Log............................................................................................................................................................................................................ 3
Poker-Wahrscheinlichkeiten................................................................................................................................................ 4
Poker (Encyclopædia Britannica).................................................................................................................................... 4
Poker wahrscheinlichkeit im 52
Kartensatz-poker............................................................................................. 6
Wahrscheinlichkeit....................................................................................................................................................................... 6
Q: Was ist die
Wahrscheinlichkeit, die gerades Straight Flush, Vierling, Full House etc.
erhält, wenn es fünf Karten gegeben wird?........................................................................................................................................................................................................... 6
Conditional probabilities.................................................................................................................................................. 11
Q: What is the probability to get
a House or Four of a kind if you have Three of a kind, and choose to change two
or one of the remaining cards?......................................................................................................................................................................... 11
Q: What is the probability to get
a house if you have Two Pair, and choose to change the remaining card?.......... 12
Q: What is the probability to get
a house, four of a kind, three of a kind or Two Pair if you have a Pair, and
choose to change 2 or 3 of the remaining Cards?............................................................................................................................................................. 13
Q: What is the probability that
the other players get at least one pair, two pair etc. when the 5 cards are
dealt?. 17
Q: What is the probability that
the other players get at least one pair, two pair etc. when all have changed
cards? 17
Poker probability in 32-card deck
poker................................................................................................................... 18
Introduction............................................................................................................................................................................... 18
Probabilities................................................................................................................................................................................ 18
Q: What is the probability
getting Straight Flush, Four of a Kind, House etc. when given five Cards?.................... 18
References........................................................................................................................................................................................ 20
Dieses Dokument
wurde ursprünglich nach irgendeiner kräftigen Diskussion zwischen mir und
einigen meiner Freunde erstellt. Was die beste Klage ist: Flush oder Straße?
Wenn du Standard
five-card abgehobenen Betrag spielst und du hast ein Paar und ein As.
Solltest du das Paar/Zwilling halten oder solltest du das As auch halten? Was
über das Spielen mit 32 Kartensatz? Dieses Dokument tut diese Berechnungen und
viel mehr. Während des Momentes hat dieses Dokument nicht von dem Texas Hold’em
Schürhakenvariante gehandelt.
Ich habe auch
irgendeine Preisangabe von anderen Quellen getan, in denen ich Fehlberechnung
behoben habe. Dieses wurde getan, bevor die erste Version dieses Dokumentes bei
2000-01-17 veröffentlicht wurde, als gute Schürhakenberechnungen nicht
allgemein vorhanden waren. Heute erklärt Einstellung wie die Wikipedia Artikel Poker Probability
(English) und gibt die Standardwahrscheinlichkeiten sehr gut an.
Ich Wille in
diesem Abschnittgeben, das etwas Bemerkung von einem on-line-Artikel über
Schürhaken veröffentlichte, bei
http://www.britannica.com/bcom/eb/article/6/0,5716,62116+1,00.html, aber jetzt
entfernte.
Der Text im
Kursiv ist von diesem Artikel in Encyclopædia
Britannica.
Poker
a family of card games, almost invariably
played as gambling games. Although played internationally, Poker is most
popular in North America.
Heute wird Poker auf der ganzen Erde gespielt und ist überall,
hauptsächlich wegen des Internets und der Möglichkeit extrem populär, äußere
Zustandon-line-steuerung zu spielen, deren sehr einschränkend in den meisten
Ländern.
A Poker hand usually consists of five cards. Players
try for combinations of two or more cards of a kind, five-card sequences, or
five cards of the same suit. (See below Rank of Hands.)
Heute ist Texas Hold’em die populärste Hauptströmungsvariante,
hauptsächlich weil es auf der Fernsehen gelüfteten World Series of Poker (abgekürztes WSOP) gespielt hat, weit erkannt
als die Weltmeisterschaft des Spiels. Die meisten on-line-Kasinos bieten auch
Varianten des Texas Hold’em als das Standardspiel im on-line-Schürhaken an.
Texas Hold’em besteht zwei Karten unten und aus fünf Karten oben, d.h. sieben
Karten, um für jeden Spieler an der runden Endrunde zu verwenden.
Poker is played with a standard 52-card deck
in which all suits are of equal value, the cards ranking from the ace high,
downward through king, queen, jack, and the numbered cards 10 to the deuce. The
ace may also be considered low to form a straight (sequence) ace through five
as well as high with king-queen-jack-10.
Ich habe Poker mit 32 Kartensatz in Deutschland gespielt und während du
unterhalb der Klassifizierung der Hand sehen kannst, ändere!
Rank of
hands.
