Study 7Z77 Main

Against All Odds

by Richard Pavlicek

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 Criteria

Hand Patterns

Suit Lengths

Specific Cards

Each table lists a variety of hand types, the method of calculation*, the number of possible bridge hands, the percent probability (rounded to four 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 more meaningful to show how the number is derived.

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: A-K A-K-Q-J-10-9-2 A-K-Q A, but note that changing the 9 to the eight would disqualify it.

Hand Type	Calculation	Hands	Percent	Odds Against
Highest card 5	16c13	560	0.0000	1133952784:1
Highest card 6	4c1·16c12+4c2·16c11+4c3·16c10+4c4·16c9	76,960	0.0000	8251215:1
Highest card 7	4c1·20c12+4c2·20c11+4c3·20c10+4c4·20c9	2,418,624	0.0004	262551:1
Highest card 8	4c1·24c12+4c2·24c11+4c3·24c10+4c4·24c9	34,946,016	0.0055	18171:1
Highest card 9	4c1·28c12+4c2·28c11+4c3·28c10+4c4·28c9	309,931,440	0.0488	2048:1
Highest card 10	4c1·32c12+4c2·32c11+4c3·32c10+4c4·32c9	1,963,416,000	0.3092	323:1
Highest card J	4c1·36c12+4c2·36c11+4c3·36c10+4c4·36c9	9,722,433,280	1.5311	65:1
Highest card Q	4c1·40c12+4c2·40c11+4c3·40c10+4c4·40c9	39,882,303,552	6.2805	15:1
Highest card K	4c1·44c12+4c2·44c11+4c3·44c10+4c4·44c9	141,012,722,864	22.2063	4:1
Yarborough	32c13	347,373,600	0.0547	1827:1
Flat Yarborough	4·8c33·8c4	49,172,480	0.0077	12913:1
No card over 8	28c13	37,442,160	0.0059	16959:1
Flat no card over 8	4·7c33·7c4	6,002,500	0.0009	105791:1
Aceless	48c13	192,928,249,296	30.3818	7:3
One card each rank	(4c1)13	67,108,864	0.0106	9462:1
Balanced	4·13c3(13c4·13c32+3·13c42·13c2+3·13c5·13c3·13c2)=A	302,292,822,832	47.6042	13:12
Semibalanced	12·13c22(13c5·13c4+13c6·13c3)=B	103,012,900,848	16.2222	5:1
Quasibalanced	4·13c4·13c1(13c42+6·13c5·13c3)=C	101,119,078,060	15.9239	5:1
Balanced or semi-	A+B	405,305,723,680	63.8263	(favored)
Semi- or quasi-	B+C	204,131,978,908	32.1461	2:1
Bal- semi- or quasi-	A+B+C	506,424,801,740	79.7502	(favored)
All black cards	26c13	10,400,600	0.0016	61054:1
Guaranteed 7 NT	sum of all cases	3756	0.0000	169066442:1
5 honors in any suit	4·47c8-6·42c3	1,257,761,100	0.1981	504:1
4 honors in any suit	5c4(4·47c9-6·5c4·42c5+4·5c42·37c1)	27,125,401,200	4.2716	22:1
3 honors in any suit	5c3(4·47c10-6·5c3·42c7+4·5c32·37c4-5c33·32c1)	191,199,533,240	30.1095	7:3
2 honors in any suit	5c2(4·47c11-6·5c2·42c9+4·5c22·37c7-5c23·32c5)	468,318,386,680	73.7494	(favored)
1 honor in any suit	5c1(4·47c12-6·5c1·42c11+4·5c12·37c10-5c13·32c9)	559,578,378,620	88.1207	(favored)
2 tripletons	6·13c32(26c7-2·13c3·13c4)	122,114,884,320	19.2303	4:1
1 tripleton	4·13c3(39c10-3·13c3(26c7-13c3·13c4))	282,345,395,904	44.4629	5:4
At least 1 tripleton	4·13c3·39c10-6·13c32·26c7+4·13c33·13c4	471,366,136,384	74.2293	(favored)
2 doubletons	6·13c22(26c9-2·13c2·13c7)	104,286,598,416	16.4227	5:1
1 doubleton	4·13c2(39c11-3·13c2(26c9-13c2·13c7))	304,584,314,112	47.9650	13:12
At least 1 doubleton	4·13c2·39c11-6·13c22·26c9+4·13c23·13c7	412,128,237,456	64.9007	(favored)
2 singletons	6·13c12(26c11-2·13c1·13c10)	7,826,786,136	1.2325	80:1
1 singleton	4·13c1(39c12-3·13c1(26c11-13c1·13c10))	187,700,354,296	29.5585	7:3
At least 1 singleton	4·13c1·39c12-6·13c12·26c11+4·13c13·13c10	195,529,653,800	30.7914	9:4
2 voids	6(26c13-2)	62,403,588	0.0098	10175:1
1 void	4(39c13-3(26c13-13c13))	32,364,894,588	5.0967	19:1
At least 1 void	4·39c13-6·26c13+4·13c13	32,427,298,180	5.1066	19:1
At least 1 sing./void	not 4333 4432 5332 5422 6322 7222	226,450,510,992	35.6607	9:5

