Durango Bill's
Bridge Probabilities and Combinatorics
Bridge Probabilities and Statistics - Suit Splits
What is the probability of various possible splits given that you and dummy have "N" cards?
Math notation is
generally the same as that used in Microsoft's Excel. The MathNotation
link will also give examples of the notation as used
here.
The tables below show the number of possible combinations and the probability of having the opponent's cards split in any possible way given that you and dummy have a combined total of "N" in a suit. These calculations are only valid for random hands. Both the bidding and play of the hand reveal addition knowledge and will modify the results shown here.
Each line has 2 COMBIN() functions. The first is for the split in the suit of interest while the second fills out the remainder of the hand using the 3 other suits in the remainder of the deck.
Note: For all combinations, there are COMBIN(26,13) = 10,400,600 possible hands for a specific opponent after 26 cards have been removed from the deck for you and dummy.
You and dummy have a combined total of 11 cards in the suit.
You and dummy have a combined total of 10 cards in the suit.
You and dummy have a combined total of 9 cards in the suit.
You and dummy have a combined total of 8 cards in a suit.
You and dummy have a combined total of 7 cards in a suit.
You and dummy have a combined total of 6 cards in a suit.
You and dummy have a combined total of 5 cards in a suit.
You and dummy have a combined total of 4 cards in a suit.
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The tables below show the number of possible combinations and the probability of having the opponent's cards split in any possible way given that you and dummy have a combined total of "N" in a suit. These calculations are only valid for random hands. Both the bidding and play of the hand reveal addition knowledge and will modify the results shown here.
Each line has 2 COMBIN() functions. The first is for the split in the suit of interest while the second fills out the remainder of the hand using the 3 other suits in the remainder of the deck.
Note: For all combinations, there are COMBIN(26,13) = 10,400,600 possible hands for a specific opponent after 26 cards have been removed from the deck for you and dummy.
You and dummy have a combined total of 11 cards in the suit.
Split
Number
of possible
hands
Probability
-------------------------------------------------------------
"2,0" COMBIN(2,0) * COMBIN(24,13) = 2,496,144 .2400
"1,1" COMBIN(2,1) * COMBIN(24,12) = 5,408,312 .5200
"0,2" COMBIN(2,2) * COMBIN(24,11) = 2,496,144 .2400
Total COMBIN(26,13) = 10,400,600 1.0000
-------------------------------------------------------------
"2,0" COMBIN(2,0) * COMBIN(24,13) = 2,496,144 .2400
"1,1" COMBIN(2,1) * COMBIN(24,12) = 5,408,312 .5200
"0,2" COMBIN(2,2) * COMBIN(24,11) = 2,496,144 .2400
Total COMBIN(26,13) = 10,400,600 1.0000
You and dummy have a combined total of 10 cards in the suit.
Split
Number
of possible
hands
Probability
-------------------------------------------------------------
"3,0" COMBIN(3,0) * COMBIN(23,13) = 1,144,066 .1100
"2,1" COMBIN(3,1) * COMBIN(23,12) = 4,056,234 .3900
"1,2" COMBIN(3,2) * COMBIN(23,11) = 4,056,234 .3900
"0,3" COMBIN(3,3) * COMBIN(23,10) = 1,144,066 .1100
Total COMBIN(26,13) = 10,400,600 1.0000
-------------------------------------------------------------
"3,0" COMBIN(3,0) * COMBIN(23,13) = 1,144,066 .1100
"2,1" COMBIN(3,1) * COMBIN(23,12) = 4,056,234 .3900
"1,2" COMBIN(3,2) * COMBIN(23,11) = 4,056,234 .3900
"0,3" COMBIN(3,3) * COMBIN(23,10) = 1,144,066 .1100
Total COMBIN(26,13) = 10,400,600 1.0000
You and dummy have a combined total of 9 cards in the suit.
Split
Number
of possible
hands
Probability
-------------------------------------------------------------
"4,0" COMBIN(4,0) * COMBIN(22,13) = 497,420 .0478
"3,1" COMBIN(4,1) * COMBIN(22,12) = 2,586,584 .2487
"2,2" COMBIN(4,2) * COMBIN(22,11) = 4,232,592 .4070
"1,3" COMBIN(4,3) * COMBIN(22,10) = 2,586,584 .2487
"0,4" COMBIN(4,4) * COMBIN(22, 9) = 497,420 .0478
Total COMBIN(26,13 = 10,400,600 1.0000
-------------------------------------------------------------
"4,0" COMBIN(4,0) * COMBIN(22,13) = 497,420 .0478
"3,1" COMBIN(4,1) * COMBIN(22,12) = 2,586,584 .2487
"2,2" COMBIN(4,2) * COMBIN(22,11) = 4,232,592 .4070
"1,3" COMBIN(4,3) * COMBIN(22,10) = 2,586,584 .2487
"0,4" COMBIN(4,4) * COMBIN(22, 9) = 497,420 .0478
Total COMBIN(26,13 = 10,400,600 1.0000
You and dummy have a combined total of 8 cards in a suit.
