Poker Odds

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:

http://www.poker-gaming-zone.com/ provides the following:

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 45 = 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 * 42 = 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 * 62 * 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 * 43 = 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 45=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 jack-high. 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 9-8-7-6-5 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 jack-high hand. For ace-high remember to subtract 2 rather than 1 from the total number of ways to arrange the ranks since A-K-Q-J-10 and 5-4-3-2-A 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 pre-calculus 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(n-1)(n-2)* ... * 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 1067.)

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!*(10-4)!) = 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!*(x-y)!) 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%

 

 

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