Main     Study 7Z77 by Richard Pavlicek    

Against All Odds

What are the odds against being dealt a Yarborough? Or a singleton? Or 4-3-3-3 shape? Or an eight-card suit? Or all four aces? Or about anything else of interest or not. Check out the following tables for answers.

General CriteriaHand PatternsSuit LengthsSpecific Cards
Each table lists a variety of hand types, the method of calculation*, the number of possible bridge hands, the percent probability (rounded to 12 decimal places) and the approximate odds against being dealt such a hand. In the few cases in which the odds favor a holding it is indicated simply as “favored.” For cases related to HCP, see the separate work on High Card Expectancy.

*Combinatorial notation (NcR) is frequently used. Think of this as “N choose R” which means, given N items, the number of ways to select R of them. For example, 26c2 means “26 choose 2” which can be done in 325 ways; I could have just use the number 325, but the notation is meaningful to show how the number is derived. When N=13 (common) it is omitted to save space; e.g., c4 = 13c4 = 715.

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General Criteria

This table lists hand types based on general characteristics. Clarification of terms: A “Yarborough” has no card above a nine. “Flat” means 4-3-3-3 shape; “balanced” means 4-3-3-3, 4-4-3-2 or 5-3-3-2; “semibalanced” means 5-4-2-2 or 6-3-2-2; and “quasibalanced” means 4-4-4-1 or 5-4-3-1. (OK, I invented the last one.)

A “guaranteed 7 NT” hand must have all four aces and no possible loser regardless of the remaining distribution and opening lead; in other words, you can claim 13 tricks before the lead. Example: S A-K H A-K-Q-J-10-9-2 D A-K-Q C A, but note that changing the H 9 to the eight would disqualify it.

Hand TypeCalculationHandsPercentOdds Against
Highest card 516c135600.0000000881871133952784 to 1
Highest card 620c13775200.0000122076138191608 to 1
Highest card 724c1324961440.000393085149254397 to 1
Highest card 828c13374421600.00589627724216959 to 1
Flat highest card 84·7c3³·7c460025000.000945255406105791 to 1
Yarborough32c133473736000.0547033358181827 to 1
Flat Yarborough4·8c3³·8c4491724800.00774353228512913 to 1
Highest card 1036c1323107896000.363896103487274 to 1
Highest card J40c13120332228801.89495526482652 to 1
Highest card Q44c13519155264328.17549887670111 to 1
Aceless48c1319292824929630.3817527010807 to 3
Balanced4·c3(c4·c3²+3·c4²·c2+3·c5·c3·c2)=A30229282283247.60415242509413 to 12
Semibalanced12·c2²(c5·c4+c6·c3)=B10301290084816.2221576674505 to 1
Quasibalanced4·c4·c1(c4²+6·c5·c3)=C10111907806015.9239242267045 to 1
Balanced or semi-A+B40530572368063.826310092544(favored)
Semi- or quasi-B+C20413197890832.1460818941542 to 1
Bal- semi- or quasi-A+B+C50642480174079.750234319248(favored)
All black cards26c13104006000.00163785479061054 to 1
Guaranteed 7 NTsum of all cases37560.000000591483169066442 to 1
5 honors in any suit4·47c8-6·42c312577611000.198068384680504 to 1
4 honors in any suit5c4(4·47c9-6·5c4·42c5+4·5c4²·37c1)271254012004.27162550939622 to 1
3 honors in any suit5c3(4·47c10-6·5c3·42c7+4·5c3²·37c4-5c3³·32c1)19119953324030.1095197652857 to 3
2 honors in any suit5c2(4·47c11-6·5c2·42c9+4·5c2²·37c7-5c2³·32c5)46831838668073.749352214620(favored)
1 honor in any suit5c1(4·47c12-6·5c1·42c11+4·5c1²·37c10-5c1³·32c9)55957837862088.120697607226(favored)
2 tripletons6·c3²(26c7-2·c3·c4)12211488432019.2302798064534 to 1
1 tripleton4·c3(39c10-3·c3(26c7-c3·c4))28234539590444.4628924273455 to 4
At least 1 tripleton4·c3·39c10-6·c3²·26c7+4·c3³·c447136613638474.229302549211(favored)
2 doubletons6·c2²(26c9-2·c2·c7)10428659841616.4227356785415 to 1
1 doubleton4·c2(39c11-3·c2(26c9-c2·c7))30458431411247.96500948796413 to 12
At least 1 doubleton4·c2·39c11-6·c2²·26c9+4·c2³·c741212823745664.900698768638(favored)
2 singletons6·c1²(26c11-2·c1·c10)78267861361.23253842657080 to 1
1 singleton4·c1(39c12-3·c1(26c11-c1·c10))18770035429629.5584797298247 to 3
At least 1 singleton4·c1·39c12-6·c1²·26c11+4·c1³·c1019552965380030.7914139539279 to 4
2 voids6(26c13-2)624035880.00982712684710175 to 1
1 void4(39c13-3(26c13-c13))323648945885.09672495944619 to 1
At least 1 void4·39c13-6·26c13+4·c13324272981805.10655208692319 to 1
At least 1 sing./voidnot 4333 4432 5332 5422 6322 722222645051099235.6607363053239 to 5