Study 7Z77 Main

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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.)

Pattern	Freakness	Calculation	Hands	Percent	Odds Against
4-4-3-2	1	12·13c42·13c3·13c2	136,852,887,600	21.5512	7:2
5-3-3-2	2	12·13c5·13c32·13c2	98,534,079,072	15.5168	11:2
5-4-3-1	4	24·13c5·13c4·13c3·13c1	82,111,732,560	12.9307	13:2
5-4-2-2	3	12·13c5·13c4·13c22	67,182,326,640	10.5797	8:1
4-3-3-3	0	4·13c4·13c33	66,905,856,160	10.5361	17:2
6-3-2-2	4	12·13c6·13c3·13c22	35,830,574,208	5.6425	17:1
6-4-2-1	6	24·13c6·13c4·13c2·13c1	29,858,811,840	4.7021	20:1
6-3-3-1	5	12·13c6·13c32·13c1	21,896,462,016	3.4482	28:1
5-5-2-1	6	12·13c52·13c2·13c1	20,154,697,992	3.1739	31:1
4-4-4-1	3	4·13c43·13c1	19,007,345,500	2.9932	32:1
7-3-2-1	7	24·13c7·13c3·13c2·13c1	11,943,524,736	1.8808	52:1
6-4-3-0	7	24·13c6·13c4·13c3	8,421,716,160	1.3262	74:1
5-4-4-0	6	12·13c5·13c42	7,895,358,900	1.2433	79:1
5-5-3-0	7	12·13c52·13c3	5,684,658,408	0.8952	111:1
6-5-1-1	8	12·13c6·13c5·13c12	4,478,821,776	0.7053	141:1
6-5-2-0	9	24·13c6·13c5·13c2	4,134,297,024	0.6511	153:1
7-2-2-2	6	4·13c7·13c23	3,257,324,928	0.5130	194:1
7-4-1-1	8	12·13c7·13c4·13c12	2,488,234,320	0.3918	254:1
7-4-2-0	9	24·13c7·13c4·13c2	2,296,831,680	0.3617	275:1
7-3-3-0	8	12·13c7·13c32	1,684,343,232	0.2652	376:1
8-2-2-1	9	12·13c8·13c22·13c1	1,221,496,848	0.1924	519:1
8-3-1-1	9	12·13c8·13c3·13c12	746,470,296	0.1176	850:1
7-5-1-0	11	24·13c7·13c5·13c1	689,049,504	0.1085	921:1
8-3-2-0	10	24·13c8·13c3·13c2	689,049,504	0.1085	921:1
6-6-1-0	11	12·13c62·13c1	459,366,336	0.0723	1381:1
8-4-1-0	11	24·13c8·13c4·13c1	287,103,960	0.0452	2211:1
9-2-1-1	11	12·13c9·13c2·13c12	113,101,560	0.0178	5614:1
9-3-1-0	12	24·13c9·13c3·13c1	63,800,880	0.0100	9952:1
9-2-2-0	12	12·13c9·13c22	52,200,720	0.0082	12164:1
7-6-0-0	13	12·13c7·13c6	35,335,872	0.0056	17970:1
8-5-0-0	13	12·13c8·13c5	19,876,428	0.0031	31947:1
10-2-1-0	14	24·13c10·13c2·13c1	6,960,096	0.0011	91235:1
9-4-0-0	13	12·13c9·13c4	6,134,700	0.0010	103511:1
10-1-1-1	13	4·13c10·13c13	2,513,368	0.0004	252653:1
10-3-0-0	14	12·13c10·13c3	981,552	0.0002	646947:1
11-1-1-0	16	12·13c11·13c12	158,184	0.0000	4014397:1
11-2-0-0	16	12·13c11·13c2	73,008	0.0000	8697862:1
12-1-0-0	18	12·13c12·13c1	2028	0.0000	313123056:1
13-0-0-0	20	4·13c13	4	0.0000	158753389899:1
Check totals:			635,013,559,600	100

