Embed Size (px)
Citation preview
Counting Poker Hands
George Ballinger
In a standard deck of cards there are 13 kinds of cards: Ace (A), 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack (J),Queen (Q) and King (K). Each of these kinds comes in four suits: Spade (♠), Heart (r), Diamond(q) and Club (♣). There are 52 cards altogether:
A♠
♠A
♠
2♠
♠2
♠
♠
3♠
♠3
♠
♠
♠
4♠
♠4
♠
♠
♠
♠
5♠
♠5
♠
♠
♠
♠
♠
6♠
♠6
♠
♠
♠
♠
♠
♠
7♠
♠7
♠
♠
♠
♠
♠
♠
♠
8♠
♠8
♠
♠
♠
♠
♠
♠
♠
♠
9♠
♠9
♠
♠
♠
♠
♠
♠
♠
♠
♠
10♠
♠10
♠
♠
♠
♠
♠
♠
♠
♠
♠
♠
J♠
♠J
♠
♠
Q♠
♠Q
♠
♠
K♠
♠K
♠
♠
Ar
rA
r
2r
r2
r
r
3r
r3
r
r
r
4r
r4
r
r
r
r
5r
r5
r
r
r
r
r
6r
r6
r
r
r
r
r
r
7r
r7
r
r
r
r
r
r
r
8r
r8
r
r
r
r
r
r
r
r
9r
r9
r
r
r
r
r
r
r
r
r
10r
r10
r
r
r
r
r
r
r
r
r
r
Jr
rJ
r
r
Qr
rQ
r
r
Kr
rK
r
r
Aq
qA
q
2q
q2
q
q
3q
q3
q
q
q
4q
q4
q
q
q
q
5q
q5
q
q
q
q
q
6q
q6
q
q
q
q
q
q
7q
q7
q
q
q
q
q
q
q
8q
q8
q
q
q
q
q
q
q
q
9q
q9
q
q
q
q
q
q
q
q
q
10q
q10
q
q
q
q
q
q
q
q
q
q
Jq
qJ
q
q
q
q
Kq
qK
q
q
A♣
♣A
♣
2♣
♣2
♣
♣
3♣
♣3
♣
♣
♣
4♣
♣4
♣
♣
♣
♣
5♣
♣5
♣
♣
♣
♣
♣
6♣
♣6
♣
♣
♣
♣
♣
♣
7♣
♣7
♣
♣
♣
♣
♣
♣
♣
8♣
♣8
♣
♣
♣
♣
♣
♣
♣
♣
9♣
♣9
♣
♣
♣
♣
♣
♣
♣
♣
♣
10♣
♣10
♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
J♣
♣J
♣
♣
Q♣
♣Q
♣
♣
K♣
♣K
♣
♣
A poker hand consists of an unordered selection of five cards chosen from a standard deck of 52cards. There areC(52, 5) = 2,598,960 distinct poker hands.
Poker hands belong to one of ten categories ranging from the highest ranking, a royal flush, to thelowest ranking, a high card. We describe these categories and use counting techniques to countthe number of possible hands belonging to each category. Finally, we calculate the probability ofbeing dealt a 5-card hand from each category.