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Hand Patterns

The following table lists the 39 generic hand patterns in their order of frequency. In each calculation the first number (4, 12 or 24) is the permutation factor for that pattern. For example, the generic pattern 4-4-3-2 can be permuted 12 different ways to form specific patterns. (See the second table for specific patterns.)

PatternFreaknessCalculationHandsPercentOdds Against
4-4-3-2112·c4²·c3·c213685288760021.5511756451637 to 2
5-3-3-2212·c5·c3²·c29853407907215.51684646451811 to 2
5-4-3-1424·c5·c4·c3·c18211173256012.93070538709813 to 2
5-4-2-2312·c5·c4·c2²6718232664010.5796680439898 to 1
4-3-3-304·c4·c3³6690585616010.53613031541317 to 2
6-3-2-2412·c6·c3·c2²358305742085.64248962346117 to 1
6-4-2-1624·c6·c4·c2·c1298588118404.70207468621720 to 1
6-3-3-1512·c6·c3²·c1218964620163.44818810322628 to 1
5-5-2-1612·c5²·c2·c1201546979923.17390041319731 to 1
4-4-4-134·c4³·c1190073455002.99321883960632 to 1
7-3-2-1724·c7·c3·c2·c1119435247361.88082987448752 to 1
6-4-3-0724·c6·c4·c384217161601.32622619354974 to 1
5-4-4-0612·c5·c4²78953589001.24333705645279 to 1
5-5-3-0712·c5²·c356846584080.895202680645111 to 1
6-5-1-1812·c6·c5·c1²44788217760.705311202933141 to 1
6-5-2-0924·c6·c5·c241342970240.651056495015153 to 1
7-2-2-264·c7·c2³32573249280.512953602133194 to 1
7-4-1-1812·c7·c4·c1²24882343200.391839557185254 to 1
7-4-2-0924·c7·c4·c222968316800.361698052786275 to 1
7-3-3-0812·c7·c3²16843432320.265245238710376 to 1
8-2-2-1912·c8·c2²·c112214968480.192357600800519 to 1
8-3-1-1912·c8·c3·c1²7464702960.117551867155850 to 1
7-5-1-01124·c7·c5·c16890495040.108509415836921 to 1
8-3-2-01024·c8·c3·c26890495040.108509415836921 to 1
6-6-1-01112·c6²·c14593663360.0723396105571381 to 1
8-4-1-01124·c8·c4·c12871039600.0452122565982211 to 1
9-2-1-11112·c9·c2·c1²1131015600.0178108889635614 to 1
9-3-1-01224·c9·c3·c1638008800.0100471681339952 to 1
9-2-2-01212·c9·c2²522007200.00822041029112164 to 1
7-6-0-01312·c7·c6353358720.00556458542717970 to 1
8-5-0-01312·c8·c5198764280.00313007930331947 to 1
10-2-1-01424·c10·c2·c169600960.00109605470591235 to 1
9-4-0-01312·c9·c461347000.000966073859103511 to 1
10-1-1-1134·c10·c1³25133680.000395797533252653 to 1
10-3-0-01412·c10·c39815520.000154571817646947 to 1
11-1-1-01612·c11·c1²1581840.0000249103344014397 to 1
11-2-0-01612·c11·c2730080.0000114970778697862 to 1
12-1-0-01812·c12·c120280.000000319363313123056 to 1
13-0-0-0204·c1340.000000000630158753389899 to 1
Check totals:635013559600100.000000000000 

The 39 generic patterns produce 560 specific patterns, which are listed below in order of frequency. (Only one entry is shown for each specific pattern because others would be the same except for suit identity.) The notation 4=4=3=2 means four spades, four hearts, three diamonds and two clubs; whereas 4-4-3-2 means any two four-card suits, any tripleton and any doubleton.

