Poker Odds
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Mathematics is what card games are all about. The combination possibilities are
numerous. Here are the math odds for various combinations with a details about
how to
identify the math techniques which allow you to calculate them.
After looking at the odds, try and calculate them with the formulas given on
the links.
Obviously the odds are only part of the game. Strategy has everything to do
with winning. That will not be discussed since this page is intended to help a
student calculate odds.
Find more information at:
Poker Odds
The table below shows the ways to draw a poker hand in 5 card stud and the
odds. 
Five
Card Stud 
Hand 
Combinations

Odds

Royal flush 
4 
0.00000154 
Straight flush 
36 
0.00001385 
Four of a kind 
624 
0.00024010 
Full house 
3,744 
0.00144058 
Flush 
5,108 
0.00196540 
Straight 
10,200 
0.00392465 
Three of a kind 
54,912 
0.02112845 
Two pair 
123,552 
0.04753902 
Pair 
1,098,240 
0.42256903 
Nothing 
1,302,540 
0.501177394 


Please note that these are
just statistics, there are no guarantees when you are playing cards. Many
factors come into play such as: The more people at the table, the greater
the chance that one or more will be dealt a pair. Less people at the
table, there is a greater chance of getting a hand of higher value.

Explanation of
the above chart is below.
Royal Flush
There
are four different ways to draw a royal flush (one for each suit).
Straight Flush
The
highest card in a straight flush can be 5,6,7,8,9,10,Jack,Queen, or King.
Thus there are 9 possible high cards, and 4 possible suits, creating 9 * 4
= 36 different possible straight flushes.
Four of a Kind
There
are 13 different possible ranks of the 4 of a kind. The fifth card could
be anything of the remaining 48. Thus there are 13 * 48 = 624 different
four of a kinds.
Full House
There
are 13 different possible ranks for the three of a kind, and 12 left for
the two of a kind. There are 4 ways to arrange three cards of one rank (4
different cards to leave out), and combin(4,2) = 6 ways to arrange two
cards of one rank. Thus there are 13 * 12 * 4 * 6 = 3,744 ways to create a
full house.
Flush
There
are 4 suits to choose from and combination (13,5) = 1,287 ways to arrange
five cards in the same suit. From 1,287 subtract 10 for the ten high cards
that can lead a straight, resulting in a straight flush, leaving 1,277.
Then multiply for 4 for the four suits, resulting in 5,108 ways to form a
flush.
Straight
The
highest card in a straight flush can be 5,6,7,8,9,10,Jack,Queen,King, or
Ace. Thus there are 10 possible high cards. Each card may be of four
different suits. The number of ways to arrange five cards of four
different suits is 4^{5} = 1024. Next subtract 4 from 1024 for the
four ways to form a flush, resulting in a straight flush, leaving 1020.
The total number of ways to form a straight is 10*1020=10,200.
Three of a Kind
There
are 13 ranks to choose from for the three of a kind and 4 ways to arrange
3 cards among the four to choose from. There are combination(12,2) = 66
ways to arrange the other two ranks to choose from for the other two
cards. In each of the two ranks there are four cards to choose from. Thus
the number of ways to arrange a three of a kind is 13 * 4 * 66 * 4^{2}
= 54,912.
Two Pair
There
are (13:2) = 78 ways to arrange the two ranks represented. In both ranks
there are (4:2) = 6 ways to arrange two cards. There are 44 cards left for
the fifth card. Thus there are 78 * 6^{2} * 44 = 123,552 ways to
arrange a two pair.
One Pair
There
are 13 ranks to choose from for the pair and combination(4,2) = 6 ways to
arrange the two cards in the pair. There are combin(12,3) = 220 ways to
arrange the other three ranks of the singletons, and four cards to choose
from in each rank. Thus there are 13 * 6 * 220 * 4^{3} = 1,098,240
ways to arrange a pair.
Nothing
First
find the number of ways to choose five different ranks out of 13, which is
combination(13,5) = 1287. Then subtract 10 for the 10 different high cards
that can lead a straight, leaving you with 1277. Each card can be of 1 of
4 suits so there are 4^{5}=1024 different ways to arrange the
suits in each of the 1277 combinations. However we must subtract 4 from
the 1024 for the four ways to form a flush, leaving 1020. So the final
number of ways to arrange a high card hand is 1277*1020=1,302,540.
Specific High Card.
Let's find the odds of drawing a
jackhigh. There must be four different cards in the hand all less than a
jack, of which there are 9 to choose from. The number of ways to arrange 4
ranks out of 9 is combin(9,4) = 126. We must then subtract 1 for the
98765 combination which would form a straight, leaving 125. From above
we know there are 1020 ways to arrange the suits. Multiplying 125 by 1020
yields 127,500 which the number of ways to form a jackhigh hand. For
acehigh remember to subtract 2 rather than 1 from the total number of
ways to arrange the ranks since AKQJ10 and 5432A are both valid
straights.