The traditional ranking is (1) straight flush
(five cards of the same suit in sequence, the highest ace-king-queen-jack-ten
being called a royal flush; (2) four of a kind, plus any fifth card; (3) full
house; (4) flush; (5) straight; (6) three of a kind; (7) two pair; (8) one
pair; (9) no pair, highest card determining the winner.
Unter habe mir alle Möglichkeiten, nicht gerechtes Geben der Rank
errechnet. Wie ich oben angab, 32 Kartensatz, Änderungen dieser Rank.
To determine the winner in hands
in which there are hands of the same rank, the one containing the highest card
wins. If the high cards are identical, the second highest wins, and so on. With
full houses, the higher three of a kind wins; with two pairs, the highest pair
wins, or if the pairs are identical, the odd high card wins, as in the case of
identical pairs. For the occasion when none of the above applies, e.g., two
flushes with identical cards in different suits, house rules may apply (the
winning hand being the one in the higher bridge suit, or the two hands may
split the pot). There is no universally accepted code of Poker rules. A code
prepared by Oswald Jacoby in 1940 and a set of rules in the United States
Playing Card Company's Official Rules of Card Games, published from 1945, are
the ones usually adopted subject to house rules in the United States.
Viele Leute, die Schürhaken spielen, fangen an, was zu argumentieren zu
tun, wenn zwei Leute dasselbe bündig mit identische Karten in den
unterschiedlichen Klagen haben. Eine
Richtlinie aufstellen, bevor das Spiel den Topf 50/50 als Rückstellung
angestellt oder aufgespaltet hat, wenn nichts sonst vereinbart worden ist.
Wild cards. Most serious poker players decry
the use of wild cards, but among the less serious, a wild card can be declared
by the dealer (the deuce is most popular, but any rank can be used, or a
distinguishable face card: the one-eyed jack). When there are wild cards in the
game, the highest hand becomes five of a kind, though some house rules preserve
the sanctity of the royal flush
beachten, dass wilde Karte die Wahrscheinlichkeit ernsthaft ändern kann und
ich mich wirklich wilde, empfehle Karten nicht in jedem ernsten Spiel zu
verwenden. Du solltest die Wahrscheinlichkeiten mindestens berücksichtigen und
den korrekten Rank errechnen lassen (wenn er ihn ändert - ich habe nicht dieses
errechnet, also kann ich nicht wirklich Sagen). Aber einen Durchgang in der
wilden Karte des Artikels Wikipedia Wild card (poker)
(English).
Another
issue with wild cards is that they distort the hand frequencies. In 5-card
stud, the stronger hands are less frequent than the weaker hands; i.e., no pair
is most common, followed by one pair, two pair, three of a kind, etc. When you
add wild cards, the stronger hands gain frequency while the weaker hands lose
frequency. For example, if you have a pair and a wild card, you will always
choose three of a kind rather than two pair. This causes three of a kind to be more common than two pair.
Über den Rank.
A: Wahrscheinlichkeit,
die unterschiedliche Hände erhält, wenn Sie fünf Karten behandelt werden
|
(Rank der Hände)
Wahrscheinlichkeit, die diese Hand erhält:
|
Aus COMB (52;5) = 2598960 Möglichkeiten, fünf Karten zu zeichnen erhältst
du den folgenden Händen diese Zahl von Zeiten
|
Genaue Wahrscheinlichkeit
|
Ca. Wahrscheinlichkeit 1/
|
Ca. Wahrscheinlichkeit numerisch.
|
|
Royal Flush
|
4
|
1/649740
|
1/649740
|
0,00000154
|
|
Straight Flush
|
36
|
9/649740
|
1/72193
|
0,0000139
|
|
Poker/Vierling
(Four Of A Kind)
|
624 [5]
|
1/4165
|
1/4165
|
0,000240
|
|
Full House
|
3744[11]
|
6/4165
|
1/694
|
0,00144
|
|
Flush
|
5108 [8]
|
1277/649740
|
1/509
|
0,00197
|
|
Straße
|
10200
|
5/1274
|
1/255
|
0,00392
|
|
Drilling (Three
Of A Kind)
|
54912
|
88/4165
|
1/47
|
0,0211
|
|
Zwei Paare (Two
Pairs)
|
123552
|
198/4165
|
1/21
|
0,0475
|
|
Paar/Zwilling
(One Pair)
|
1098240[11]
|
352/833
|
1/2.4
|
0,423
|
|
Höchste Karte
(High Card)
|
1302540
|
1302540/2598960
|
1/2
|
0,501
|
|
Wahrscheinlichkeit, die mindestens diese Hand erhält:
|
Aus COMB (52;5) = 2598960 Möglichkeiten, fünf Karten zu zeichnen erhältst
du den folgenden Händen diese Zahl von Zeiten
|
Genaue Wahrscheinlichkeit
|
Ca. Wahrscheinlichkeit 1/
|
Ca. Wahrscheinlichkeit numerisch.