PatternFreaknessCalculationHandsPercentOdds Against
4=3=3=30c4·c3³167264640402.63403257885337 to 1
4=4=3=21c4²·c3·c2114044073001.79593130376455 to 1
5=3=3=22c5·c3²·c282111732561.29307053871076 to 1
5=4=2=23c5·c4·c2²55985272200.881639003666112 to 1
4=4=4=13c4³·c147518363750.748304709902133 to 1
5=4=3=14c5·c4·c3·c134213221900.538779391129185 to 1
6=3=2=24c6·c3·c2²29858811840.470207468622212 to 1
6=3=3=15c6·c3²·c118247051680.287349008602347 to 1
5=5=2=16c5²·c2·c116795581660.264491701100377 to 1
6=4=2=16c6·c4·c2·c112441171600.195919778592509 to 1
7=2=2=26c7·c2³8143312320.128238400533779 to 1
5=4=4=06c5·c4²6579465750.103611421371964 to 1
7=3=2=17c7·c3·c2·c14976468640.0783679114371275 to 1
5=5=3=07c5²·c34737215340.0746002233871339 to 1
6=5=1=18c6·c5·c1²3732351480.0587759335781700 to 1
6=4=3=07c6·c4·c33509048400.0552594247311809 to 1
7=4=1=18c7·c4·c1²2073528600.0326532964323061 to 1
6=5=2=09c6·c5·c21722623760.0271273539593685 to 1
7=3=3=08c7·c3²1403619360.0221037698924523 to 1
8=2=2=19c8·c2²·c11017914040.0160298000676237 to 1
7=4=2=09c7·c4·c2957013200.0150707521996634 to 1
8=3=1=19c8·c3·c1²622058580.00979598893010207 to 1
6=6=1=011c6²·c1382805280.00602830088016587 to 1
7=5=1=011c7·c5·c1287103960.00452122566022117 to 1
8=3=2=010c8·c3·c2287103960.00452122566022117 to 1
8=4=1=011c8·c4·c1119626650.00188384402553082 to 1
9=2=1=111c9·c2·c1²94251300.00148424074767374 to 1
9=2=2=012c9·c2²43500600.000685034191145977 to 1
7=6=0=013c7·c629446560.000463715452215648 to 1
9=3=1=012c9·c3·c126583700.000418632006238872 to 1
8=5=0=013c8·c516563690.000260839942383376 to 1
10=1=1=113c10·c1³6283420.0000989493831010617 to 1
9=4=0=013c9·c45112250.0000805061551242140 to 1
10=2=1=014c10·c2·c12900040.0000456689462189671 to 1
10=3=0=014c10·c3817960.0000128809857763381 to 1
11=1=1=016c11·c1²131820.00000207586148172777 to 1
11=2=0=016c11·c260840.000000958090104374351 to 1
12=1=0=018c12·c11690.0000000266143757476683 to 1
13=0=0=020c1310.000000000157635013559599 to 1

Theoretical note: The above tables also apply to suit breaks among four hands. For example, the a priori chance of a suit splitting 4-4-3-2 is identical to that generic hand pattern; and the chance of a 4=4=3=2 split (say N=E=S=W) is identical to that specific hand pattern.

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Suit Lengths

This table lists cases pertaining to suit lengths. For example, the chance of being dealt at least 5-4 shape (any two suits) is shown to be about 37 percent, or 5-to-3 odds against.

The notation “6+ card suit” means at least a six-card suit.