Split
Number
of possible
hands
Probability
-------------------------------------------------------------
"5,0" COMBIN(5,0) * COMBIN(21,13) = 203,490 .0196
"4,1" COMBIN(5,1) * COMBIN(21,12) = 1,469,650 .1413
"3,2" COMBIN(5,2) * COMBIN(21,11) = 3,527,160 .3391
"2,3" COMBIN(5,3) * COMBIN(21,10) = 3,527,160 .3391
"1,4" COMBIN(5,4) * COMBIN(21, 9) = 1,469,650 .1413
"0,5" COMBIN(5,5) * COMBIN(21, 8) = 203,490 .0196
Total COMBIN(26,13) = 10,400,600 1.0000
-------------------------------------------------------------
"5,0" COMBIN(5,0) * COMBIN(21,13) = 203,490 .0196
"4,1" COMBIN(5,1) * COMBIN(21,12) = 1,469,650 .1413
"3,2" COMBIN(5,2) * COMBIN(21,11) = 3,527,160 .3391
"2,3" COMBIN(5,3) * COMBIN(21,10) = 3,527,160 .3391
"1,4" COMBIN(5,4) * COMBIN(21, 9) = 1,469,650 .1413
"0,5" COMBIN(5,5) * COMBIN(21, 8) = 203,490 .0196
Total COMBIN(26,13) = 10,400,600 1.0000
You and dummy have a combined total of 7 cards in a suit.
Split
Number
of possible
hands
Probability
-------------------------------------------------------------
"6,0" COMBIN(6,0) * COMBIN(20,13) = 77,520 .0075
"5,1" COMBIN(6,1) * COMBIN(20,12) = 755,820 .0727
"4,2" COMBIN(6,2) * COMBIN(20,11) = 2,519,400 .2422
"3,3" COMBIN(6,3) * COMBIN(20,10) = 3,695,120 .3553
"2,4" COMBIN(6,4) * COMBIN(20, 9) = 2,519,400 .2422
"1,5" COMBIN(6,5) * COMBIN(20, 8) = 755,820 .0727
"0,6" COMBIN(6,6) * COMBIN(20, 7) = 77,520 .0075
Total COMBIN(26,13) = 10,400,600 1.0000
-------------------------------------------------------------
"6,0" COMBIN(6,0) * COMBIN(20,13) = 77,520 .0075
"5,1" COMBIN(6,1) * COMBIN(20,12) = 755,820 .0727
"4,2" COMBIN(6,2) * COMBIN(20,11) = 2,519,400 .2422
"3,3" COMBIN(6,3) * COMBIN(20,10) = 3,695,120 .3553
"2,4" COMBIN(6,4) * COMBIN(20, 9) = 2,519,400 .2422
"1,5" COMBIN(6,5) * COMBIN(20, 8) = 755,820 .0727
"0,6" COMBIN(6,6) * COMBIN(20, 7) = 77,520 .0075
Total COMBIN(26,13) = 10,400,600 1.0000
You and dummy have a combined total of 6 cards in a suit.
Split
Number
of possible
hands
Probability
-------------------------------------------------------------
"7,0" COMBIN(7,0) * COMBIN(19,13) = 27,132 .0026
"6,1" COMBIN(7,1) * COMBIN(19,12) = 352,716 .0339
"5,2" COMBIN(7,2) * COMBIN(19,11) = 1,587,222 .1526
"4,3" COMBIN(7,3) * COMBIN(19,10) = 3,233,230 .3109
"3,4" COMBIN(7,4) * COMBIN(19, 9) = 3,233,230 .3109
"2,5" COMBIN(7,5) * COMBIN(19, 8) = 1,587,222 .1526
"1,6" COMBIN(7,6) * COMBIN(19, 7) = 352,716 .0339
"0,7" COMBIN(7,7) * COMBIN(19, 6) = 27,132 .0026
Total COMBIN(26,13) = 10,400,600 1.0000
-------------------------------------------------------------
"7,0" COMBIN(7,0) * COMBIN(19,13) = 27,132 .0026
"6,1" COMBIN(7,1) * COMBIN(19,12) = 352,716 .0339
"5,2" COMBIN(7,2) * COMBIN(19,11) = 1,587,222 .1526
"4,3" COMBIN(7,3) * COMBIN(19,10) = 3,233,230 .3109
"3,4" COMBIN(7,4) * COMBIN(19, 9) = 3,233,230 .3109
"2,5" COMBIN(7,5) * COMBIN(19, 8) = 1,587,222 .1526
"1,6" COMBIN(7,6) * COMBIN(19, 7) = 352,716 .0339
"0,7" COMBIN(7,7) * COMBIN(19, 6) = 27,132 .0026
Total COMBIN(26,13) = 10,400,600 1.0000
You and dummy have a combined total of 5 cards in a suit.
