Durango Bill's
Bridge Probabilities and Combinatorics
Bridge Probabilities and Statistics - Point Count
Computer Program by Bill Butler
What is the probability that you will be dealt a hand worth "N" points?
One of the best
methods of evaluating the strength of a hand is by adding up
the point count. This method awards 4 points for every Ace in
a hand, 3 points for each King, 2 points for each Queen, and 1
point for each Jack. Additional points are awarded for
distributional strength such as a short suit, but here we will
just give the probabilities for raw honor count power.
The highest possible honor count that can exist in a hand would have all four Aces, four Kings, four Queens, and one of the four Jacks for a total of 37 points. At the other end of the scale there are over 2 billion hands that have a zero point count (10 high or worse). Of these, COMBIN(32, 13) = 347,373,600 are Yarboroughs (9 high or less). We also note there are COMBIN(52, 13) = 635,013,599,600 different hands that could be dealt.
For each row in the table below, column 1 gives the point count via the 4, 3, 2, 1 analysis. Note that there are many combinations that can produce a given count. For example a hand that has 2 Aces and a Queen, or another hand that has 4 Queens and 2 Jacks would both be included in the "10" row.
The second column shows the exact number of possible hands that will produce the given point count (Honors only - distribution is not counted). For the third column we divide the total combinations in the second column by the total number of all hands (635,013,559,600) to get the probability of being dealt this particular point count. The fourth column gives the cumulative probability of receiving a particular point count or higher. Finally the fifth column shows the average number of honor cards that a hand will have, given that the hand has a particular honor count.
High point count hands have a very low probability, and hence scientific notation is used. For example, to express the probability of getting a 37 point hand as a fixed point decimal number, you have to move the decimal point 12 places further to the left (e.g. .00000000000629908)
Bridge Honor Count Combinations for 13 cards (1 hand)
The total number of possible 13 card hands is: COMBIN(52,13) = 635,013,559,600
Honor Total Honor Count Cumulative Avg. Nbr.
Count Hands Probability Probability of Honors
---------------------------------------------------------------------
37 4 6.29908e-012 6.29908e-012 13.0000
36 60 9.44862e-011 1.00785e-010 12.4000
35 624 9.82656e-010 1.08344e-009 12.0769
34 4,484 7.06127e-009 8.14471e-009 11.4585
33 22,360 3.52118e-008 4.33566e-008 11.2161
32 109,156 1.71896e-007 2.15252e-007 10.6851
31 388,196 6.11319e-007 8.26571e-007 10.4401
30 1,396,068 2.19849e-006 3.02506e-006 10.0376
29 4,236,588 6.67165e-006 9.69671e-006 9.7116
28 11,790,760 1.85677e-005 2.82644e-005 9.4187
27 31,157,940 4.90666e-005 7.7331e-005 9.0614
26 74,095,248 0.000116683 0.000194014 8.7857
25 167,819,892 0.000264278 0.000458292 8.4670
24 354,993,864 0.000559034 0.00101733 8.1655
23 710,603,628 0.00111904 0.00213636 7.8697
22 1,333,800,036 0.00210043 0.00423679 7.5769
21 2,399,507,844 0.00377867 0.00801546 7.2797
20 4,086,538,404 0.00643536 0.0144508 6.9817
19 6,579,838,440 0.0103617 0.0248125 6.7023
18 10,192,504,020 0.0160508 0.0408634 6.3982
17 14,997,082,848 0.0236169 0.0644803 6.1113
16 21,024,781,756 0.0331092 0.0975895 5.8196
15 28,090,962,724 0.0442368 0.14183 5.5275
14 36,153,374,224 0.0569332 0.19876 5.2273
13 43,906,944,752 0.0691433 0.2679 4.9381
12 50,971,682,080 0.0802687 0.34817 4.6450
11 56,799,933,520 0.0894468 0.43762 4.3279
10 59,723,754,816 0.0940511 0.53167 4.0415
9 59,413,313,872 0.0935623 0.62523 3.7356
8 56,466,608,128 0.0889219 0.71415 3.4192
7 50,979,441,968 0.0802809 0.79443 3.0811
6 41,619,399,184 0.065541 0.85998 2.8059
5 32,933,031,040 0.0518619 0.91184 2.4620
4 24,419,055,136 0.0384544 0.95029 2.0525
3 15,636,342,960 0.0246236 0.97492 1.7448
2 8,611,542,576 0.0135612 0.98848 1.4186
1 5,006,710,800 0.00788442 0.99636 1.0000
0 2,310,789,600 0.00363896 1 0.0000
Similar calculations can be made for the combined point count totals if you add the point count total in your hand to the point count total in your partner’s hand. The table below shows the number of ways any possible point count total could occur and the probability of each of these possibilities.