|
|
Royal Flush
|
4
|
|
1/649740
|
0,00000154
|
|
Straight Flush
|
40
|
|
1/64974
|
0,0000154
|
|
Poker/Vierling
(Four Of A Kind)
|
664
|
|
1/3914
|
0,000256
|
|
Full House
|
4408
|
|
1/590 [1]
|
0,00170
|
|
Flush
|
9516
|
|
1/273
|
0,00366
|
|
Straße
|
19716
|
|
1/132
|
0,00759
|
|
Drilling (Three
Of A Kind)
|
74628
|
|
1/35
|
0,0287
|
|
Zwei Paare (Two
Pairs)
|
198180
|
|
1/13
|
0,0763
|
|
Paar/Zwilling
(One Pair)
|
1296420
|
|
1/2 [1]
|
0,499
|
|
Höchste Karte
(High Card)
|
2598960
|
|
1
|
1
|
1 in 5 für mindestens ein Paar Steckfassungen (11) [1]
Warum:
P(House) =
13*12*COMBIN(4;3)*COMBIN(4;2) = 3744
Warum?
13*12, weil: 13 Spalten horizontal und 12
Reihen vertikal
1-1-1-2-2 2-2-2-1-1 ...
13-13-13-1-1
1-1-1-3-3 2-2-2-3-3 13-13-13-2-2
1-1-1-4-4 2-2-2-4-4 13-13-13-3-3
...
1-1-1-13-13-13 2-2-2-13-13-13 13-13-13-12-12
Lässt Blick bei
einem von diesem 1-1-1-2-2
Du hast Combin
(4; 3) zum 1-1-1 und Combin zu zeichnen (4; 2) zum 2-2 zu zeichnen
P(Drilling) =
13*COMBIN(48;2)*(COMBIN(4;3) - P(House)
= 13*1128*4 - 3744
= 54912
Warum?
13*COMBIN (48; 2) 13 Spalten horizontal und
Combin (48; 2) Reihen Vertikale als oben
1-1-1-2-2 2-2-2-1-1 ...
13-13-13-1-1
1-1-1-2-3 2-2-2-1-3 13-13-13-1-2
1-1-1-2-4
1-1-1-2-5
...
1-1-1-3-3 2-2-2-3-3 13-13-13-2-2
1-1-1-3-4 2-2-2-3-4 13-13-13-2-3
1-1-1-3-5
...
1-1-1-4-4 2-2-2-4-4 13-13-13-3-3
...
1-1-1-13-13-13 2-2-2-13-13-13 13-13-13-12-12
Lässt Blick bei
1-1-1-2-3. Du hast Combin (4; 3) Möglichkeiten, 1-1-1 zu zeichnen. Da wir jetzt
alles 1-1-1-2-2.1-1-1-3-3 etc. gezählt haben, müssen wir Substract die
Hausberechnung.
- Eine andere
Weise, sie zu betrachten ist so
Du hast 13
Spalten. Lässt Blick an 1-1-1-y-x. Du hast Combin (4; 3) Möglichkeiten, 1-1-1
und dich zu zeichnen Combin haben (48; 2) zum von y-x zu zeichnen. Da wir jetzt
alles 1-1-1-2-2.1-1-1-3-3 etc. gezählt haben, müssen wir Substract die
Hausberechnung.
Von [14] : http://www.thewizardofodds.com/game/pokerodd.html
Royal Flush
The number of different royal
flushes are four (one for each suit).
Straight Flush
The highest card in a straight
flush can be 5,6,7,8,9,10,Jack,Queen, or King. Thus there are 9 possible high
cards, and 4 possible suits, creating 9 * 4 = 36 different possible straight
flushes.
Four of a Kind
There are 13 different possible
ranks of the 4 of a kind. The fifth card could be anything of the remaining 48.
Thus there are 13 * 48 = 624 different four of a kinds.
Full House
There are 13 different possible
ranks for the three of a kind, and 12 left for the two of a kind. There are 4
ways to arrange three cards of one rank (4 different cards to leave out), and
combin(4,2) = 6 ways to arrange two cards of one rank. Thus there are 13 * 12 *
4 * 6 = 3,744 ways to create a full house.
Flush
There are 4 suits to choose from
and combin(13,5) = 1,287 ways to arrange five cards in the same suit. From
1,287 subtract 10 for the ten high cards that can lead a straight, resulting in
a straight flush, leaving 1,277. Then multiply for 4 for the four suits,
resulting in 5,108 ways to form a flush.
