ProbabilitiesProbabilities
in PokerBelow are the number of ways to draw each hand
and the probability of drawing for the first draw in five-card
draw and in seven-card stud.
Another good source of Gambling Odds, Gambling
Probabilities and Top Casino Guide is the House of Odds.
|
Five Card Stud
|
|
Hand |
Combinations |
Probability |
|
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
|
|
Seven Card Stud
|
|
Hand |
Combinations |
Probability |
|
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
|
Derivations for Five Card Draw
|
Five Card Draw High Card Hands
|
|
Hand |
Combinations |
Probability |
|
Ace high
|
502,860 |
0.19341583 |
|
King high
|
335,580 |
0.12912088 |
|
Queen high
|
213,180 |
0.08202512 |
|
Jack high
|
127,500 |
0.04905808 |
|
10 high
|
70,380 |
0.02708006 |
|
9 high
|
34,680 |
0.01334380 |
|
8 high
|
14,280 |
0.00549451 |
|
7 high
|
4,080 |
0.00156986 |
|
Total
|
1,302,540 |
0.501177394
|
Probabilities in Bingo
The following table shows the probability of
forming a bingo, black out, or four corners within a specified
number of calls. For example the probability of a single player
forming a bingo within 25 calls is 0.06396106, or about 6.4%.
|
Probabilities in Bingo
|
|
Number
of Calls |
Bingo |
Black Out |
Four Corners |
X |
|
1 |
0.00000000 |
0.00000000 |
0.00000000 |
0.00000000 |
|
2 |
0.00000000 |
0.00000000 |
0.00000000 |
0.00000000 |
|
3 |
0.00000000 |
0.00000000 |
0.00000000 |
0.00000000 |
|
4 |
0.00000329 |
0.00000000 |
0.00000082 |
0.00000000 |
|
5 |
0.00001692 |
0.00000000 |
0.00000411 |
0.00000000 |
|
6 |
0.00005215 |
0.00000000 |
0.00001234 |
0.00000000 |
|
7 |
0.00012492 |
0.00000000 |
0.00002880 |
0.00000000 |
|
8 |
0.00025632 |
0.00000000 |
0.00005759 |
0.00000000 |
|
9 |
0.00047305 |
0.00000000 |
0.00010367 |
0.00000000 |
|
10 |
0.00080783 |
0.00000000 |
0.00017278 |
0.00000000 |
|
11 |
0.00129986 |
0.00000000 |
0.00027150 |
0.00000001 |
|
12 |
0.00199521 |
0.00000000 |
0.00040726 |
0.00000003 |
|
13 |
0.00294715 |
0.00000000 |
0.00058826 |
0.00000008 |
|
14 |
0.00421648 |
0.00000000 |
0.00082356 |
0.00000018 |
|
15 |
0.00587167 |
0.00000000 |
0.00112304 |
0.00000038 |
|
16 |
0.00798905 |
0.00000000 |
0.00149739 |
0.00000076 |
|
17 |
0.01065272 |
0.00000000 |
0.00195812 |
0.00000144 |
|
18 |
0.01395440 |
0.00000000 |
0.00251759 |
0.00000259 |
|
19 |
0.01799309 |
0.00000000 |
0.00318894 |
0.00000448 |
|
20 |
0.02287445 |
0.00000000 |
0.00398618 |
0.00000747 |
|
21 |
0.02871003 |
0.00000000 |
0.00492410 |
0.00001206 |
|
22 |
0.03561614 |
0.00000000 |
0.00601835 |
0.00001895 |
|
23 |
0.04371249 |
0.00000000 |
0.00728537 |
0.00002906 |
|
24 |
0.05312045 |
0.00000000 |
0.00874244 |
0.00004359 |
|
25 |
0.06396106 |
0.00000000 |
0.01040767 |
0.00006411 |
|
26 |
0.07635261 |
0.00000000 |
0.01229997 |
0.00009260 |
|
27 |
0.09040799 |
0.00000000 |
0.01443910 |
0.00013159 |
|
28 |
0.10623163 |
0.00000000 |
0.01684561 |
0.00018423 |
|
29 |
0.12391628 |
0.00000000 |
0.01954091 |
0.00025441 |
|
30 |
0.14353947 |
0.00000000 |
0.02254720 |