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.

Pattern	Freakness	Calculation	Hands	Percent	Odds Against
4=3=3=3	0	13c4·13c33	16,726,464,040	2.6340	37:1
4=4=3=2	1	13c42·13c3·13c2	11,404,407,300	1.7959	55:1
5=3=3=2	2	13c5·13c32·13c2	8,211,173,256	1.2931	76:1
5=4=2=2	3	13c5·13c4·13c22	5,598,527,220	0.8816	112:1
4=4=4=1	3	13c43·13c1	4,751,836,375	0.7483	133:1
5=4=3=1	4	13c5·13c4·13c3·13c1	3,421,322,190	0.5388	185:1
6=3=2=2	4	13c6·13c3·13c22	2,985,881,184	0.4702	212:1
6=3=3=1	5	13c6·13c32·13c1	1,824,705,168	0.2873	347:1
5=5=2=1	6	13c52·13c2·13c1	1,679,558,166	0.2645	377:1
6=4=2=1	6	13c6·13c4·13c2·13c1	1,244,117,160	0.1959	509:1
7=2=2=2	6	13c7·13c23	814,331,232	0.1282	779:1
5=4=4=0	6	13c5·13c42	657,946,575	0.1036	964:1
7=3=2=1	7	13c7·13c3·13c2·13c1	497,646,864	0.0784	1275:1
5=5=3=0	7	13c52·13c3	473,721,534	0.0746	1339:1
6=5=1=1	8	13c6·13c5·13c12	373,235,148	0.0588	1700:1
6=4=3=0	7	13c6·13c4·13c3	350,904,840	0.0553	1809:1
7=4=1=1	8	13c7·13c4·13c12	207,352,860	0.0327	3061:1
6=5=2=0	9	13c6·13c5·13c2	172,262,376	0.0271	3685:1
7=3=3=0	8	13c7·13c32	140,361,936	0.0221	4523:1
8=2=2=1	9	13c8·13c22·13c1	101,791,404	0.0160	6237:1
7=4=2=0	9	13c7·13c4·13c2	95,701,320	0.0151	6634:1
8=3=1=1	9	13c8·13c3·13c12	62,205,858	0.0098	10207:1
6=6=1=0	11	13c62·13c1	38,280,528	0.0060	16587:1
7=5=1=0	11	13c7·13c5·13c1	28,710,396	0.0045	22117:1
8=3=2=0	10	13c8·13c3·13c2	28,710,396	0.0045	22117:1
8=4=1=0	11	13c8·13c4·13c1	11,962,665	0.0019	53082:1
9=2=1=1	11	13c9·13c2·13c12	9,425,130	0.0015	67374:1
9=2=2=0	12	13c9·13c22	4,350,060	0.0007	145977:1
7=6=0=0	13	13c7·13c6	2,944,656	0.0005	215648:1
9=3=1=0	12	13c9·13c3·13c1	2,658,370	0.0004	238872:1
8=5=0=0	13	13c8·13c5	1,656,369	0.0003	383376:1
10=1=1=1	13	13c10·13c13	628,342	0.0001	1010617:1
9=4=0=0	13	13c9·13c4	511,225	0.0001	1242140:1
10=2=1=0	14	13c10·13c2·13c1	290,004	0.0000	2189671:1
10=3=0=0	14	13c10·13c3	81,796	0.0000	7763381:1
11=1=1=0	16	13c11·13c12	13,182	0.0000	48172777:1
11=2=0=0	16	13c11·13c2	6084	0.0000	104374351:1
12=1=0=0	18	13c12·13c1	169	0.0000	3757476683:1
13=0=0=0	20	13c13	1	0.0000	635013559599:1