Hand TypeCalculationHandsPercentOdds Against
13 card suit4·c13=A40.000000000630158753389899 to 1
12 card suit4·c12·39c1=B20280.000000319363313123056 to 1
11 card suit4·c11·39c2=C2311920.0000364074122746692 to 1
10 card suit4·c10·39c3=D104550160.00164642405560737 to 1
9 card suit4·c9·39c4=E2352378600.0370445412452698 to 1
8 card suit4·c8·39c5=F29639970360.466761219692213 to 1
7 card suit4·c7·39c6=G223946442723.52664032656427 to 1
6 card suit (max)4·c6(39c7-3·c7-3·c6·c1)=H10508004936016.5476859149585 to 1
5 card suit (max)4·c5(39c8-3·c8-3·c7·26-3·c6·26c2)-6·c5²·26c3=I28156285357244.3396600458995 to 4
4 card suit (max)4·c4·c3³+12·c4²·c3·c2+4·c4³·c1=J22276608926035.08052480018313 to 7
12+ card suitA+B20320.000000319993312506672 to 1
11+ card suitA+B+C2332240.0000367274052722761 to 1
10+ card suitA+B+C+D106882400.00168315146059411 to 1
9+ card suitA+B+C+D+E2459261000.0387276927052581 to 1
8+ card suitA+B+C+D+E+F32099231360.505488912398197 to 1
7+ card suitA+B+C+D+E+F+G256045674084.03212923896124 to 1
6+ card suitA+B+C+D+E+F+G+H13068461676820.5798151539194 to 1
5+ card suitA+B+C+D+E+F+G+H+I41224747034064.919475199817(favored)
4+ card suitA+B+C+D+E+F+G+H+I+J635013559600100.000000000000(bet on it)
9-4 two suits12·c9·c4=A61347000.000966073859103511 to 1
8-5 two suits12·c8·c5=B198764280.00313007930331947 to 1
8-4 two suits12·c8·c4·26c1=C2871039600.0452122565982211 to 1
7-6 two suits12·c7·c6=D353358720.00556458542717970 to 1
7-5 two suits12·c7·c5·26c1=E6890495040.108509415836921 to 1
7-4 two suits12·c7·c4·26c2=F47850660000.753537609971132 to 1
6-6 two suits6·c6²·26c1=G4593663360.0723396105571381 to 1
6-5 two suits12·c6·c5·26c2=H86131188001.35636769794773 to 1
6-4 two suits12·c6·c4·26c3=I382805280006.02830087976616 to 1
5-5 two suits6·c5²·26c3=J258393564004.06910309384224 to 1
5-4 two suits12·c5·c4(26c4-c4)=K15718941810024.7537104875393 to 1
4-4 two suits (no 5)12·c4²·c3·c2+4·c4³·c1=L15586023310024.5443944847693 to 1
At least 8-4A+B+C3131150880.0493084097602027 to 1
At least 7-5B+D+E7442618040.117204080566852 to 1
At least 7-4A+B+C+D+E+F58225664640.916920020994108 to 1
At least 6-6D+G4947022080.0779041959851283 to 1
At least 6-5B+D+G+H91276974361.43740197323569 to 1
At least 6-4A+B+C+D+E+F+G+H+I531755796008.37392820926511 to 1
At least 5-5B+D+E+G+H+J356561033405.61501448291317 to 1
At least 5-4A+B+C+D+E+F+G+H+I+J+K23620435410037.1967417906465 to 3
At least 4-4A+B+C+D+E+F+G+H+I+J+K+L39206458720061.741136275415(favored)
Three-suiter12·c5·c4²+4·c4³·c1=M269027044004.23655589605823 to 1
Two-suiter (not 3)A+B+C+D+E+F+G+H+I+J+K+L-M36516188280057.504580379357(favored)

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Specific Cards

The following 13 tables show the cases for any number of specific cards from 1 to 13. For example, to find the odds against holding all four aces, look at the table for “4 Cards” and the line “4 of 4” gives the answer as 378 to 1.

The notation “1+ of 4” means at least one of four.

1 CardCalculationHandsPercentOdds Against
1 of 11c1·51c1215875338990025.0000000000003 to 1
0 of 151c1347626016970075.000000000000(favored)

2 CardsCalculationHandsPercentOdds Against
2 of 22c2·50c11=A373537388005.88235294117616 to 1
1 of 22c1·50c12=B24279930220038.2352941176478 to 5
0 of 250c1335486051860055.882352941176(favored)
1+ of 2A+B28015304100044.1176470588245 to 4

3 CardsCalculationHandsPercentOdds Against
3 of 33c3·49c10=A82178225361.29411764705976 to 1
2 of 33c2·49c11=B8740774879213.7647058823536 to 1
1 of 33c1·49c12=C27679120450843.5882352941185 to 4
0 of 349c1326259678376441.3529411764713 to 2
2+ of 3A+B9562557132815.05882352941211 to 2
1+ of 3A+B+C37241677583658.647058823529(favored)