Bridge Honor Count Combinations for 26 cards (2 hands)
The total number of possible combined hands is: COMBIN(52, 26) = 495,918,532,948,104
Honor Total Honor Count Cumulative Avg. Nbr.
Count Hands Probability Probability of Honors
--------------------------------------------------------------------------
40 254,186,856 5.12558e-007 0.00000051 16.0000
39 2,403,221,184 4.846e-006 0.00000536 15.0000
38 9,913,287,384 1.99897e-005 0.00002535 14.2424
37 31,673,222,784 6.38678e-005 0.00008922 13.7840
36 89,195,378,184 0.000179859 0.00026908 13.3201
35 211,712,342,400 0.00042691 0.00069598 12.9023
34 459,808,617,240 0.000927186 0.00162317 12.5337
33 920,662,591,680 0.00185648 0.00347965 12.1553
32 1,691,764,828,380 0.00341138 0.00689103 11.8072
31 2,914,543,903,680 0.00587706 0.01276809 11.4616
30 4,734,398,485,800 0.00954673 0.02231481 11.1296
29 7,257,585,574,080 0.0146346 0.03694945 10.8019
28 10,533,038,026,200 0.0212395 0.05818890 10.4784
27 14,596,737,921,600 0.0294337 0.08762264 10.1614
26 19,258,439,527,560 0.0388339 0.12645652 9.8462
25 24,259,718,677,440 0.0489188 0.17537528 9.5360
24 29,295,317,098,380 0.0590728 0.23444812 9.2246
23 33,876,647,618,880 0.0683109 0.30275903 8.9182
22 37,522,340,994,600 0.0756623 0.37842134 8.6110
21 39,905,485,171,200 0.0804678 0.45888916 8.3055
20 40,775,251,597,080 0.0822217 0.54111084 8.0000
19 39,905,485,171,200 0.0804678 0.62157866 7.6945
18 37,522,340,994,600 0.0756623 0.69724097 7.3890
17 33,876,647,618,880 0.0683109 0.76555188 7.0818
16 29,295,317,098,380 0.0590728 0.82462472 6.7754
15 24,259,718,677,440 0.0489188 0.87354348 6.4640
14 19,258,439,527,560 0.0388339 0.91237736 6.1538
13 14,596,737,921,600 0.0294337 0.94181110 5.8386
12 10,533,038,026,200 0.0212395 0.96305055 5.5216
11 7,257,585,574,080 0.0146346 0.97768519 5.1981
10 4,734,398,485,800 0.00954673 0.98723191 4.8704
9 2,914,543,903,680 0.00587706 0.99310897 4.5384
8 1,691,764,828,380 0.00341138 0.99652035 4.1928
7 920,662,591,680 0.00185648 0.99837683 3.8447
6 459,808,617,240 0.000927186 0.99930402 3.4663
5 211,712,342,400 0.00042691 0.99973092 3.0977
4 89,195,378,184 0.000179859 0.99991078 2.6799
3 31,673,222,784 6.38678e-005 0.99997465 2.2160
2 9,913,287,384 1.99897e-005 0.99999464 1.7576
1 2,403,221,184 4.846e-006 0.99999949 1.0000
0 254,186,856 5.12558e-007 1.00000000 0.0000
The symmetry that exists above vs. below the 20-point line is not just a coincidence. For example the numbers for a 26 point hand are the same as for a 14 point hand. (If your side has only 14 high card points, then the opponents have 26 high card points. It’s a 50-50 toss-up as to which side gets the good hands.