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 Type	Calculation	Hands	Percent	Odds Against
13 card suit	4·13c13=A	4	0.0000	158753389899:1
12 card suit	4·13c12·39c1=B	2028	0.0000	313123056:1
11 card suit	4·13c11·39c2=C	231,192	0.0000	2746692:1
10 card suit	4·13c10·39c3=D	10,455,016	0.0016	60737:1
9 card suit	4·13c9·39c4=E	235,237,860	0.0370	2698:1
8 card suit	4·13c8·39c5=F	2,963,997,036	0.4668	213:1
7 card suit	4·13c7·39c6=G	22,394,644,272	3.5266	27:1
6 card suit (max)	4·13c6(39c7-3·13c7-3·13c6·13c1)=H	105,080,049,360	16.5477	5:1
5 card suit (max)	4·13c5(39c8-3·13c8-3·13c7·26-3·13c6·26c2)-6·13c52·26c3=I	281,562,853,572	44.3397	5:4
4 card suit (max)	4·13c4·13c33+12·13c42·13c3·13c2+4·13c43·13c1=J	222,766,089,260	35.0805	13:7
12+ card suit	A+B	2032	0.0000	312506672:1
11+ card suit	A+B+C	233,224	0.0000	2722761:1
10+ card suit	A+B+C+D	10,688,240	0.0017	59411:1
9+ card suit	A+B+C+D+E	245,926,100	0.0387	2581:1
8+ card suit	A+B+C+D+E+F	3,209,923,136	0.5055	197:1
7+ card suit	A+B+C+D+E+F+G	25,604,567,408	4.0321	24:1
6+ card suit	A+B+C+D+E+F+G+H	130,684,616,768	20.5798	4:1
5+ card suit	A+B+C+D+E+F+G+H+I	412,247,470,340	64.9195	(favored)
4+ card suit	A+B+C+D+E+F+G+H+I+J	635,013,559,600	100	(bet on it)
9-4 two suits	12·13c9·13c4=A	6,134,700	0.0010	103511:1
8-5 two suits	12·13c8·13c5=B	19,876,428	0.0031	31947:1
8-4 two suits	12·13c8·13c4·26c1=C	287,103,960	0.0452	2211:1
7-6 two suits	12·13c7·13c6=D	35,335,872	0.0056	17970:1
7-5 two suits	12·13c7·13c5·26c1=E	689,049,504	0.1085	921:1
7-4 two suits	12·13c7·13c4·26c2=F	4,785,066,000	0.7535	132:1
6-6 two suits	6·13c62·26c1=G	459,366,336	0.0723	1381:1
6-5 two suits	12·13c6·13c5·26c2=H	8,613,118,800	1.3564	73:1
6-4 two suits	12·13c6·13c4·26c3=I	38,280,528,000	6.0283	16:1
5-5 two suits	6·13c52·26c3=J	25,839,356,400	4.0691	24:1
5-4 two suits	12·13c5·13c4(26c4-13c4)=K	157,189,418,100	24.7537	3:1
4-4 two suits (no 5)	12·13c42·13c3·13c2+4·13c43·13c1=L	155,860,233,100	24.5444	3:1
At least 8-4	A+B+C	313,115,088	0.0493	2027:1
At least 7-5	B+D+E	744,261,804	0.1172	852:1
At least 7-4	A+B+C+D+E+F	5,822,566,464	0.9169	108:1
At least 6-6	D+G	494,702,208	0.0779	1283:1
At least 6-5	B+D+G+H	9,127,697,436	1.4374	69:1
At least 6-4	A+B+C+D+E+F+G+H+I	53,175,579,600	8.3739	11:1
At least 5-5	B+D+E+G+H+J	35,656,103,340	5.6150	17:1
At least 5-4	A+B+C+D+E+F+G+H+I+J+K	236,204,354,100	37.1967	5:3
At least 4-4	A+B+C+D+E+F+G+H+I+J+K+L	392,064,587,200	61.7411	(favored)
Three-suiter	12·13c5·13c42+4·13c43·13c1=M	26,902,704,400	4.2366	23:1
Two-suiter (not 3)	A+B+C+D+E+F+G+H+I+J+K+L-M	365,161,882,800	57.5046	(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 378:1.

