# Problem of the Month (January 2010)

When 5 cards are dealt from a normal 52-card deck, it is fairly straightforward to calculate the probabilities of getting a hand in various poker categories (5-of-a-kind, straight flush, 4-of-a-kind, full house, flush, straight, 3-of-a-kind, two pair, one pair, or nothing). What if the deck is changed, and we still deal 5 cards? Can we make the probability of obtaining one of these hands a fraction whose numerator and denominator are small? For example, if the deck contains only 13♠'s and 2's, the probability of dealing a 5-card flush is 3/7.

Berend van der Zwaag sent many results, and even proved some of them impossible (shown with an X below). Here are the smallest decks known that will make the probability of dealing these hands a fraction with small numerator and denominator.

5-of-a-Kind
Den \ Num123456789
2 10 cards:
9/1
3 1315 cards:
1056/36/22/17/11/
10/7/7/6/5/5/5/5/1123
15 cards:
14/1
4 320 cards:
243/28/9/7/5/128
20 cards:
19/1
5 725 cards:
526/93/31/20/
11/9/8/8/5/114
1925 cards:
1603/52/12/11/
6/6/6/5/5/1219
3925 cards:
3544/139/70/35/14/
9/8/7/6/5/5/5/5/173
25 cards:
24/1
6 6 cards:
5/1
30 cards:
29/1
7 368 cards:
250/45/15/11/
10/6/5/5/121
7 cards:
6/1
15 cards:
13/2
21 cards:
19/2
?35 cards:
34/1
8 99 cards:
66/15/11/17
8 cards:
7/1
9008 cards:
8200/428/170/40/9/
8/7/6/6/5/5/5/1119
40 cards:
39/1
9 522 cards:
337/90/47/13/
7/7/6/5/5/15
10 cards:
8/2
9 cards:
8/1
3879 cards:
3449/170/74/37/15/
10/8/7/7/6/6/5/185
?45 cards:
44/1
10 576 cards:
364/110/32/13/
10/7/6/6/5/5/118
1002 cards:
788/94/49/16/
12/10/8/7/7/6/15
?50 cards:
49/1

Straight Flush
Den \ Num123456789
2 X
3 6 cards:
A23455
X
4 XX
5 XXXX
6 6 cards:
A23457
X
7 7 cards:
A234555
XXXXX
8 XXXX
9 XXXXXX
10 XXXX

4-of-a-Kind
Den \ Num123456789
2 10 cards:
9/1
3 6 cards:
4/2
(BZ)
?
4 20 cards:
19/1
?
5 25 cards:
24/1
???
6 30 cards:
29/1
6 cards:
5/1
7 7 cards:
4/3
(BZ)
67 cards:
41/17/9
??7 cards:
6/1
?
8 40 cards:
39/1
?8 cards:
7/1
?
9 45 cards:
44/1
28 cards:
16/112
?9 cards:
7/2
or 8/1
??
10 50 cards:
49/1
???
(BZ) = Berend van der Zwaag

Full House
Den \ Num123456789
2 10 cards:
7/3
3 ?6 cards:
4/2
4 ??
5 77 cards:
29/29/14/3/12
???
6 6 cards:
3/2/1
(BZ)
?
7 8 cards:
4/2/2
or 3/3/2
(BZ)
7 cards:
3/3/1
(BZ)
??8 cards:
5/3
7 cards:
4/3
8 64 cards:
22/20/14/18
???
9 ?10 cards:
4/4/2
or 8/2
????
10 ????
(BZ) = Berend van der Zwaag