1. A royal flush consists of an Ace, King, Queen, Jack and 10 of the same suit. There are fourroyal flush hands:
A♠
♠A♠
K♠
♠K
♠
♠
Q♠
♠Q
♠
♠
J♠
♠J
♠
♠
10♠
♠10
♠
♠
♠
♠
♠
♠
♠
♠
♠
♠
Ar
rAr
Kr
rK
r
r
Qr
rQ
r
r
Jr
rJ
r
r
10r
r10
r
r
r
r
r
r
r
r
r
r
Aq
qAq
Kq
qK
q
q
q
q
Jq
qJ
q
q
10q
q10
q
q
q
q
q
q
q
q
q
q
A♣
♣A♣
K♣
♣K
♣
♣
Q♣
♣Q
♣
♣
J♣
♣J
♣
♣
10♣
♣10
♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
1
2. A straight flush is a consecutive sequence of five cards of the same suit, excluding a royalflush. For each of the four suits there are nine straight flushes. For example in hearts thestraight flushes are:
Kr
rK
r
r
Qr
rQ
r
r
Jr
rJ
r
r
10r
r10
r
r
r
r
r
r
r
r
r
r
9r
r9
r
r
r
r
r
r
r
r
rQr
rQ
r
r
Jr
rJ
r
r
10r
r10
r
r
r
r
r
r
r
r
r
r
9r
r9
r
r
r
r
r
r
r
r
r
8r
r8
r
r
r
r
r
r
r
r
Jr
rJ
r
r
10r
r10
r
rr
r
r
rr
r
r
r
9r
r9
r
r
r
r
r
r
r
r
r
8r
r8
r
r
r
r
r
r
r
r
7r
r7
r
r
r
r
r
r
r 10r
r10
r
r r
r
r
r r
r
r
r
9r
r9
r
rr
r
r
rr
r
r
8r
r8
r
r
r
r
r
r
r
r
7r
r7
r
r
r
r
r
r
r
6r
r6
r
r
r
r
r
r
9r
r9
r
r r
r
r
r r
rr
8r
r8
r
r
r
r
r
r
r
r
7r
r7
r
r
r
r
r
r
r
6r
r6
r
r
r
r
r
r
5r
r5
r
r
r
r
r
8r
r8
r
r
r
r
r
r
r
r
7r
r7
r
r
r
r
r
r
r
6r
r6
r
r
r
r
r
r
5r
r5
r
r
r
r
r
4r
r4
r
r
r
r
7r
r7
r
r
r
r
r
r
r
6r
r6
r
r
r
r
r
r
5r
r5
r
r
r
r
r
4r
r4
r
r
r
r
3r
r3
r
r
r
6r
r6
r
r
r
r
r
r
5r
r5
r
r
r
r
r
4r
r4
r
r
r
r
3r
r3
r
r
r
2r
r2
r
r
5r
r5
r
r
r
rr
4r
r4
r
r
r
r
3r
r3
r
r
r
2r
r2
r
r
Ar
rA
r
There are 9· 4 = 36 straight flushes in total.
3. A four of a kind consists of four cards that are all of the same kind together with a fifth cardof a different kind. Examples of a four of a kind poker hand are:
8♠
♠8
♠
♠
♠
♠
♠
♠
♠
♠
8r
r8
r
r
r
r
r
r
r
r
8q
q8
q
q
q
q
q
q
q
q
8♣
♣8
♣
♣
♣
♣
♣
♣
♣
♣
Qr
rQ
r
r
J♠
♠J
♠
♠
Jr
rJ
r
r
Jq
qJ
q
q
J♣
♣J
♣
♣
4r
r4
r
r
r
r
3♠
♠3
♠
♠
♠
3r
r3
r
r
r
3q
q3
q
q
q
3♣
♣3
♣
♣
♣
7q
q7
q
q
q
q
q
q
q
There areC(13, 1) · C(4, 4) ways of choosing one of the 13 kinds and all four suits followedthen byC(12, 1)·C(4, 1) ways of choosing one of the remaining 12 kinds and one of thefoursuits for the fifth card. By the product rule there areC(13, 1)·C(4, 4)·C(12, 1)·C(4, 1) = 624such hands.
4. A full houseconsists of three cards that are all of the same kind (a “threeof a kind”) togetherwith two cards of another kind (a “pair”). Examples of a full house are:
A♠
♠A♠
Ar
rA
r
A♣
♣A
♣
5r
r5
r
r
r
r
r
5♣
♣5
♣
♣
♣
♣
♣10r
r10
r
r r
r
r
r r
r
r
r
10q
q10
q
q
q
q
q
q
q
q
q
q
10♣
♣10
♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
Jr
rJ
r
r
J♣
♣J
♣
♣
4♠
♠4
♠
♠
♠
♠
4r
r4
r
r
r
r
4q
q4
q
q
q
q
9♠
♠9
♠
♠
♠
♠
♠
♠
♠
♠
♠
9q
q9
q
q
q
q
q
q
q
q
q
There areC(13, 1)·C(4, 3) ways of choosing one of the 13 kinds and three of the four suits forthe three of a kind followed then byC(12, 1) ·C(4, 2) ways of choosing one of the remaining12 kinds and two of the four suits for the pair. In total there areC(13, 1) · C(4, 3) · C(12, 1) ·C(4, 2) = 3,744 such hands.