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

1 Card	Calculation	Hands	Percent	Odds Against
1 of 1	1c1·51c12	158,753,389,900	25.0000	3:1
0 of 1	51c13	476,260,169,700	75.0000	(favored)

2 Cards	Calculation	Hands	Percent	Odds Against
2 of 2	2c2·50c11=A	37,353,738,800	5.8824	16:1
1 of 2	2c1·50c12=B	242,799,302,200	38.2353	8:5
0 of 2	50c13	354,860,518,600	55.8824	(favored)
1+ of 2	A+B	280,153,041,000	44.1176	5:4

3 Cards	Calculation	Hands	Percent	Odds Against
3 of 3	3c3·49c10=A	8,217,822,536	1.2941	76:1
2 of 3	3c2·49c11=B	87,407,748,792	13.7647	6:1
1 of 3	3c1·49c12=C	276,791,204,508	43.5882	5:4
0 of 3	49c13	262,596,783,764	41.3529	3:2
2+ of 3	A+B	95,625,571,328	15.0588	11:2
1+ of 3	A+B+C	372,416,775,836	58.6471	(favored)

4 Cards	Calculation	Hands	Percent	Odds Against
4 of 4	4c4·48c9=A	1,677,106,640	0.2641	378:1
3 of 4	4c3·48c10=B	26,162,863,584	4.1200	23:1
2 of 4	4c2·48c11=C	135,571,202,208	21.3493	4:1
1 of 4	4c1·48c12=D	278,674,137,872	43.8848	5:4
0 of 4	48c13	192,928,249,296	30.3818	7:3
3+ of 4	A+B	27,839,970,224	4.3842	22:1
2+ of 4	A+B+C	163,411,172,432	25.7335	3:1
1+ of 4	A+B+C+D	442,085,310,304	69.6182	(favored)

5 Cards	Calculation	Hands	Percent	Odds Against
5 of 5	5c5·47c8=A	314,457,495	0.0495	2018:1
4 of 5	5c4·47c9=B	6,813,245,725	1.0729	92:1
3 of 5	5c3·47c10=C	51,780,667,510	8.1543	11:1
2 of 5	5c2·47c11=D	174,171,336,170	27.4280	8:3
1 of 5	5c1·47c12=E	261,257,004,255	41.1420	3:2
0 of 5	47c13	140,676,848,445	22.1534	7:2
4+ of 5	A+B	7,127,703,220	1.1224	88:1
3+ of 5	A+B+C	58,908,370,730	9.2767	10:1
2+ of 5	A+B+C+D	233,079,706,900	36.7047	5:3
1+ of 5	A+B+C+D+E	494,336,711,155	77.8466	(favored)

6 Cards	Calculation	Hands	Percent	Odds Against
6 of 6	6c6·46c7=A	53,524,680	0.0084	11863:1
5 of 6	6c5·46c8=B	1,565,596,890	0.2465	405:1
4 of 6	6c4·46c9=C	16,525,744,950	2.6024	37:1
3 of 6	6c3·46c10=D	81,527,008,420	12.8386	13:2
2 of 6	6c2·46c11=E	200,111,747,940	31.5130	2:1
1 of 6	6c1·46c12=F	233,463,705,930	36.7652	5:3
0 of 6	46c13	101,766,230,790	16.0258	5:1
5+ of 6	A+B	1,619,121,570	0.2550	391:1
4+ of 6	A+B+C	18,144,866,520	2.8574	34:1
3+ of 6	A+B+C+D	99,671,874,940	15.6960	5:1
2+ of 6	A+B+C+D+E	299,783,622,880	47.2090	9:8
1+ of 6	A+B+C+D+E+F	533,247,328,810	83.9742	(favored)