Straight
The highest card in a straight
flush can be 5,6,7,8,9,10,Jack,Queen,King, or Ace. Thus there are 10 possible
high cards. Each card may be of four different suits. The number of ways to
arrange five cards of four different suits is 45 = 1024. Next
subtract 4 from 1024 for the four ways to form a flush, resulting in a straight
flush, leaving 1020. The total number of ways to form a straight is
10*1020=10,200.
Three of a Kind
There are 13 ranks to choose
from for the three of a kind and 4 ways to arrange 3 cards among the four to
choose from. There are combin(12,2) = 66 ways to arrange the other two ranks to
choose from for the other two cards. In each of the two ranks there are four
cards to choose from. Thus the number of ways to arrange a three of a kind is
13 * 4 * 66 * 42 = 54,912.
Two Pair
There are (13:2) = 78 ways to arrange
the two ranks represented. In both ranks there are (4:2) = 6 ways to arrange
two cards. There are 44 cards left for the fifth card. Thus there are 78 * 62
* 44 = 123,552 ways to arrange a two pair.
One Pair
There are 13 ranks to choose
from for the pair and combin(4,2) = 6 ways to arrange the two cards in the
pair. There are combin(12,3) = 220 ways to arrange the other three ranks of the
singletons, and four cards to choose from in each rank. Thus there are 13 * 6 *
220 * 43 = 1,098,240 ways to arrange a pair.
Nothing
First find the number of ways to
choose five different ranks out of 13 which is combin(13,5) = 1287. Then
subtract 10 for the 10 different high cards that can lead a straight, to be
left with 1277. Each card can be of 1 of 4 suits so there are 45=1024
different ways to arrange the suits in each of the 1277 combinations. However
we must subtract 4 from the 1024 for the four ways to form a flush, leaving
1020. So the final number of ways to arrange a high card hand is
1277*1020=1,302,540.
Specific High Card Lets find the probability of drawing a jack high, for example. There
must be four different cards in the hand all less than a jack, of which there
are 9 to choose from. The number of ways to arrange 4 ranks out of 9 is
combin(9,4) = 126. We must then subtract 1 for the 9-8-7-6-5 combination which
would form a straight, leaving 125. From above we know there are 1020 ways to
arrange the suits. Multiplying 125 by 1020 yields 127,500 which the number of
ways to form a jack high hand. For ace high remember to subtract 2 rather than
1 from the total number of ways to arrange the ranks since A-K-Q-J-10 and
5-4-3-2-A are both valid straights.
|
Five Card Draw High Card Hands
|
|
Hand
|
Combinations
|
Probability
|
|
Ace high
|
502,860
|
0.19341583
|
|
King high
|
335,580
|
0.12912088
|
|
Queen high
|
213,180
|
0.08202512
|
|
Jack high
|
127,500
|
0.04905808
|
|
10 high
|
70,380
|
0.02708006
|
|
9 high
|
34,680
|
0.01334380
|
|
8 high
|
14,280
|
0.00549451
|
|
7 high
|
4,080
|
0.00156986
|
|
Total
|
1,302,540
|
0.501177394
|
Von [11] http://www.sscnet.ucla.edu:80/soc/faculty/campbell/210a_Fall1997/210a_notes_10_14_97.htm
The first thing we need to
know is how many elementary events there are that can occur. We just calculated
it, it's 52!/(47!*5!)=2598960. Now all we have to do is work out how many hands
correspond to each of the above three situations, and divide by this number.
PAIR:
To dealing with the
probability of the pairs first, the first thing is to work out how many
possible pairs there are. Well for any given value, there are (4 2) pairs that
can be drawn, and there are 13 possible values, so there are 13 (4 2) ways of
having a pair. How many combinations of the remaining 12 values are there that
do not result in a pair among the remaining three cards? (12 3) Thus given 12
remaining values, there are (12 3) of picking three distinct ones from them,
for example, 2 3 4, 2 3 5, 2 3 6, .... K Q A. Of course, for each of the three
cards any suit is OK, we can have any combination of the 4 suits, so we have to
multiply by 4^3. Thus the probability of having one pair, and three distinct
remaining cards, is 13 (4 2) (12 3) 4^3
/ (52 5). If we work it out, it's about 0.40.
FULL HOUSE:
A similar approach can be taken for the full
house. There are 13 (4 3) 12 (4 2) ways of having a full house, so the total
probability
of a full house is 13 (4 3) 12 ( 4 2) / (52 5) =
0.0014.