4 CardsCalculationHandsPercentOdds Against
4 of 44c4·48c9=A16771066400.264105642257378 to 1
3 of 44c3·48c10=B261628635844.12004801920823 to 1
2 of 44c2·48c11=C13557120220821.3493397358944 to 1
1 of 44c1·48c12=D27867413787243.8847539015615 to 4
0 of 448c1319292824929630.3817527010807 to 3
3+ of 4A+B278399702244.38415366146522 to 1
2+ of 4A+B+C16341117243225.7334933973593 to 1
1+ of 4A+B+C+D44208531030469.618247298920(favored)

5 CardsCalculationHandsPercentOdds Against
5 of 55c5·47c8=A3144574950.0495198079232018 to 1
4 of 55c4·47c9=B68132457251.07292917166992 to 1
3 of 55c3·47c10=C517806675108.15426170468211 to 1
2 of 55c2·47c11=D17417133617027.4279711884758 to 3
1 of 55c1·47c12=E26125700425541.1419567827133 to 2
0 of 547c1314067684844522.1533613445387 to 2
4+ of 5A+B71277032201.12244897959288 to 1
3+ of 5A+B+C589083707309.27671068427410 to 1
2+ of 5A+B+C+D23307970690036.7046818727495 to 3
1+ of 5A+B+C+D+E49433671115577.846638655462(favored)

6 CardsCalculationHandsPercentOdds Against
6 of 66c6·46c7=A535246800.00842890347611863 to 1
5 of 66c5·46c8=B15655968900.246545426681405 to 1
4 of 66c4·46c9=C165257449502.60242394830337 to 1
3 of 66c3·46c10=D8152700842012.83862481162713 to 2
2 of 66c2·46c11=E20011174794031.5129881739932 to 1
1 of 66c1·46c12=F23346370593036.7651528696595 to 3
0 of 646c1310176623079016.0258358662615 to 1
5+ of 6A+B16191215700.254974330158391 to 1
4+ of 6A+B+C181448665202.85739827846034 to 1
3+ of 6A+B+C+D9967187494015.6960230900875 to 1
2+ of 6A+B+C+D+E29978362288047.2090112640809 to 8
1+ of 6A+B+C+D+E+F53324732881083.974164133739(favored)

7 CardsCalculationHandsPercentOdds Against
7 of 77c7·45c6=A81450600.00128265922577962 to 1
6 of 77c6·45c7=B3176573400.0500237097611998 to 1
5 of 77c5·45c8=C45266170950.712837864100139 to 1
4 of 77c4·45c9=D310157097254.88425943920619 to 1
3 of 77c3·45c10=E11165655501017.5833339811419 to 2
2 of 77c2·45c11=F21316251411033.5681830549062 to 1
1 of 77c1·45c12=G20132015221531.7032839963002 to 1
0 of 745c137300620904511.4967952953618 to 1
6+ of 7A+B3258024000.0513063689861948 to 1
5+ of 7A+B+C48524194950.764144233086130 to 1
4+ of 7A+B+C+D358681292205.64840367229217 to 1
3+ of 7A+B+C+D+E14752468423023.2317376534337 to 2
2+ of 7A+B+C+D+E+F36068719834056.799920708339(favored)
1+ of 7A+B+C+D+E+F+G56200735055588.503204704639(favored)

8 CardsCalculationHandsPercentOdds Against
8 of 88c8·44c5=A10860080.000171021230584722 to 1
7 of 88c7·44c6=B564724160.00889310395811244 to 1
6 of 88c6·44c7=C10729759040.168968975194591 to 1
5 of 88c5·44c8=D99250271121.56296302054663 to 1
4 of 88c4·44c9=E496251355607.81481510272912 to 1
3 of 88c3·44c10=F13895037956821.8814822876427 to 2
2 of 88c2·44c11=G21474149569633.8168362627202 to 1
1 of 88c1·44c12=H16872546090426.5703713492808 to 3
0 of 844c13519155264328.17549887670111 to 1
7+ of 8A+B575584240.00906412518811032 to 1
6+ of 8A+B+C11305343280.178033100382561 to 1
5+ of 8A+B+C+D110555614401.74099612092856 to 1
4+ of 8A+B+C+D+E606806970009.5558112236579 to 1
3+ of 8A+B+C+D+E+F19963107656831.4372935112999 to 4
2+ of 8A+B+C+D+E+F+G41437257226465.254129774019(favored)
1+ of 8A+B+C+D+E+F+G+H58309803316891.824501123299(favored)

