Curio 8J77 Main

Deals with a Straight Flush

by Richard Pavlicek

Avon Wilsmore of Sidney, Australia, inquired: “What is the percent chance that a bridge deal has a straight flush in at least one hand?” Well, that’s easy: Pretty low! But of course it’s not easy to get a true answer when there are 53+ octillion bridge deals, or to be exact: 53,644,737,765,488,792,839,237,440,000.

One way to get an answer would be empirically: Create millions of random deals, count the ones that contain a straight flush, then divide by the number of attempts (times 100 for percent). This would give a good approximation, but it could never be theoretically correct. To get an exact answer would require checking every deal, which is far beyond the scope of any iterative means. For an insight see October Octillions.

This does not mean that an exact answer is unobtainable, but only that it requires some ingenuity. Many years ago I discovered there are 37,478,624 specific dealprints, however, if you ignore suit identity and rotations, this can be reduced to 393,197 generic dealprints. I have listed all of these in order of frequency in a data file, which will provide the groundwork for my calculations.

For each generic dealprint, I will consider the four suit patterns (not hand patterns). If a suit is distributed 4-4-3-2, 4-3-3-3 or 4-4-4-1 around the table (in any order) it cannot contain a straight flush.* Every other suit pattern (5+ in any hand) must be examined to determine how many of its combinations contain a straight flush. With only one 5+ suit this is straightforward. For example, the combinations of 5-3-3-2 are 13c5 × 8c3 × 5c3 = 1287 × 56 × 10 = 720,720. Of the 1287 possible 5-card suits exactly 10 comprise a straight flush, so the number of combinations with a straight flush is 10 × 56 × 10 = 5600.

*Five consecutive cards (e.g., Q-J-10-9-8) in the same suit, or because the term is from poker where an ace can be low, including 5-4-3-2-A.

Suit patterns with 5+ cards in two hands (tinted pink in the table below) are more complicated. Besides counting combinations with a straight flush in the first hand, you must count those that do not have a straight flush when the remaining cards form a straight flush in the second hand. Awkward to calculate intuitively, but easily found by computer.

Generic Suit Patterns
Pattern Combin. with SF Pattern Combin. with SF Pattern Combin. with SF
4-4-3-2 900,900 0 5-5-3-0 72,072 1092 9-2-1-1 8580 4608
5-3-3-2 720,720 5600 6-5-1-1 72,072 3410 9-3-1-0 2860 1536
5-4-3-1 360,360 2800 6-5-2-0 36,036 1705 9-2-2-0 4290 2304
5-4-2-2 540,540 4200 7-2-2-2 154,440 19,530 7-6-0-0 1716 248
4-3-3-3 1,201,200 0 7-4-1-1 51,480 6510 8-5-0-0 1287 371
6-3-2-2 360,360 14,910 7-4-2-0 25,740 3255 9-4-0-0 715 384
6-4-2-1 180,180 7455 7-3-3-0 34,320 4340 10-2-1-0 858 702
6-3-3-1 240,240 9940 8-2-2-1 38,610 11,130 10-3-0-0 286 234
5-5-2-1 216,216 3276 8-3-1-1 25,740 7420 10-1-1-1 1716 1404
4-4-4-1 450,450 0 7-5-1-0 10,296 1334 11-1-1-0 156 154
7-3-2-1 102,960 13,020 8-3-2-0 12,870 3710 11-2-0-0 78 77
6-4-3-0 60,060 2485 6-6-1-0 12,012 888 12-1-0-0 13 13
5-4-4-0 90,090 700 8-4-1-0 6435 1855 13-0-0-0 1 1

Generic Suit Patterns
Pattern	Combin.	with SF	Pattern	Combin.	with SF	Pattern	Combin.	with SF
4-4-3-2	900,900	0	5-5-3-0	72,072	1092	9-2-1-1	8580	4608
5-3-3-2	720,720	5600	6-5-1-1	72,072	3410	9-3-1-0	2860	1536
5-4-3-1	360,360	2800	6-5-2-0	36,036	1705	9-2-2-0	4290	2304
5-4-2-2	540,540	4200	7-2-2-2	154,440	19,530	7-6-0-0	1716	248
4-3-3-3	1,201,200	0	7-4-1-1	51,480	6510	8-5-0-0	1287	371
6-3-2-2	360,360	14,910	7-4-2-0	25,740	3255	9-4-0-0	715	384
6-4-2-1	180,180	7455	7-3-3-0	34,320	4340	10-2-1-0	858	702
6-3-3-1	240,240	9940	8-2-2-1	38,610	11,130	10-3-0-0	286	234
5-5-2-1	216,216	3276	8-3-1-1	25,740	7420	10-1-1-1	1716	1404
4-4-4-1	450,450	0	7-5-1-0	10,296	1334	11-1-1-0	156	154
7-3-2-1	102,960	13,020	8-3-2-0	12,870	3710	11-2-0-0	78	77
6-4-3-0	60,060	2485	6-6-1-0	12,012	888	12-1-0-0	13	13
5-4-4-0	90,090	700	8-4-1-0	6435	1855	13-0-0-0	1	1

Now that I know the straight flush count of each generic suit pattern, it only remains to apply this to each suit layout of each generic dealprint. To simplify the math, for each generic suit pattern I used the combinations without a straight flush, then direct multiplication gives the number of generic deals that are free of a straight flush. This must be multiplied by the number of permutations of the dealprint (16, 24, 48 or 96) to obtain the number of deals without a straight flush. Finally, subtracting from the number of deals gives those with at least one straight flush.