0.00034692 |
|
31 |
0.16515993 |
0.00000000 |
0.02588753 |
0.00046759 |
|
32 |
0.18881391 |
0.00000000 |
0.02958575 |
0.00062345 |
|
33 |
0.21451154 |
0.00000000 |
0.03366654 |
0.00082296 |
|
34 |
0.24223348 |
0.00000000 |
0.03815542 |
0.00107617 |
|
35 |
0.27192783 |
0.00000000 |
0.04307870 |
0.00139504 |
|
36 |
0.30350759 |
0.00000000 |
0.04846353 |
0.00179362 |
|
37 |
0.33684876 |
0.00000000 |
0.05433790 |
0.00228842 |
|
38 |
0.37178933 |
0.00000000 |
0.06073059 |
0.00289866 |
|
39 |
0.40812916 |
0.00000000 |
0.06767123 |
0.00364670 |
|
40 |
0.44563111 |
0.00000000 |
0.07519026 |
0.00455838 |
|
41 |
0.48402328 |
0.00000001 |
0.08331894 |
0.00566344 |
|
42 |
0.52300269 |
0.00000001 |
0.09208935 |
0.00699602 |
|
43 |
0.56224021 |
0.00000003 |
0.10153441 |
0.00859511 |
|
44 |
0.60138685 |
0.00000007 |
0.11168785 |
0.01050513 |
|
45 |
0.64008123 |
0.00000015 |
0.12258423 |
0.01277651 |
|
46 |
0.67795818 |
0.00000031 |
0.13425892 |
0.01546630 |
|
47 |
0.71465810 |
0.00000063 |
0.14674812 |
0.01863888 |
|
48 |
0.74983686 |
0.00000125 |
0.16008886 |
0.02236665 |
|
49 |
0.78317588 |
0.00000245 |
0.17431898 |
0.02673088 |
|
50 |
0.81439191 |
0.00000472 |
0.18947715 |
0.03182247 |
|
51 |
0.84324614 |
0.00000891 |
0.20560286 |
0.03774293 |
|
52 |
0.86955207 |
0.00001654 |
0.22273644 |
0.04460528 |
|
53 |
0.89318170 |
0.00003023 |
0.24091900 |
0.05253511 |
|
54 |
0.91406974 |
0.00005441 |
0.26019252 |
0.06167165 |
|
55 |
0.93221528 |
0.00009654 |
0.28059978 |
0.07216896 |
|
56 |
0.94768080 |
0.00016894 |
0.30218438 |
0.08419712 |
|
57 |
0.96058846 |
0.00029180 |
0.32499074 |
0.09794358 |
|
58 |
0.97111353 |
0.00049778 |
0.34906413 |
0.11361456 |
|
59 |
0.97947539 |
0.00083912 |
0.37445061 |
0.13143645 |
|
60 |
0.98592639 |
0.00139853 |
0.40119709 |
0.15165744 |
|
61 |
0.99073928 |
0.00230569 |
0.42935127 |
0.17454913 |
|
62 |
0.99419379 |
0.00376192 |
0.45896170 |
0.20040826 |
|
63 |
0.99656346 |
0.00607694 |
0.49007775 |
0.22955855 |
|
64 |
0.99810354 |
0.00972311 |
0.52274960 |
0.26235263 |
|
65 |
0.99904080 |
0.01541468 |
0.55702826 |
0.29917406 |
|
66 |
0.99956626 |
0.02422308 |
0.59296557 |
0.34043944 |
|
67 |
0.99983122 |
0.03774293 |
0.63061418 |
0.38660072 |
|
68 |
0.99994699 |
0.05832999 |
0.67002756 |
0.43814749 |
|
69 |
0.99998812 |
0.08943931 |
0.71126003 |
0.49560945 |
|
70 |
0.99999861 |
0.13610330 |
0.75436670 |
0.55955906 |
|
71 |
1.00000000 |
0.20560286 |
0.79940351 |
0.63061418 |
|
72 |
1.00000000 |
0.30840429 |
0.84642725 |
0.70944095 |
|
73 |
1.00000000 |
0.45945946 |
0.89549550 |
0.79675676 |
|
74 |
1.00000000 |
0.68000000 |
0.94666667 |
0.89333333 |
|
75 |
1.00000000 |
1.00000000 |
1.00000000 |
1.00000000
|
Dice Probability Basics
The Probabilities of Two Dice Totals
Before you play any dice game it is good to
know the probability of any given total to be thrown. First lets
look at the possibilities of the total of two dice. The table
below shows the six possibilities for die 1 along the left
column and the six possibilities for die 2 along the top column.