7 Cards	Calculation	Hands	Percent	Odds Against
7 of 7	7c7·45c6=A	8,145,060	0.0013	77962:1
6 of 7	7c6·45c7=B	317,657,340	0.0500	1998:1
5 of 7	7c5·45c8=C	4,526,617,095	0.7128	139:1
4 of 7	7c4·45c9=D	31,015,709,725	4.8843	19:1
3 of 7	7c3·45c10=E	111,656,555,010	17.5833	9:2
2 of 7	7c2·45c11=F	213,162,514,110	33.5682	2:1
1 of 7	7c1·45c12=G	201,320,152,215	31.7033	2:1
0 of 7	45c13	73,006,209,045	11.4968	8:1
6+ of 7	A+B	325,802,400	0.0513	1948:1
5+ of 7	A+B+C	4,852,419,495	0.7641	130:1
4+ of 7	A+B+C+D	35,868,129,220	5.6484	17:1
3+ of 7	A+B+C+D+E	147,524,684,230	23.2317	7:2
2+ of 7	A+B+C+D+E+F	360,687,198,340	56.7999	(favored)
1+ of 7	A+B+C+D+E+F+G	562,007,350,555	88.5032	(favored)

8 Cards	Calculation	Hands	Percent	Odds Against
8 of 8	8c8·44c5=A	1,086,008	0.0002	584722:1
7 of 8	8c7·44c6=B	56,472,416	0.0089	11244:1
6 of 8	8c6·44c7=C	1,072,975,904	0.1690	591:1
5 of 8	8c5·44c8=D	9,925,027,112	1.5630	63:1
4 of 8	8c4·44c9=E	49,625,135,560	7.8148	12:1
3 of 8	8c3·44c10=F	138,950,379,568	21.8815	7:2
2 of 8	8c2·44c11=G	214,741,495,696	33.8168	2:1
1 of 8	8c1·44c12=H	168,725,460,904	26.5704	8:3
0 of 8	44c13	51,915,526,432	8.1755	11:1
7+ of 8	A+B	57,558,424	0.0091	11032:1
6+ of 8	A+B+C	1,130,534,328	0.1780	561:1
5+ of 8	A+B+C+D	11,055,561,440	1.7410	56:1
4+ of 8	A+B+C+D+E	60,680,697,000	9.5558	9:1
3+ of 8	A+B+C+D+E+F	199,631,076,568	31.4373	9:4
2+ of 8	A+B+C+D+E+F+G	414,372,572,264	65.2541	(favored)
1+ of 8	A+B+C+D+E+F+G+H	583,098,033,168	91.8245	(favored)

9 Cards	Calculation	Hands	Percent	Odds Against
9 of 9	9c9·43c4=A	123,410	0.0000	5145559:1
8 of 9	9c8·43c5=B	8,663,382	0.0014	73298:1
7 of 9	9c7·43c6=C	219,472,344	0.0346	2892:1
6 of 9	9c6·43c7=D	2,706,825,576	0.4263	234:1
5 of 9	9c5·43c8=E	18,271,072,638	2.8773	34:1
4 of 9	9c4·43c9=F	71,054,171,370	11.1894	8:1
3 of 9	9c3·43c10=G	161,056,121,772	25.3626	3:1
2 of 9	9c2·43c11=H	207,072,156,564	32.6091	2:1
1 of 9	9c1·43c12=I	138,048,104,376	21.7394	7:2
0 of 9	43c13	36,576,848,168	5.7600	16:1
8+ of 9	A+B	8,786,792	0.0014	72268:1
7+ of 9	A+B+C	228,259,136	0.0359	2781:1
6+ of 9	A+B+C+D	2,935,084,712	0.4622	215:1
5+ of 9	A+B+C+D+E	21,206,157,350	3.3395	29:1
4+ of 9	A+B+C+D+E+F	92,260,328,720	14.5289	11:2
3+ of 9	A+B+C+D+E+F+G	253,316,450,492	39.8915	3:2
2+ of 9	A+B+C+D+E+F+G+H	460,388,607,056	72.5006	(favored)
1+ of 9	A+B+C+D+E+F+G+H+I	598,436,711,432	94.2400	(favored)