FLUSH
A flush is 4 (13
5)/(52 5)
Was über Royal
Straight flush? Substract 40! benötigen Sie!
FOUR OF A KIND
What about four of
a kind? There are 13 ways of four of a kind, 12 choices for the remaining card,
so 13*12 / (52 5). Pretty
unlikely!
UNRECHT! Nicht
12 Wahlen für die restlichen Karten ABER 48. Sehen [ 5 ] wer mit mir
übereinstimmt.
Flush: (From http://www.schoolnet.ca/vp-pv/amof/e_combI.htm)
We give now a simple
question that can be answered with a knowledge combinations and binomial
coefficients. What is the probability of getting a flush in a five card poker
hand on the initial deal? (A flush means that all five cards are in the same
suit.) First, we have to recognize that a five card poker hand is a combination
of 5 cards chosen from 52 cards. Thus the total number of possible hands is the
binomial coefficient C(52,5) = 2,598,960. The ranks of the cards making up the
flush is a combination of 5 ranks chosen from 13 rank. The suit of the cards
making up the flush is a combination of 1 suit chosen from 4 suits. Multiplying,
there are thus
C(13,5)*C(4,1) = 1287*4 = 5148 ways of getting a flush. The probability of
getting a flush is the ratio of the number of ways of getting a flush divided
by the total number of hands; it is 5148/2598960 = 33/16660 = .001980792317.
Not very high
odds --- about 2 in
every 1000 hands!
Müssen Sie Erröten 4 (Royal flush) und 36 Straight
flush.
Bedingte Wahrscheinlichkeiten
Zusammenfassung
|
Sie haben Drilling und ändern Karten
|
Erhält Haus
|
Erhält
Vierling
|
Gesamtmenge
|
|
2
|
0,0611
|
0,0425
|
0,1036 (ca.. 1/9)
|
|
1
|
0,0638
|
0,021
|
0,0851 (ca.
1/12)
|
Sie sollten zwei
Karten immer ändern. Dann erhalten Sie ein Haus oder ein Vierling in 1 aus 9mal
heraus.
Zusammenfassung mit Brüchen
|
Sie haben Drilling und
|
Ändern Sie 2 Karten - Wahrscheinlichkeit
|
Ändern Sie 1 Karten - Wahrscheinlichkeit
|
|
Selben (Drilling)
|
969/1081 (ca.
1/1,1)
|
43/47 (ca.
1/1.09)
|
|
House oder
besseres
|
112/1081 (ca.
1/9)
|
4/47 (ca.
1/12)
|
|
|
House
|
|
66/1081 (ca.
1/16)
|
|
3/47 (ca. 1/16)
|
|
|
Vierling
|
|
46/1081
(1/23.5)
|
|
1/47
|
WARUM:
Ändern Sie 2
Karten
Es gibt
Comb(47;2) = 1081 mögliche Möglichkeiten, 2 Karten von den restlichen (52-5 =)
47 unbekannten Karten zu zeichnen.
P(House | Drilling und UND ändern 2) =
(2*Combin(3;2) +
10*Combin(4;2))/Comb (47;2) =
66/1081 (ca. 1/6)
Warum? Sie erhielten z.B. diese
Hand 7-7-7-6-8, werfen Sie weg sechs und acht. 2*Combin(3;2) : (7-7-7-6-6 oder 7-7-7-8-8) ist es 3 sechs '
oder 3 acht in den restlichen 47 Karten. 10*Combin(4;2) : (7-7-7-1-1 oder 7-7-7-2-2 oder usw.) ist es
4 irgendjemandes, 4 zwei, usw. in den restlichen 47 Karten.
P(Vierling | Drilling UND ändern 2) =
(Combin(1;1)*(47-1))/Comb(47;2) =
46/1081 (ca. 1/23.5)
Warum? Sie erhielten z.B. diese
Hand 7-7-7-6-8 und Sie werfen weg 6-8.
Sie haben 1 sieben unter den restlichen 47 unbekannten Karten. Es ist
möglich, diese sieben mit allen anderen 46 unbekannten Karten zu kombinieren.
P(House ODER Vierling | Drilling UND ändern 2) =
(66+46)/Comb(47;2) =
112/1081 (ca. 1/9)
Ändern Sie 1
Karte
Es gibt Comb(47;1) = 47 mögliche Möglichkeiten, Karten 1 von den
restlichen (52-5 =) 47 unbekannten Karten zu zeichnen.
P(House | Drilling UND ändern
1) =
Comb(3;1)/Comb(47;1) =
3/47 (ca.. 1/15)
Warum? Sie erhielten z.B. diese Hand 7-7-7-6-8 und Sie
werfen weg die acht. Sie haben 3 sechs unter den restlichen 47 unbekannten
Karten.