Flush
Den \ Num123456789
2 10 cards:
A2346789J,A
(BZ)
or AAAAAAA234
(BZ)
3 13 cards:
AAAAA345678,AA
(BZ)
6 cards:
A23456
(BZ)
4 8 cards:
AAAA223,A
(BZ)
16 cards:
AAAAAAAAA2234678
(BZ)
5 25 cards:
AAAAAA2222223467
89J,AAAA,AA (BZ)
25 cards:
AA223344555555
55555678,AAA (BZ)
25 cards:
AAAAAAAA222233
456778JQK,AA (BZ)
25 cards:
24/1
6 6 cards:
A2346,A
(BZ)
6 cards:
A23457
(BZ)
7 9 cards:
A234567,AA
(BZ)
7 cards:
A23467,A
(BZ)
9 cards:
A2345689,A
(BZ)
14 cards:
AAAAAA2234578,A
(BZ)
7 cards:
AAA2223
(BZ)
7 cards:
A234567
(BZ)
8 20 cards:
AA223344556688,
AAAA,AA (BZ)
8 cards:
A234678,A
(BZ)
16 cards:
AAAAAAA
222222333 (BZ)
8 cards:
AAA22345
(BZ)
9 9 cards:
AAAA223,AA
(BZ)
10 cards:
A2346789,AA
(BZ)
or AAAAAAAA23
(BZ)
9 cards:
A2346789,A
(BZ)
12 cards:
A2233445678,A
(BZ)
9 cards:
AAAA22233
(BZ)
9 vards:
AAA223345
(BZ)
10 25 cards:
AAAAAA22222333346
AAAA,AAAA (BZ)
25 cards:
AAAAAAAAAAA
222222222333,AA (BZ)
25 cards:
AAAAAAAAAAA223
456799JJQK,A (BZ)
25 cards:
AAAAAAAAA22222
34444456788 (BZ)
(BZ) = Berend van der Zwaag

Straight
Den \ Num123456789
2 X
3 6 cards:
A23455
X
4 ?X
5 XXXX
6 6 cards:
A23457
X
7 7 cards:
A234555
?XXXX
8 8 cards:
A2345/A23
(BZ)
XXX
9 9 cards:
AA2345/234
(BZ)
XXXXX
10 XXXX
(BZ) = Berend van der Zwaag

3-of-a-Kind
Den \ Num123456789
2 6 cards:
3/13 (BZ)
or AAA23,A (BZ)
3 6 cards:
3/2/1
AAA23,4 (BZ)
6 cards:
4/12
4 8 cards:
3/3/12 (BZ)
or AA2223,AA (BZ)
or AAA223,A3 (BZ)
?
5 25 cards:
AAAAAAAA2222222
33345,6667,8 (BZ)
???
6 6 cards:
AAA23,2 (BZ)
?
7 7 cards:
AAA234,5 (BZ)
or AAA223,4 (BZ)
or AAA23,23 (BZ)
7 cards:
3/3/1
or 3/14 (BZ)
or AAAA23,A (BZ)
or AAA223,A (BZ)
7 cards:
AAA234,A (BZ)
or AAA23,AA (BZ)
7 cards:
4/13
??
8 8 cards:
AAA234,56 (BZ)
or AAA223,45 (BZ)
or AAA23,244 (BZ)
8 cards:
AAA234,A5 (BZ)
??
9 9 cards:
3/2/14 (BZ)
or AAA23,4567 (BZ)
or AA222334,A (BZ)
9 cards:
AAA,A22223 (BZ)
or AA,AA22233 (BZ)
or AAA2233,A4 (BZ)
or AAAA223,45 (BZ)
or AAAA234,56 (BZ)
9 cards:
AAAA23,A45 (BZ)
or AAAA23,AA4 (BZ)
???
10 25 cards:
AAAAA22222,33333444,
555678,9 (BZ)
25 cards:
AAAAAAAAAAAA222,
333456,7789 (BZ)
??
(BZ) = Berend van der Zwaag

2 Pair
Den \ Num123456789
2 6 cards:
3/2/1
3 6 cards:
2/2/12
7 cards:
AA,A2233 (BZ)
4 10 cards:
AAA,2233,445 (BZ)
or A,AAA222333 (BZ)
or AAA223,AAA3 (BZ)
or AAA233,AA24 (BZ)
or AAA2223,A24 (BZ)
or AAA223,4445 (BZ)
?
5 25 cards:
AAAAAAAAA2222,
22223334,5678 (BZ)
25 cards:
AAAAAAAAA222222223,
A333333 (BZ)
??
6 6 cards:
AA223,4 (BZ)
6 cards:
AA223,3 (BZ)
7 7 cards:
2/2/13
or AA2233,4 (BZ)
or AAA223,4 (BZ)
or A,AAA223 (BZ)
7 cards:
3/2/12
or 4/2/1
or A,AA2223 (BZ)
7 cards:
2/2/2/1
or 3/3/1
or A,AA2233 (BZ)
10 cards:
4/3/3 (BZ)
7 cards:
3/2/2
?
8 8 cards:
AA,AAA223 (BZ)
or AAA2233,4 (BZ)
or AA2234,A5 (BZ)
8 cards:
A,AA22233 (BZ)
or AA223,334 (BZ)
8 cards:
AA2,A2233 (BZ)
?
9 9 cards:
AAAA,AA223 (BZ)
or AAAA223,A4 (BZ)
or AAA2233,45 (BZ)
or AA223,3456 (BZ)
9 cards:
AA223,3345 (BZ)
or AA2,233445 (BZ)
13 cards:
AAAA2233,A2223 (BZ)
???
10 25 cards:
AAAAA2222,333444,
5556667,890 (BZ)
25 cards:
10/9/3/3 (BZ)
??
(BZ) = Berend van der Zwaag