10 Cards	Calculation	Hands	Percent	Odds Against
10 of 10	10c10·42c3=A	11,480	0.0000	55314769:1
9 of 10	10c9·42c4=B	1,119,300	0.0002	567330:1
8 of 10	10c8·42c5=C	38,280,060	0.0060	16588:1
7 of 10	10c7·42c6=D	629,494,320	0.0991	1008:1
6 of 10	10c6·42c7=E	5,665,448,880	0.8922	111:1
5 of 10	10c5·42c8=F	29,743,606,620	4.6839	20:1
4 of 10	10c4·42c9=G	93,637,280,100	14.7457	11:2
3 of 10	10c3·42c10=H	176,573,156,760	27.8062	8:3
2 of 10	10c2·42c11=I	192,625,261,920	30.3340	7:3
1 of 10	10c1·42c12=J	110,581,168,880	17.4140	5:1
0 of 10	42c13	25,518,731,280	4.0186	24:1
9+ of 10	A+B	1,130,780	0.0002	561570:1
8+ of 10	A+B+C	39,410,840	0.0062	16112:1
7+ of 10	A+B+C+D	668,905,160	0.1053	948:1
6+ of 10	A+B+C+D+E	6,334,354,040	0.9975	99:1
5+ of 10	A+B+C+D+E+F	36,077,960,660	5.6814	17:1
4+ of 10	A+B+C+D+E+F+G	129,715,240,760	20.4272	4:1
3+ of 10	A+B+C+D+E+F+G+H	306,288,397,520	48.2334	13:12
2+ of 10	A+B+C+D+E+F+G+H+I	498,913,659,440	78.5674	(favored)
1+ of 10	A+B+C+D+E+F+G+H+I+J	609,494,828,320	95.9814	(favored)

11 Cards	Calculation	Hands	Percent	Odds Against
11 of 11	11c11·41c2=A	820	0.0000	774406779:1
10 of 11	11c10·41c3=B	117,260	0.0000	5415431:1
9 of 11	11c9·41c4=C	5,569,850	0.0009	114008:1
8 of 11	11c8·41c5=D	123,650,670	0.0195	5135:1
7 of 11	11c7·41c6=E	1,483,808,040	0.2337	427:1
6 of 11	11c6·41c7=F	10,386,656,280	1.6357	60:1
5 of 11	11c5·41c8=G	44,143,289,190	6.9516	13:1
4 of 11	11c4·41c9=H	115,613,376,450	18.2064	9:2
3 of 11	11c3·41c10=I	184,981,402,320	29.1303	5:2
2 of 11	11c2·41c11=J	173,770,408,240	27.3648	8:3
1 of 11	11c1·41c12=K	86,885,204,120	13.6824	6:1
0 of 11	41c13	17,620,076,360	2.7748	35:1
10+ of 11	A+B	118,080	0.0000	5377824:1
9+ of 11	A+B+C	5,687,930	0.0009	111641:1
8+ of 11	A+B+C+D	129,338,600	0.0204	4909:1
7+ of 11	A+B+C+D+E	1,613,146,640	0.2540	393:1
6+ of 11	A+B+C+D+E+F	11,999,802,920	1.8897	52:1
5+ of 11	A+B+C+D+E+F+G	56,143,092,110	8.8412	10:1
4+ of 11	A+B+C+D+E+F+G+H	171,756,468,560	27.0477	8:3
3+ of 11	A+B+C+D+E+F+G+H+I	356,737,870,880	56.1780	(favored)
2+ of 11	A+B+C+D+E+F+G+H+I+J	530,508,279,120	83.5428	(favored)
1+ of 11	A+B+C+D+E+F+G+H+I+J+K	617,393,483,240	97.2252	(favored)