Poker Odds
Figuring Odds for Five Card Stud
I have
created this section to explain how I arrived at the odds of drawing poker
hands. I am not a mathematical genius, and you don't have to be either to
understand the concepts below. These math formulas come out of an old basic
statistics book and a precalculus textbook of mine. The skills used here can be
applied to a wide range of calculating odds.
Factorials
A factorial
means that you simply multiply the integers in a number. For example, for the
number 4, you multiply 4x3x2x1=24. Imagine that you have 4 coffee cups.
How many combinations can you arrange them in? The answer is 4!, or 24. There
are obviously 4 positions to put the first cup , then there will be 3 positions
left to put the second cup, 2 positions for the third cup, and only 1 for the
fourth cup, or 4x3x2x1 = 24. If you had n cups there would be n(n1)(n2)* ... *
1 = n! ways to arrange them. Any scientific calculator should have a factorial
button, usually denoted as x!, and the factor (x) function in Excel will give
the factorial of x. (The total number of ways to arrange 52 cards would be 52! =
8.065818 x 10^{67}.)
The Combinatorial Function
Now imagine
that you have 10 coffee cups each of which is a different color. Imagine that
you want to see how many different groups of 4 coffee cups out of the 10 coffee
cups you could have. How many different combinations of coffee cups are
there to choose from? The answer is 10! / (4!*(104)!) = 210. The general case
is if you have to form groups of y coffee cups out of a total of x then there
are x!/(y!*(xy)!) combinations to choose from. Why? For the example given there
would be 10! = 3,628,800 ways to put the 10 coffee cups in order. However you
don't have to establish an order of the coffee cups or those that aren't in the
group of 4. There are 4! = 24 ways to arrange the coffee cups in each grouping
of 4 and 6! = 720 ways to arrange the other 6. By dividing 10! by the product of
4! and 6! you will divide out the order of coffee cups in and out of the total
and be left with only the number of combinations, specifically
(1*2*3*4*5*6*7*8*9*10)/((1*2*3*4)*(1*2*3*4*5*6)) = 210. The combination (x,y)
function in Excel will tell you the number of ways you can arrange a group of y
out of x.
Now we can determine the number of possible
five card hands out of a 52 card deck. The answer is combine (52,5), or
52!/(5!*47!) = 2,598,960. If you're doing this by hand because your calculator
doesn't have a factorial button and you don't have a copy of Excel, then realize
that all the factors of 47! cancel out those in 52! leaving
(52*51*50*49*48)/(1*2*3*4*5). The probability of forming any given hand is the
number of ways it can be arranged divided by the total number of combinations of
2,598.960. On page 1 are the number of combinations for each hand. Just divide
by 2,598,960 to get the odds
Poker
Odds
Six
Card Stud 
Hand

Combinations

Odds

Royal flush 
376 
0.000018 
Straight
flush 
1468 
0.000072 
Four of a
kind 
14664 
0.000720 
Full house 
165984 
0.008153 
Flush 
205792 
0.010108 
Straight 
361620 
0.017763 
Three of a
kind 
732160 
0.035963 
Two pair 
2532816 
0.124411 
Pair 
9730740 
0.477969 
Nothing 
6612900 
0.324822 
Total 
20358520 
1 


Seven
Card Stud 
Hand 
Combinations

Odds

Royal flush 
4,324 
0.00003232 
Straight flush 
37,260 
0.00027851 
Four of a kind 
224,848 
0.00168067 
Full house 
3,473,184 
0.02596102 
Flush 
4,047,644 
0.03025494 
Straight 
6,180,020 
0.04619382 
Three of a kind 
6,461,620 
0.04829870 
Two pair 
31,433,400 
0.23495536 
Pair 
58,627,800 
0.43822546 
Ace high or less 
23,294,460 
0.17411920 
Total 
133,784,560 
1.00000000 


Poker
Odds 
ODDS
OF BEING DEALT THESE HANDS (5 CARDS) 

ROYAL
FLUSH 
1
IN 650,000 
STRAIGHT
FLUSH 
1
IN 72,200 
FOUR
OF A KIND 
1
IN 4,200 
FULL
HOUSE 
1
IN 700 
FLUSH 
1
IN 510 
STRAIGHT 
1
IN 250 
THREE
OF A KIND 
1
IN 48 
TWO
PAIR 
1
IN 21 
ONE
PAIR 
1
IN 2.4 

NO
PAIR 
1
IN 2 

ODDS OF BEING DEALT
THESE HANDS (7 CARDS)


ROYAL
FLUSH 
0.0002% 

STRAIGHT
FLUSH 
0.0012% 
FOUR
OF A KIND 
0.0240% 
FULL
HOUSE 
0.1441% 
FLUSH 
0.1967% 
STRAIGHT 
0.3532% 
3
OF A KIND 
2.1128% 
2
PAIR 
4.7539% 
1
PAIR 
42.2569% 

NOTHING 
50.1570% 