P(Vierling | Drilling UND ändern1) =
1/Comb(47;1) =
1/47
P(House OR Vierling | Drilling UND
ändern 1) =
(3+1)/Comb(47;1) =
4/47 (ca.. 1/12)
A:
P(House | Zwei
Paare) = (2+2)/Comb(47;1) = 4/47 (ca. 1/12)
Warum? Z.B. Sie
haben dieses zwei Paare (12, 12) und (3, 3) und Sie werfen die fünfte Karte weg
(5). Dann haben Sie 47 restliche (52-5) Karten, in denen 2 von ihnen 12's sind
und zwei von ihnen 3 sind. Dann ist sie 4 aus 47 heraus, zum entweder dritte 12
oder dritte 3 zu erhalten.
A: Es gibt vier Strategien. Halten Sie alle Karten, Änderung eine, zwei oder
drei der restlichen Karten. Es liegt beendigtes auf der Hand, dass Sie entweder
zwei oder drei Karten immer ändern sollten, wenn Sie Ihre Wahrscheinlichkeit
maximieren, um bessere Karten zu erhalten (ausgenommen, wenn Sie
"täuschen").
Zusammenfassung
|
Sie haben ein Paar und ändern x Karten
|
Erhalten Sie Zwei Paare
|
Erhalten Sie Drilling
|
Erhält House
|
Erhält Vierling
|
Gesamtmenge
|
|
3
|
0,160
|
0,11
|
0,01
|
0,0028
|
0,29
|
|
2
|
0,172
|
0,078
|
0,0083
|
0,00093
|
0,26
|
Zusammenfassung
mit Brüchen
|
Sie haben ein Paar und
|
Ändern Sie 3
Karten - Wahrscheinlichkeit
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Ändern Sie 2 Karten - Wahrscheinlichkeit
|
|
Selben (ein
Paar)
|
11559/16215 (ca.
1/1,4)
|
801/1081 (ca.
1/1,3)
|
|
Zwei Paar oder
besser
|
4656/16215 (ca.
1/3,5)
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280/1081 (ca.
¼)
|
|
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Zwei Paare
|
|
2592/16215 (ca.
1/6)
|
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186/1081 (ca.
1/6)
|
|
|
Drilling
|
|
1854/16215 (ca.
1/9)
|
|
84/1081 (ca.
1/13)
|
|
|
House
|
|
165/16215 (ca.
1/98)
|
|
9/1081
|
|
|
Vierling
|
|
45/16215 (ca.
1/360)
|
|
1/1081[1]
|
WARUM:
Ändern Sie 3
Karten
Es gibt
Comb(47;3) = 16215 mögliche Möglichkeiten, 3 Karten von den restlichen (52-5 =)
47 unbekannten Karten zu zeichnen.
P(Two pair | one pair AND change 3) =
(Combin(3;2)*3*(47-2-1-2) +
Combin(4;2)*9*(47-2-2-2))/Combin(47;3)=
(378+2214)/16215=
2592/16215 (approx. 1/6)
Why? You have 5 known Cards where two of them are a pair, and the rest is
different (ex. 7-7-5-6-8). You have 47 remaining unknown Cards. This 47 unknown
Cards contains a pair (7-7), 3 three of a kind (5-5-5, 6-6-6, 8-8-8) and 9 Four
of a kind (1-1-1-1,2-2-2-2,3-3-3-3,4-4-4-4,9-9-9-9,…,13-13-13-13).
You are not interested in the other pair. This card will give you three or
four of a kind.
The 3 Three of a kind can be
combined in 3*Combin(3;2) ways. This again can be combined with 47 (all
unknown) – 3 unknown cards used in Combin(3;2) – 2 other cards belonging to the
pair (other two 7’s).
The 9 Four of a kind can be
combined in 9*Combin(4;2) ways. This again can be combined 47 (all unknown) – 4
unknown cards used in Combin(4;2) ) – 2 other cards belonging to the pair
(other two 7’s).
You still don’t believe me?
You got
this hand 7-7-5-6-8 and you throw away 5-6-8. Then the possibilities to get two
pair with either 5-5, 6-6 or 8-8 combined with 7-7 is:
5-5-x
5-x-5
x-5-5
In these 3 combinations the last single Card can be substituted with all
remaining 47 Cards except the three 5’s and the two other 7’s.
=3*(47-2-1-2)
added by
6-6-x
6-x-6
x-6-6
In these 3 combinations the last single Card can be substituted with all
remaining 47 Cards except the three 6’s and the two other 7’s.