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The highest possible honor count that can exist in a hand would have all four Aces, four Kings, four Queens, and one of the four Jacks for a total of 37 points. At the other end of the scale there are over 2 billion hands that have a zero point count (10 high or worse). Of these, COMBIN(32, 13) = 347,373,600 are Yarboroughs (9 high or less). We also note there are COMBIN(52, 13) = 635,013,599,600 different hands that could be dealt.
For each row in the table below, column 1 gives the point count via the 4, 3, 2, 1 analysis. Note that there are many combinations that can produce a given count. For example a hand that has 2 Aces and a Queen, or another hand that has 4 Queens and 2 Jacks would both be included in the "10" row.
The second column shows the exact number of possible hands that will produce the given point count (Honors only - distribution is not counted). For the third column we divide the total combinations in the second column by the total number of all hands (635,013,559,600) to get the probability of being dealt this particular point count. The fourth column gives the cumulative probability of receiving a particular point count or higher. Finally the fifth column shows the average number of honor cards that a hand will have, given that the hand has a particular honor count.
High point count hands have a very low probability, and hence scientific notation is used. For example, to express the probability of getting a 37 point hand as a fixed point decimal number, you have to move the decimal point 12 places further to the left (e.g. .00000000000629908)
Bridge Honor Count Combinations for 13 cards (1 hand)
The total number of possible 13 card hands is: COMBIN(52,13) = 635,013,559,600
Honor Total Honor Count Cumulative Avg. Nbr.
Count Hands Probability Probability of Honors
---------------------------------------------------------------------
37 4 6.29908e-012 6.29908e-012 13.0000
36 60 9.44862e-011 1.00785e-010 12.4000
35 624 9.82656e-010 1.08344e-009 12.0769
34 4,484 7.06127e-009 8.14471e-009 11.4585
33 22,360 3.52118e-008 4.33566e-008 11.2161
32 109,156 1.71896e-007 2.15252e-007 10.6851
31 388,196 6.11319e-007 8.26571e-007 10.4401
30 1,396,068 2.19849e-006 3.02506e-006 10.0376
29 4,236,588 6.67165e-006 9.69671e-006 9.7116
28 11,790,760 1.85677e-005 2.82644e-005 9.4187
27 31,157,940 4.90666e-005 7.7331e-005 9.0614
26 74,095,248 0.000116683 0.000194014 8.7857
25 167,819,892 0.000264278 0.000458292 8.4670
24 354,993,864 0.000559034 0.00101733 8.1655
23 710,603,628 0.00111904 0.00213636 7.8697
22 1,333,800,036 0.00210043 0.00423679 7.5769
21 2,399,507,844 0.00377867 0.00801546 7.2797
20 4,086,538,404 0.00643536 0.0144508 6.9817
19 6,579,838,440 0.0103617 0.0248125 6.7023
18 10,192,504,020 0.0160508 0.0408634 6.3982
17 14,997,082,848 0.0236169 0.0644803 6.1113
16 21,024,781,756 0.0331092 0.0975895 5.8196
15 28,090,962,724 0.0442368 0.14183 5.5275
14 36,153,374,224 0.0569332 0.19876 5.2273
13 43,906,944,752 0.0691433 0.2679 4.9381
12 50,971,682,080 0.0802687 0.34817 4.6450
11 56,799,933,520 0.0894468 0.43762 4.3279
10 59,723,754,816 0.0940511 0.53167 4.0415
9 59,413,313,872 0.0935623 0.62523 3.7356
8 56,466,608,128 0.0889219 0.71415 3.4192
7 50,979,441,968 0.0802809 0.79443 3.0811
6 41,619,399,184 0.065541 0.85998 2.8059
5 32,933,031,040 0.0518619 0.91184 2.4620
4 24,419,055,136 0.0384544 0.95029 2.0525
3 15,636,342,960 0.0246236 0.97492 1.7448
2 8,611,542,576 0.0135612 0.98848 1.4186
1 5,006,710,800 0.00788442 0.99636 1.0000
0 2,310,789,600 0.00363896 1 0.0000
Similar calculations can be made for the combined point count totals if you add the point count total in your hand to the point count total in your partner’s hand. The table below shows the number of ways any possible point count total could occur and the probability of each of these possibilities.