9 CardsCalculationHandsPercentOdds Against
9 of 99c9·43c4=A1234100.0000194342315145559 to 1
8 of 99c8·43c5=B86633820.00136428299373298 to 1
7 of 99c7·43c6=C2194723440.0345618358352892 to 1
6 of 99c6·43c7=D27068255760.426262641967234 to 1
5 of 99c5·43c8=E182710726382.87727283327834 to 1
4 of 99c4·43c9=F7105417137011.1893943516358 to 1
3 of 99c3·43c10=G16105612177225.3626271970403 to 1
2 of 99c2·43c11=H20707215656432.6090921104802 to 1
1 of 99c1·43c12=I13804810437621.7393947403207 to 2
0 of 943c13365768481685.76001057222116 to 1
8+ of 9A+B87867920.00138371722472268 to 1
7+ of 9A+B+C2282591360.0359455530592781 to 1
6+ of 9A+B+C+D29350847120.462208195026215 to 1
5+ of 9A+B+C+D+E212061573503.33948102830429 to 1
4+ of 9A+B+C+D+E+F9226032872014.52887537993911 to 2
3+ of 9A+B+C+D+E+F+G25331645049239.8915025769793 to 2
2+ of 9A+B+C+D+E+F+G+H46038860705672.500594687459(favored)
1+ of 9A+B+C+D+E+F+G+H+I59843671143294.239989427779(favored)

10 CardsCalculationHandsPercentOdds Against
10 of 1010c10·42c3=A114800.00000180783555314769 to 1
9 of 1010c9·42c4=B11193000.000176263953567330 to 1
8 of 1010c8·42c5=C382800600.00602822718116588 to 1
7 of 1010c7·42c6=D6294943200.0991308469691008 to 1
6 of 1010c6·42c7=E56654488800.892177622722111 to 1
5 of 1010c5·42c8=F297436066204.68393251928920 to 1
4 of 1010c4·42c9=G9363728010014.74571348665111 to 2
3 of 1010c3·42c10=H17657315676027.8062025748288 to 3
2 of 1010c2·42c11=I19262526192030.3340391725397 to 3
1 of 1010c1·42c12=J11058116888017.4139854509025 to 1
0 of 1042c13255187312804.01861202713124 to 1
9+ of 10A+B11307800.000178071788561570 to 1
8+ of 10A+B+C394108400.00620629896916112 to 1
7+ of 10A+B+C+D6689051600.105337145938948 to 1
6+ of 10A+B+C+D+E63343540400.99751476865999 to 1
5+ of 10A+B+C+D+E+F360779606605.68144728794917 to 1
4+ of 10A+B+C+D+E+F+G12971524076020.4271607746004 to 1
3+ of 10A+B+C+D+E+F+G+H30628839752048.23336334942713 to 12
2+ of 10A+B+C+D+E+F+G+H+I49891365944078.567402521967(favored)
1+ of 10A+B+C+D+E+F+G+H+I+J60949482832095.981387972869(favored)

11 CardsCalculationHandsPercentOdds Against
11 of 1111c11·41c2=A8200.000000129131774406779 to 1
10 of 1111c10·41c3=B1172600.0000184657475415431 to 1
9 of 1111c9·41c4=C55698500.000877123002114008 to 1
8 of 1111c8·41c5=D1236506700.0194721306555135 to 1
7 of 1111c7·41c6=E14838080400.233665567856427 to 1
6 of 1111c6·41c7=F103866562801.63565897499060 to 1
5 of 1111c5·41c8=G441432891906.95155064370713 to 1
4 of 1111c4·41c9=H11561337645018.2064421620909 to 2
3 of 1111c3·41c10=I18498140232029.1303074593435 to 2
2 of 1111c2·41c11=J17377040824027.3648342799898 to 3
1 of 1111c1·41c12=K8688520412013.6824171399956 to 1
0 of 1141c13176200763602.77475592349535 to 1
10+ of 11A+B1180800.0000185948795377824 to 1
9+ of 11A+B+C56879300.000895717881111641 to 1
8+ of 11A+B+C+D1293386000.0203678485364909 to 1
7+ of 11A+B+C+D+E16131466400.254033416391393 to 1
6+ of 11A+B+C+D+E+F119998029201.88969239138152 to 1
5+ of 11A+B+C+D+E+F+G561430921108.84124303508810 to 1
4+ of 11A+B+C+D+E+F+G+H17175646856027.0476851971788 to 3
3+ of 11A+B+C+D+E+F+G+H+I35673787088056.177992656521(favored)
2+ of 11A+B+C+D+E+F+G+H+I+J53050827912083.542826936510(favored)
1+ of 11A+B+C+D+E+F+G+H+I+J+K61739348324097.225244076505(favored)