The following table lists every dealprint in order of frequency; well, not really. For practicality only the first and last and 13 in between are shown, but all 393,197 were calculated and summed. Pleasingly, the total number of deals matches the expected 52!/(13!)⁴.

Generic Dealprints
Number Dealprint Perm Deals Deals with SF
1 4432-4333-3334-2344 96 112,422,969,894,409,574,400,000,000 0
65 5431-2344-3334-3334 96 44,969,187,957,763,829,760,000,000 349,410,939,842,764,800,000,000
129 5422-3343-2353-3325 96 32,377,815,329,589,957,427,200,000 748,878,578,906,185,728,000,000
257 5422-3244-3433-2344 48 25,295,168,226,242,154,240,000,000 196,543,653,661,555,200,000,000
513 5431-2335-3424-3253 96 16,188,907,664,794,978,713,600,000 374,439,289,453,092,864,000,000
1025 5431-3235-3523-2254 96 9,713,344,598,876,987,228,160,000 298,390,698,964,824,883,200,000
2049 6421-4243-1345-2434 96 5,059,033,645,248,430,848,000,000 247,001,311,694,225,088,000,000
4097 6331-2524-4144-1444 96 2,529,516,822,624,215,424,000,000 123,500,655,847,112,544,000,000
8193 6331-1534-2263-4315 96 1,079,260,510,986,331,914,240,000 102,814,523,217,602,058,240,000
16,385 6421-0553-3145-4324 96 404,722,691,619,874,467,840,000 25,719,188,160,704,855,040,000
32,769 6520-3046-3433-1444 96 112,422,969,894,409,574,400,000 9,913,345,591,163,414,400,000
65,537 7510-1327-1363-4243 96 22,025,724,714,006,773,760,000 6,038,986,080,932,098,560,000
131,073 7420-0535-3415-3073 96 2,202,572,471,400,677,376,000 560,165,888,428,263,936,000
262,145 A210-0616-0274-3343 96 30,591,284,325,009,408,000 26,126,334,242,543,808,000
393,197 D000-0D00-00D0-000D 24 24 24
Totals 53,644,737,765,488,792,839,237,440,000 3,477,843,772,978,196,776,686,538,296

Generic Dealprints
Number	Dealprint	Perm	Deals	Deals with SF
1	4432-4333-3334-2344	96	112,422,969,894,409,574,400,000,000	0
65	5431-2344-3334-3334	96	44,969,187,957,763,829,760,000,000	349,410,939,842,764,800,000,000
129	5422-3343-2353-3325	96	32,377,815,329,589,957,427,200,000	748,878,578,906,185,728,000,000
257	5422-3244-3433-2344	48	25,295,168,226,242,154,240,000,000	196,543,653,661,555,200,000,000
513	5431-2335-3424-3253	96	16,188,907,664,794,978,713,600,000	374,439,289,453,092,864,000,000
1025	5431-3235-3523-2254	96	9,713,344,598,876,987,228,160,000	298,390,698,964,824,883,200,000
2049	6421-4243-1345-2434	96	5,059,033,645,248,430,848,000,000	247,001,311,694,225,088,000,000
4097	6331-2524-4144-1444	96	2,529,516,822,624,215,424,000,000	123,500,655,847,112,544,000,000
8193	6331-1534-2263-4315	96	1,079,260,510,986,331,914,240,000	102,814,523,217,602,058,240,000
16,385	6421-0553-3145-4324	96	404,722,691,619,874,467,840,000	25,719,188,160,704,855,040,000
32,769	6520-3046-3433-1444	96	112,422,969,894,409,574,400,000	9,913,345,591,163,414,400,000
65,537	7510-1327-1363-4243	96	22,025,724,714,006,773,760,000	6,038,986,080,932,098,560,000
131,073	7420-0535-3415-3073	96	2,202,572,471,400,677,376,000	560,165,888,428,263,936,000
262,145	A210-0616-0274-3343	96	30,591,284,325,009,408,000	26,126,334,242,543,808,000
393,197	D000-0D00-00D0-000D	24	24	24
Totals	53,644,737,765,488,792,839,237,440,000	3,477,843,772,978,196,776,686,538,296

Dividing the number of deals with a straight flush by the number of deals, then multiplying by 100, gives the percent chance that a deal has at least one straight flush = 6.4831033160825590, and that’s my final answer.

Curio 8J77 Main Top Deals with a Straight Flush