=3*(47-2-1-2)
added by
8-8-x
8-x-8
x-8-8
In these 3 combinations the last single Card can be substituted with all
remaining 47 Cards except the three 6’s and the two other 7’s.
=3*(47-2-1-2)
=3*3*(47-2-1-2)
=3*Combin(3;2)*(47-2-1-2)
=378
In the same manner
You got this hand 7-7-5-6-8 and you throw away 5-6-8. Then the
possibilities to get two pair with either 1-1,2-2,3-3,4-4,
9-9,10-10,11-11,12-12 or 13-13 combined with 7-7 is:
1-1-x-x
1-x-1-x
1-x-x-1
x-1-1-x
x-1-x-1
x-x-1-1
In these 6 combinations the last Card can be substituted with all remaining
47 Cards except the four 1’s and the two other 7’s.
=6*(47-2-2-2)
added by
2-2-x-x
….
etc..
=9*6*(47-4)
=9*Combin(4;2)*(47-2-2)
=2214
Q.E.D.
P(Three of a kind | one pair AND change 3) =
[Combin(2;1)*Combin((47-2);2)
-Combin(2;1)*3*Combin(3;2)
-Combin(2;1)*9*Combin(4;2)]/16215=
2*(990-9-54)/16215=
1854/16215 (approx. 1/9)
Why? You have 5 known Cards where
two of them are a pair, and the rest is different(f.ex. 7-7-5-6-8). You have 47
remaining unknown Cards. This 47 unknown Cards contains a pair (7-7), 3 three
of a kind (5-5-5, 6-6-6, 8-8-8) and 9 Four of a kind
(1-1-1-1,2-2-2-2,3-3-3-3,4-4-4-4,9-9-9-9,…,13-13-13-13).
You have two 7’s that will give you the Third 7 (Combin(2;1))and 47-2 other
cards to fill the Combin(45;2) remaining hand.
You need to subtract the possible
house you can get with either 5-5, 6-6 or 8-8. F.ex. 7-7-7-5-5. You have 3 pair like
this, and each pair can be drawn out of three 5’s, 6’s or 8’s (Combin(3;2)) . The
two remaining 7’s Combin(2;1).
You also need to subtract the house you can get with
either 1-1, 2-2, 3-3,4-4,9-9,…or 13-13. F.ex.
7-7-7-1-1You have 9 pair like this, and each pair can be drawn out of four 1’s,
2’s, 3’s etc.(Combin(4;2)) . The two
remaining 7’s Combin(2;1).
P(House | one pair AND change 3) =
(3*Combin(3;3) +
9*Combin(4;3) +
Combin(2;1)*3*Combin(3;2) +
Combin(2;1)*9*Combin(4;2) )/16215=
165/16215 (approx. 1/98)
Why? You got this hand 7-7-5-6-8 and you throw away 5-6-8.
Add the bullet points:
House with the pair (7-7)
·
You
have three 5-5-5, 6-6-6, 8-8-8 (3*Combin(3;3))
·
and
nine 1-1-1-1,2-2-2-2, etc (9*Combin(4;3))
House with an extra card in the pair (7-7-7)
You can draw the extra 7 in Combin(2;1) ways.
·
You have three 5-5-5, 6-6-6, 8-8-8. You can draw
2 out of 3 of these (Combin(2;1)*3*Combin(3;2)).
·
And nine 1-1-1-1,2-2-2-2, … , etc You can draw 2
out of 4 of these (Combin(2;1)*9*Combin(4;2)).
P(Four of a kind | one pair AND change 3) =
(Combin(2;2)*(47-2))/16215=
45/16215 (approx. 1/360)
Why? You got this hand 7-7-5-6-8 and you throw away 5-6-8. Then the possibility
to get the two other 7’s is all the 47 remaining unknown Cards except the two
last 7’s.
Change 2 Card
There are Comb(47;2) = 1081 possible ways to draw 2 Cards
from the remaining (52-5=) 47 Cards.
P(Two pair | one pair AND change 2) =
(Combin(3;1)*(47-3-2) +
9*Combin(4;2) +
2*Combin(3;2))/Combin(47;2)=
186/1081 (approx. 1/6)
Why? You got this hand 7-7-5-6-8 and you throw away 6-8
Combin(3;1)*(47-3-2) : (7-7-5-5-*)
One out of three 5’s multiplied by the 47 remaining - three 5’s - two 7’s
P(Three of a kind | A Pair AND Change 2) =
(Combin(2;1)*(47-3-2))/Combin(47;2) =
84/1081 (approx. 1/13)
Why? You got this hand 7-7-5-6-8 and you throw away 6-8
Combin(2;1)*(47-3-2) : (7-7-5-7-*) One out of two 7’s multiplied
by the 47 remaining - three 5’s - two 7’s.