12 Cards	Calculation	Hands	Percent	Odds Against
12 of 12	12c12·40c1=A	40	0.0000	15875338989:1
11 of 12	12c11·40c2=B	9360	0.0000	67843328:1
10 of 12	12c10·40c3=C	652,080	0.0001	973827:1
9 of 12	12c9·40c4=D	20,105,800	0.0032	31583:1
8 of 12	12c8·40c5=E	325,713,960	0.0513	1949:1
7 of 12	12c7·40c6=F	3,039,996,960	0.4787	208:1
6 of 12	12c6·40c7=G	17,226,649,440	2.7128	36:1
5 of 12	12c5·40c8=H	60,908,510,520	9.5917	9:1
4 of 12	12c4·40c9=I	135,352,245,600	21.3149	4:1
3 of 12	12c3·40c10=J	186,485,316,160	29.3671	5:2
2 of 12	12c2·40c11=K	152,578,895,040	24.0277	3:1
1 of 12	12c1·40c12=L	67,042,241,760	10.5576	8:1
0 of 12	40c13	12,033,222,880	1.8950	52:1
11+ of 12	A+B	9400	0.0000	67554633:1
10+ of 12	A+B+C	661,480	0.0001	959988:1
9+ of 12	A+B+C+D	20,767,280	0.0033	30577:1
8+ of 12	A+B+C+D+E	346,481,240	0.0546	1832:1
7+ of 12	A+B+C+D+E+F	3,386,478,200	0.5333	187:1
6+ of 12	A+B+C+D+E+F+G	20,613,127,640	3.2461	30:1
5+ of 12	A+B+C+D+E+F+G+H	81,521,638,160	12.8378	13:2
4+ of 12	A+B+C+D+E+F+G+H+I	216,873,883,760	34.1526	2:1
3+ of 12	A+B+C+D+E+F+G+H+I+J	403,359,199,920	63.5198	(favored)
2+ of 12	A+B+C+D+E+F+G+H+I+J+K	555,938,094,960	87.5474	(favored)
1+ of 12	A+B+C+D+E+F+G+H+I+J+K+L	622,980,336,720	98.1050	(favored)

13 Cards	Calculation	Hands	Percent	Odds Against
13 of 13	13c13=A	1	0.0000	635013559599:1
12 of 13	13c12·39c1=B	507	0.0000	1252492227:1
11 of 13	13c11·39c2=C	57,798	0.0000	10986773:1
10 of 13	13c10·39c3=D	2,613,754	0.0004	242950:1
9 of 13	13c9·39c4=E	58,809,465	0.0093	10797:1
8 of 13	13c8·39c5=F	740,999,259	0.1167	856:1
7 of 13	13c7·39c6=G	5,598,661,068	0.8817	112:1
6 of 13	13c6·39c7=H	26,393,687,892	4.1564	23:1
5 of 13	13c5·39c8=I	79,181,063,676	12.4692	7:1
4 of 13	13c4·39c9=J	151,519,319,380	23.8608	3:1
3 of 13	13c3·39c10=K	181,823,183,256	28.6330	5:2
2 of 13	13c2·39c11=L	130,732,371,432	20.5873	4:1
1 of 13	13c1·39c12=M	50,840,366,668	8.0062	11:1
0 of 13	39c13	8,122,425,444	1.2791	77:1
12+ of 13	A+B	508	0.0000	1250026691:1
11+ of 13	A+B+C	58,306	0.0000	10891049:1
10+ of 13	A+B+C+D	2,672,060	0.0004	237648:1
9+ of 13	A+B+C+D+E	61,481,525	0.0097	10328:1
8+ of 13	A+B+C+D+E+F	802,480,784	0.1264	790:1
7+ of 13	A+B+C+D+E+F+G	6,401,141,852	1.0080	98:1
6+ of 13	A+B+C+D+E+F+G+H	32,794,829,744	5.1644	18:1
5+ of 13	A+B+C+D+E+F+G+H+I	111,975,893,420	17.6336	9:2
4+ of 13	A+B+C+D+E+F+G+H+I+J	263,495,212,800	41.4944	3:2
3+ of 13	A+B+C+D+E+F+G+H+I+J+K	445,318,396,056	70.1274	(favored)
2+ of 13	A+B+C+D+E+F+G+H+I+J+K+L	576,050,767,488	90.7147	(favored)
1+ of 13	A+B+C+D+E+F+G+H+I+J+K+L+M	626,891,134,156	98.7209	(favored)

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© 2003 Richard Pavlicek