2
5. A flush consists of five cards that are all of the same suit, excludinga straight flush or a royalflush. Examples of a flush are:
K♠
♠K
♠
♠
Q♠
♠Q
♠
♠
7♠
♠7
♠
♠
♠
♠
♠
♠
♠
6♠
♠6
♠
♠
♠
♠
♠
♠
5♠
♠5
♠
♠
♠
♠
♠10r
r10
r
r r
r
r
r r
r
r
r
8r
r8
r
r
r
r
r
r
r
r
7r
r7
r
r
r
r
r
r
r
6r
r6
r
r
r
r
r
r
2r
r2
r
r
A♣
♣A♣
J♣
♣J
♣
♣
8♣
♣8
♣
♣
♣
♣
♣
♣
♣
♣
4♣
♣4
♣
♣
♣
♣
3♣
♣3
♣
♣
♣
There areC(13, 5) · C(4, 1) ways of choosing five of the 13 kinds and one of the four suits.However, since this also includes straight and royal flushes, of which there are 40, then thetotal number of flushes isC(13, 5) · C(4, 1)− 40= 5,108.
6. A straight is a consecutive sequence of five cards that are not all of the same suit. Examplesof a straight are:
A♠
♠A♠
K♠
♠K
♠
♠
q
q
J♠
♠J
♠
♠
10♠
♠10
♠
♠
♠
♠
♠
♠
♠
♠
♠
♠
10q
q10
q
q q
q
q
q q
q
q
q
9♠
♠9
♠
♠♠
♠
♠
♠♠
♠
♠
8♠
♠8
♠
♠
♠
♠
♠
♠
♠
♠
7r
r7
r
r
r
r
r
r
r
6r
r6
r
r
r
r
r
r
5r
r5
r
r
r
rr
4♣
♣4
♣
♣
♣
♣
3♣
♣3
♣
♣
♣
2q
q2
q
q
Aq
qA
q
The smallest card in a straight can be any of ten kinds: Ace, 2,3, 4, 5, 6, 7, 8, 9 or 10.Including straight flushes and royal flushes, each of the five cards can be any of the foursuits. Therefore there are 10· 45 straights, straight flushes or royal flushes. Finally, the totalnumber of straights is 10· 45
− 40= 10,200.
7. A three of a kind is a poker hand consisting of three cards that are all the samekind togetherwith two cards of different kinds. Examples of a three of a kind are:
J♠
♠J
♠
♠
Jr
rJ
r
r
Jq
qJ
q
q
8r
r8
r
r
r
r
r
r
r
r
2r
r2
r
r
6r
r6
r
r
r
r
r
r
6q
q6
q
q
q
q
q
q
6♣
♣6
♣
♣
♣
♣
♣
♣
A♣
♣A
♣
K♣
♣K
♣
♣
3♠
♠3
♠
♠
♠
3r
r3
r
r
r
3♣
♣3
♣
♣
♣
9r
r9
r
r
r
r
r
r
r
r
r
2q
q2
q
q
There areC(13, 1) · C(4, 3) ways of choosing one of the 13 kinds and three of the foursuits for the three of a kind followed then byC(12, 2) · 42 ways of choosing two of theremaining 12 kinds and any of the four suits for the remainingtwo cards. Altogether thereareC(13, 1) · C(4, 3) · C(12, 2) · 42
= 54,912 such hands.
3
8. A two pair consists of five cards with two of one kind, two of a second kindand one of athird kind. Examples of a two pair are:
Q♠
♠Q
♠
♠
Q♣
♣Q
♣
♣
10♠
♠10
♠
♠
♠
♠
♠
♠
♠
♠
♠
♠
10r
r10
r
r
r
r
r
r
r
r
r
r
2q
q2
q
q
9r
r9
r
r r
r
r
r r
rr
9♣
♣9
♣
♣♣
♣
♣
♣♣
♣
♣
2♠
♠2
♠
♠
2♣
♣2
♣
♣
Ar
rA
r4♠
♠4
♠
♠
♠
♠
4q
q4
q
q
q
q
3r
r3
r
r
r
3♣
♣3
♣
♣
♣
7q
q7
q
q
q
q
q
q
q
There areC(13, 2) ·C(4, 2) ·C(4, 2) ways of choosing two of the 13 kinds and any two of thefour suits for each of these two kinds to produce two pairs. Then there areC(11, 1) · C(4, 1)ways of choosing one of the remaining 11 kinds and one of the four suits to make up the fifthcard. Therefore there areC(13, 2) ·C(4, 2) ·C(4, 2) ·C(11, 1) ·C(4, 1) = 123,552 such hands.