Bridge Honor Count Combinations for 26 cards (2 hands)
The total number of possible combined hands is: COMBIN(52, 26) = 495,918,532,948,104
Honor Total Honor Count Cumulative Avg. Nbr.
Count Hands Probability Probability of Honors
--------------------------------------------------------------------------
40 254,186,856 5.12558e-007 0.00000051 16.0000
39 2,403,221,184 4.846e-006 0.00000536 15.0000
38 9,913,287,384 1.99897e-005 0.00002535 14.2424
37 31,673,222,784 6.38678e-005 0.00008922 13.7840
36 89,195,378,184 0.000179859 0.00026908 13.3201
35 211,712,342,400 0.00042691 0.00069598 12.9023
34 459,808,617,240 0.000927186 0.00162317 12.5337
33 920,662,591,680 0.00185648 0.00347965 12.1553
32 1,691,764,828,380 0.00341138 0.00689103 11.8072
31 2,914,543,903,680 0.00587706 0.01276809 11.4616
30 4,734,398,485,800 0.00954673 0.02231481 11.1296
29 7,257,585,574,080 0.0146346 0.03694945 10.8019
28 10,533,038,026,200 0.0212395 0.05818890 10.4784
27 14,596,737,921,600 0.0294337 0.08762264 10.1614
26 19,258,439,527,560 0.0388339 0.12645652 9.8462
25 24,259,718,677,440 0.0489188 0.17537528 9.5360
24 29,295,317,098,380 0.0590728 0.23444812 9.2246
23 33,876,647,618,880 0.0683109 0.30275903 8.9182
22 37,522,340,994,600 0.0756623 0.37842134 8.6110
21 39,905,485,171,200 0.0804678 0.45888916 8.3055
20 40,775,251,597,080 0.0822217 0.54111084 8.0000
19 39,905,485,171,200 0.0804678 0.62157866 7.6945
18 37,522,340,994,600 0.0756623 0.69724097 7.3890
17 33,876,647,618,880 0.0683109 0.76555188 7.0818
16 29,295,317,098,380 0.0590728 0.82462472 6.7754
15 24,259,718,677,440 0.0489188 0.87354348 6.4640
14 19,258,439,527,560 0.0388339 0.91237736 6.1538
13 14,596,737,921,600 0.0294337 0.94181110 5.8386
12 10,533,038,026,200 0.0212395 0.96305055 5.5216
11 7,257,585,574,080 0.0146346 0.97768519 5.1981
10 4,734,398,485,800 0.00954673 0.98723191 4.8704
9 2,914,543,903,680 0.00587706 0.99310897 4.5384
8 1,691,764,828,380 0.00341138 0.99652035 4.1928
7 920,662,591,680 0.00185648 0.99837683 3.8447
6 459,808,617,240 0.000927186 0.99930402 3.4663
5 211,712,342,400 0.00042691 0.99973092 3.0977
4 89,195,378,184 0.000179859 0.99991078 2.6799
3 31,673,222,784 6.38678e-005 0.99997465 2.2160
2 9,913,287,384 1.99897e-005 0.99999464 1.7576
1 2,403,221,184 4.846e-006 0.99999949 1.0000
0 254,186,856 5.12558e-007 1.00000000 0.0000
The symmetry that exists above vs. below the 20-point line is not just a coincidence. For example the numbers for a 26 point hand are the same as for a 14 point hand. (If your side has only 14 high card points, then the opponents have 26 high card points. It’s a 50-50 toss-up as to which side gets the good hands.
Return to Bridge Combinatorics main page
Web page generated via Sea Monkey's Composer HTML editor
within a Linux Cinnamon Mint 18 operating system.
(Goodbye Microsoft)