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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% |
|
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 |
|
|
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.
|
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 |
|
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|