12 CardsCalculationHandsPercentOdds Against
12 of 1212c12·40c1=A400.00000000629915875338989 to 1
11 of 1212c11·40c2=B93600.00000147398467843328 to 1
10 of 1212c10·40c3=C6520800.000102687571973827 to 1
9 of 1212c9·40c4=D201058000.00316620010631583 to 1
8 of 1212c8·40c5=E3257139600.0512924417241949 to 1
7 of 1212c7·40c6=F30399969600.478729456095208 to 1
6 of 1212c6·40c7=G172266494402.71280025120336 to 1
5 of 1212c5·40c8=H609085105209.5916866024679 to 1
4 of 1212c4·40c9=I13535224560021.3148591165934 to 1
3 of 1212c3·40c10=J18648531616029.3671392273055 to 2
2 of 1212c2·40c11=K15257889504024.0276593677953 to 1
1 of 1212c1·40c12=L6704224176010.5576079040318 to 1
0 of 1240c13120332228801.89495526482652 to 1
11+ of 12A+B94000.00000148028367554633 to 1
10+ of 12A+B+C6614800.000104167854959988 to 1
9+ of 12A+B+C+D207672800.00327036796130577 to 1
8+ of 12A+B+C+D+E3464812400.0545628096851832 to 1
7+ of 12A+B+C+D+E+F33864782000.533292265780187 to 1
6+ of 12A+B+C+D+E+F+G206131276403.24609251698330 to 1
5+ of 12A+B+C+D+E+F+G+H8152163816012.83777911944913 to 2
4+ of 12A+B+C+D+E+F+G+H+I21687388376034.1526382360422 to 1
3+ of 12A+B+C+D+E+F+G+H+I+J40335919992063.519777463347(favored)
2+ of 12A+B+C+D+E+F+G+H+I+J+K55593809496087.547436831143(favored)
1+ of 12A+B+C+D+E+F+G+H+I+J+K+L62298033672098.105044735174(favored)

13 CardsCalculationHandsPercentOdds Against
13 of 13c13=A10.000000000157635013559599 to 1
12 of 13c12·39c1=B5070.0000000798411252492227 to 1
11 of 13c11·39c2=C577980.00000910185310986773 to 1
10 of 13c10·39c3=D26137540.000411606014242950 to 1
9 of 13c9·39c4=E588094650.00926113531110797 to 1
8 of 13c8·39c5=F7409992590.116690304923856 to 1
7 of 13c7·39c6=G55986610680.881660081641112 to 1
6 of 13c6·39c7=H263936878924.15639752773623 to 1
5 of 13c5·39c8=I7918106367612.4691925832077 to 1
4 of 13c4·39c9=J15151931938023.8608006221863 to 1
3 of 13c3·39c10=K18182318325628.6329607466235 to 2
2 of 13c2·39c11=L13073237143220.5873354128614 to 1
1 of 13c1·39c12=M508403666688.00618599389011 to 1
0 of 1339c1381224254441.27909480375877 to 1
12+ of 13A+B5080.0000000799981250026691 to 1
11+ of 13A+B+C583060.00000918185110891049 to 1
10+ of 13A+B+C+D26720600.000420787865237648 to 1
9+ of 13A+B+C+D+E614815250.00968192317610328 to 1
8+ of 13A+B+C+D+E+F8024807840.126372228099790 to 1
7+ of 13A+B+C+D+E+F+G64011418521.00803230974098 to 1
6+ of 13A+B+C+D+E+F+G+H327948297445.16442983747618 to 1
5+ of 13A+B+C+D+E+F+G+H+I11197589342017.6336224206839 to 2
4+ of 13A+B+C+D+E+F+G+H+I+J26349521280041.4944230428683 to 2
3+ of 13A+B+C+D+E+F+G+H+I+J+K44531839605670.127383789491(favored)
2+ of 13A+B+C+D+E+F+G+H+I+J+K+L57605076748890.714719202352(favored)
1+ of 13A+B+C+D+E+F+G+H+I+J+K+L+M62689113415698.720905196242(favored)

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© 2003 Richard Pavlicek