The body of the table shows the sum of die 1 and die 2.
|
Two dice totals
|
|
Die 1 |
Die 2 |
|
1 |
2 |
3 |
4 |
5 |
6 |
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
|
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
5 |
6 |
7 |
8 |
9 |
10 |
11 |
|
6 |
7 |
8 |
9 |
10 |
11 |
12 |
The colors of the body of the table
illustrate the number of ways to throw each total. The
probability of throwing any given total is the number of ways
to throw that total divided by the total number of
combinations (36). In the following table the specific number
of ways to throw each total and the probability of throwing
that total is shown.
|
Total |
Number of combinations |
Probability |
|
2 |
1 |
2.78% |
|
3 |
2 |
5.56% |
|
4 |
3 |
8.33% |
|
5 |
4 |
11.11% |
|
6 |
5 |
13.89% |
|
7 |
6 |
16.67% |
|
8 |
5 |
13.89% |
|
9 |
4 |
11.11% |
|
10 |
3 |
8.33% |
|
11 |
2 |
5.56% |
|
12 |
1 |
2.78% |
|
Total |
36 |
100% |
The following shows the probability of
throwing each total in a chart format. As the chart shows the
closer the total is to 7 the greater is the probability of it
being thrown.
The Field Bet Example
Now that we understand the probability of throwing each total we
can apply this information to the dice games in the casinos to
calculate the house edge. For example consider the field bet in
craps. This bet pays 1:1 (even money) if the next throw is a 3,
4, 9, 10, or 11, 2:1 (double the bet) on the 2, and 3:1 (triple
the bet) on the 12. Note that there are 7 totals that win and
only 4 that lose which might cause someone who didn't know
better to think it was a good gamble. The player's return can be
defined as the sum of the products of the probability of each
event and the net return of that event. The following table
shows each possible total, the net return, the probability of
throwing that total, and the average return. The average return
is the product of the net return and the probability. The
player's return is the sum of the average returns.
|
Total |
Net return |
Probability |
Average return |
|
2 |
2 |
0.0278 |
0.0556 |
|
3 |
1 |
0.0556 |
0.0556 |
|
4 |
1 |
0.0833 |
0.0833 |
|
5 |
-1 |
0.1111 |
-0.1111 |
|
6 |
-1 |
0.1389 |
-0.1389 |
|
7 |
-1 |
0.1667 |
-0.1667 |
|
8 |
-1 |
0.1389 |
-0.1389 |
|
9 |
1 |
0.1111 |
0.1111 |
|
10 |
1 |
0.0833 |
0.0833 |
|
11 |
1 |
0.0556 |
0.0556 |
|
12 |
3 |
0.0278 |
0.0834 |
|
Total |
|
1 |
-0.0278 |
The last row shows the player's return to
be -.0278, in other words for every $1 bet the player can
expect to lose 2.78 cents. The player's loss is the house's
gain so the house edge is the product of -1 and the player's
return, in this case 0.0278 or 2.78%.
|