Split
Number
of possible
hands
Probability
-------------------------------------------------------------
"8,0" COMBIN(8,0) * COMBIN(18,13) = 8,568 .0008
"7,1" COMBIN(8,1) * COMBIN(18,12) = 148,512 .0143
"6,2" COMBIN(8,2) * COMBIN(18,11) = 891,072 .0857
"5,3" COMBIN(8,3) * COMBIN(18,10) = 2,450,448 .2356
"4,4" COMBIN(8,4) * COMBIN(18, 9) = 3,403,400 .3272
"3,5" COMBIN(8,5) * COMBIN(18, 8) = 2,450,448 .2356
"2,6" COMBIN(8,6) * COMBIN(18, 7) = 891,072 .0857
"1,7" COMBIN(8,7) * COMBIN(18, 6) = 148,512 .0143
"0,8" COMBIN(8,8) * COMBIN(18, 5) = 8,568 .0008
Total COMBIN(26,13) = 10,400,600 1.0000
-------------------------------------------------------------
"8,0" COMBIN(8,0) * COMBIN(18,13) = 8,568 .0008
"7,1" COMBIN(8,1) * COMBIN(18,12) = 148,512 .0143
"6,2" COMBIN(8,2) * COMBIN(18,11) = 891,072 .0857
"5,3" COMBIN(8,3) * COMBIN(18,10) = 2,450,448 .2356
"4,4" COMBIN(8,4) * COMBIN(18, 9) = 3,403,400 .3272
"3,5" COMBIN(8,5) * COMBIN(18, 8) = 2,450,448 .2356
"2,6" COMBIN(8,6) * COMBIN(18, 7) = 891,072 .0857
"1,7" COMBIN(8,7) * COMBIN(18, 6) = 148,512 .0143
"0,8" COMBIN(8,8) * COMBIN(18, 5) = 8,568 .0008
Total COMBIN(26,13) = 10,400,600 1.0000
You and dummy have a combined total of 4 cards in a suit.
Split
Number
of possible
hands
Probability
-------------------------------------------------------------
"9,0" COMBIN(9,0) * COMBIN(17,13) = 2,380 .0002
"8,1" COMBIN(9,1) * COMBIN(17,12) = 55,692 .0054
"7,2" COMBIN(9,2) * COMBIN(17,11) = 445,536 .0428
"6,3" COMBIN(9,3) * COMBIN(17,10) = 1,633,632 .1571
"5,4" COMBIN(9,4) * COMBIN(17, 9) = 3,063,060 .2945
"4,5" COMBIN(9,5) * COMBIN(17, 8) = 3,063,060 .2945
"3,6" COMBIN(9,6) * COMBIN(17, 7) = 1,633,632 .1571
"2,7" COMBIN(9,7) * COMBIN(17, 6) = 445,536 .0428
"1,8" COMBIN(9,8) * COMBIN(17, 5) = 55,692 .0054
"0,9" COMBIN(9,9) * COMBIN(17, 4) = 2,380 .0002
Total COMBIN(26,13) = 10,400,600 1.0000
-------------------------------------------------------------
"9,0" COMBIN(9,0) * COMBIN(17,13) = 2,380 .0002
"8,1" COMBIN(9,1) * COMBIN(17,12) = 55,692 .0054
"7,2" COMBIN(9,2) * COMBIN(17,11) = 445,536 .0428
"6,3" COMBIN(9,3) * COMBIN(17,10) = 1,633,632 .1571
"5,4" COMBIN(9,4) * COMBIN(17, 9) = 3,063,060 .2945
"4,5" COMBIN(9,5) * COMBIN(17, 8) = 3,063,060 .2945
"3,6" COMBIN(9,6) * COMBIN(17, 7) = 1,633,632 .1571
"2,7" COMBIN(9,7) * COMBIN(17, 6) = 445,536 .0428
"1,8" COMBIN(9,8) * COMBIN(17, 5) = 55,692 .0054
"0,9" COMBIN(9,9) * COMBIN(17, 4) = 2,380 .0002
Total COMBIN(26,13) = 10,400,600 1.0000
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