9. A one pair consists of five cards where two are of the same kind and the other three are ofdifferent kinds. Examples of a one pair are:
Qr
rQ
r
r
q
q
Jq
qJ
q
q
8r
r8
r
r
r
r
r
r
r
r
4q
q4
q
q
q
q
10♠
♠10
♠
♠ ♠
♠
♠
♠ ♠
♠
♠
♠
10♣
♣10
♣
♣♣
♣
♣
♣♣
♣
♣
♣
9q
q9
q
q
q
q
q
q
q
q
q
8♣
♣8
♣
♣
♣
♣
♣
♣
♣
♣
2♣
♣2
♣
♣
4♠
♠4
♠
♠
♠
♠
4q
q4
q
q
q
q
Ar
rA
r
Jr
rJ
r
r
3♠
♠3
♠
♠
♠
There areC(13, 1) · C(4, 2) ways of choosing one of the 13 kinds and any two of the foursuits for the one pair. Then there areC(12, 3) · 43 ways of choosing three of the remaining12 kinds and any of the four suits for each of these other threecards. So in total there areC(13, 1) · C(4, 2) · C(12, 3) · 43
= 1,098,240 such hands.
10. A high card is any poker hand that does not belong to one of the above nine categories. Inother words it consists of five cards, each of a different kind, not all the same suit and not allin sequence. Examples of a high card poker hand are:
Ar
rAr
10r
r10
r
rr
r
r
rr
r
r
r
9q
q9
q
q
q
q
q
q
q
q
q
7r
r7
r
r
r
r
r
r
r
3♣
♣3
♣
♣
♣
Qr
rQ
r
r
10q
q10
q
q
q
q
q
q
q
q
q
q
6♣
♣6
♣
♣
♣
♣
♣
♣
5♣
♣5
♣
♣
♣
♣
♣
4♠
♠4
♠
♠
♠
♠
7♣
♣7
♣
♣
♣
♣
♣
♣
♣
5♣
♣5
♣
♣
♣
♣
♣
4♣
♣4
♣
♣
♣
♣
3♣
♣3
♣
♣
♣
2q
q2
q
q
There areC(13, 5) − 10 ways of choosing five different kinds of cards that are not all insequence and there are 45
−4 different suits possible for these five cards so they are not all ofthe same suit. Thus in total there are (C(13, 5)− 10) · (45
− 4) = 1,302,540 high card hands.
4
The sum of the total numbers of hands from each of these ten categories is 2,598,960, which is thetotal number of possible poker hands. Dividing each of thesecategory totals by 2,598,960 givesthe probability of being dealt a poker hand belonging to eachcategory. The results are summarizedin the following table.
Category Sample Hand Number of Hands Probability
royal flush A♠
K♠
Q♠
J♠
10♠ 4 0.0000015
straight flush Jq
10q
9q
8q
7q 36 0.0000139
four of a kind 5♠
5r
5q
5♣
10r 624 0.0002401
full house K♠
Kq
K♣
6r
6q 3,744 0.0014406
flush Ar
Jr
7r
4r
3r 5,108 0.0019654
straight 9r
8q
7q
6♣
5♣ 10,200 0.0039246
three of a kind Qr
Q♣
7q
2♣ 54,912 0.0211285
two pair 5r
5q
3r
3♣
K♠ 123,552 0.0475390
one pair 7♠
7r
A♠
4♠
3q 1,098,240 0.4225690
high card 10♠
9q
5q
3q
2r 1,302,540 0.5011774
Total 2,598,960 1.0000000
♠q♣r
♠q♣r
♠q♣r
♠q♣r
♠q♣r
♠q♣r
♠q♣r
♠q♣r
♠q♣r
♠q♣r
5
