
of Diamonds 
some
other card 
45
paths 
10
of Clubs 
some
other card 
10
of Diamonds 
45
paths 
some
other card 
10
of Clubs 
10
of Diamonds 
45
paths 
10
of Diamonds 
10
of Clubs 
some
other card 
45
paths 
10
of Diamonds 
some
other card 
10
of Clubs 
45
paths 
some
other card 
10
of Diamonds 
10
of Clubs 
45
paths 






There
are 47 x 46 x 45 paths through the cards. 


That
makes the chance of getting both of the other tens ... 











That
means the chance of getting exactly one more ten is ... 





10.0%  0.28% =
9.72% 





The
last strategy was to try to get the Royal Flush. 


To
do this, we give up the 6 of Hearts and the10 of Spades 


and
try to get the King and Ace of Hearts. 





What
is the chance that the two cards that you get, 


will
be these two cards? 





There
are two cards out there you want. 


There
are 47 cards out there total. 


The
chance that the first card you get 


is
one of the two that you want, is ... 











If
we do get it, we will still need the other one. 


There
are now 46 cards out there. 


The
chance of getting the one we need from the 46 is ... 











and
the chance of getting both of the cards we need is ... 











Let's
Review: 





You
start with ... 






10 of Hearts 

10 of Spades 

6 of Hearts 

Jack of Hearts 

Queen of
Hearts 






The
chance of getting a Heart to replace the 10 of spades, 


and
give you a flush is ... 











The
chance of getting 1 or 2 more tens 


to
replace the three cards that are not tens, is ... 











100%
 90% = 10% (a 1 in 10 chance)






The
chance that you got both of the other tens is ... 











So
the chance that you got only one other ten is ... 





10.0%
 0.28% = 9.72% (still about a 1 in 10 chance)






So
the most likely of these 


is
the chance of getting the flush. 


But
that's still only 20% 


To
go for it (or the Royal Flush) you have to give up the pair of tens. 





The
chance of getting more tens 


is
about half the chance of getting the flush. 


But
to go for the tens, you don't give up much of anything ... 





copyright 2005 Bruce Kirkpatrick 