1 Pair
Den \ Num123456789
2 6 cards:
3/13
or AA234,5 (BZ)
or A,A2234 (BZ)
3 6 cards:
A,AA234 (BZ)
6 cards:
2/14
or 2/2/12
4 8 cards:
AA2234,A2 (BZ)
?
5 25 cards:
AAAAAAAAAAAA2223,
222344445 (BZ)
25 cards:
AAAAAAAA222222233,
33445566 (BZ)
??
6 9 cards:
A,AA222334 (BZ)
or AA,AA22234 (BZ)
or AAA,AA2234 (BZ)
?
7 7 cards:
A,AAA234 (BZ)
7 cards:
4/13
or AA2345,6 (BZ)
or A,A22234 (BZ)
7 cards:
3/2/12
or AA234,56 (BZ)
or AAA234,5 (BZ)
7 cards:
3/14
or 2/2/2/1
??
8 8 cards:
AA,AAA234 (BZ)
or A,AAA2234 (BZ)
8 cards:
AAA234,A5 (BZ)
or AAA234,23 (BZ)
8 cards:
AA2,23345 (BZ)
?
9 9 cards:
AAAA,AA234 (BZ)
or A,AA222234 (BZ)
9 cards:
AA,AAA2345 (BZ)
or AA223456,7 (BZ)
or AA,AA22334 (BZ)
or A,AA223344 (BZ)
9 cards:
AA2345,A67 (BZ)
or AA2345,AA6 (BZ)
or AA2234,A25 (BZ)
or AA2234,567 (BZ)
or AA22345,A6 (BZ)
or A223345,AA (BZ)
or A23344,AA2 (BZ)
9 cards:
3/2/14
or 3/2/2/12 (BZ)
??
10 25 cards:
AAAAAAAAAAAAAA,
AA234,A22356 (BZ)
25 cards:
AAAAAAAAA222223,
A2334,33456 (BZ)
??
(BZ) = Berend van der Zwaag

Nothing
Den \ Num123456789
2 6 cards:
A2346/5
(BZ)
3 6 cards:
A234/66
6 cards:
A23456
4 8 cards:
A23466/67
(BZ)
10 cards:
A23567/A89J
(BZ)
5 25 cards:
24 unrelated cards of 1 suit
1 card of another suit

(BZ)
25 cards:
A234678/A234678/
A2346789JQ/9

(BZ)
25 cards:
A2346789/A23JQK/
A23CD/A2EF/HI

(BZ)
25 cards:
A2346789J/QKCDEF/
HHHH/IILM/OP

(BZ)
6 6 cards:
A2346/6
(BZ)
6 cards:
A23457
7 7 cards:
A234/666
7 cards:
A2346/5/5
(BZ)
7 cards:
AA23457
7 cards:
A23467/5
(BZ)
7 cards:
A23467/8
(BZ)
7 cards:
A234/567
(BZ)
8 8 cards:
A2346/346
(BZ)
8 cards:
AA24567,3
(BZ)
8 cards:
A234678/9
(BZ)
8 cards:
A23456/79
(BZ)
9 9 cards:
AA/A223457
(BZ)
9 cards:
AAA2/24567
(BZ)
9 cards:
AA22/34578
(BZ)
9 cards:
A2346789/J
(BZ)
10 cards:
A2346789/JQ
(BZ)
18 cards:
A23456789TT/
JQKBDE/F

(BZ)
10 25 cards:
AAAA/AAAA/A222/
2224/5556/6678/9

(BZ)
25 cards:
AAAA/AAA2/2223/
3445/5777/890J/Q

(BZ)
25 cards:
A2346789JQ/
A234KCDE/
A23FHIL

(BZ)
25 cards:
A2346789JQK/CCCD/
EFHI/LMOP/RS

(BZ)
(BZ) = Berend van der Zwaag

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 10/24/12.