P(House | A Pair AND Change 2) =
(Combin(3;2)+
Combin(2;1)*Combin(3;1))/Combin(47;2) =
(3+6)/1081
9/1081
Why? You got this hand 7-7-5-6-8 and you throw away 6-8
Combin(3;2) : 7-7-5-5-5 – You can draw two 5 out of tree remaining.
Combin(2;1)*Combin(3;1) :
7-7-5-7-5 One out of two 7’s and one out of three 5’s
P(Four of a kind | A Pair AND Change 2) =
Combin(2;2)/Combin(47;2) =
1/1081
A: <…missing for the moment…>
-Suppose that
people just throw away cards that don’t destroy any (one pair, two pair etc.)
A: <…missing for the moment…>
I’ve played 32
card deck poker in Germany and I discussed a lot with people what’s the correct
rank of hands. As you see below it’s more difficult to get Flush than Four of a
kind when you’re playing with 32 cards!
A: Probability getting different hands when dealt five Cards
|
(Rank of Hands)
Probability
getting this hand:
|
Out of comb(32;5) = 201376 ways to draw five
cards you will get the following hands these number of times
|
Exact
probability
|
Approx.
Probability 1/
|
Approx. Probability
num.
|
|
Royal flush
|
4
|
1/50344
|
1/50344
|
0.0000199
|
|
Straight flush
|
12
|
3/50344
|
1/16781
|
0.0000596
|
|
Flush
|
208
|
|
1/968
|
0.00103
|
|
Four of a Kind
|
224
|
|
1/899
|
0.00111
|
|
House
|
1344
|
|
1/150
|
0.00667
|
|
Straight
|
4080
|
|
1/49
|
0.0203
|
|
Three of a Kind
|
10752
|
|
1/19
|
0.0533
|
|
Two Pair
|
24192
|
|
1/8
|
0.120
|
|
One Pair
|
107520
|
|
1/1.9
|
0.534
|
|
None
|
53040
|
|
1/3.8
|
0.263
|
|
Probability
getting at least this hand:
|
Out of comb(32;5) = 201376 ways to draw five cards you will get the
following hands these number of times
|
Exact
probability
|
Approx.
Probability 1/
|
Approx.
Probability num.
|
|
Royal flush
|
4
|
|
1/50344
|
0.0000199
|
|
Straight Flush
|
16
|
|
1/12586
|
0.0000795
|
|
Flush
|
224
|
|
1/899
|
0.00111
|
|
Four of a Kind
|
448
|
|
1/449
|
0.00222
|
|
House
|
1792
|
|
1/112
|
0.00900
|
|
Straight
|
5872
|
|
1/34
|
0.0292
|
|
Three of a Kind
|
16624
|
|
1/12
|
0.0826
|
|
Two Pair
|
40816
|
|
1/5
|
0.203
|
|
One Pair
|
148336
|
|
1/1.36
|
0.737
|
|
None
|
201376
|
|
1
|
1
|
Why:
You have
Royal Straight
Flush
<…missing
for the moment…>
|
[1]
|
http://www.britannica.com/bcom/eb/article/6/0,5716,62116+3,00.html
|
|
[2]
|
http://search.britannica.com/bcom/search/results/advanced/1,5844,4,00.html?p_query0=Poker&p_phraseT0=1
|
|
[3]
|
“Profitable
Things to Watch in a Poker Game” http://www.cardplayer.com:80/caro.htm
1.
First
question: Is this game worth my time? I need to see mistakes made by others
that I wouldn't make myself. If I can't spot them, I'm probably in a bad
game.
2.
Second
question: What is my fantasy seat? By applying the criteria we've talked
about in previous lessons (sit to the left of the loose players so that they
act before you, also sit to the left of knowledgeable, aggressive players,
and sit to the right of tight nonentity players)
3.
Try
to reconstruct hands. Focus on just one opponent and – after seeing the
showdown and while the next deal is being prepared - go back mentally and try
to equate that player's hand with how he played at each stage of the action
4.
When
looking for tells, focus on just one player.
5.
When
you're out of a hand and don't feel like observing, don't.
6.
A simple, accurate way to rate your table. For 20 hands that you're not
involved in: (a) Add one point for each call; (b) Subtract one point for each
raise; and (c) Subtract one extra point for each check-raise (minus two
points total). First bets are ignored in the count. Reraises count as a
single raise (minus one point). All players' actions count, even when they
act more than once on a single betting round. The higher the score, the
better. You'll have to compare your results to other games of the same size,
type, and number of players. But soon, you'll know with surprising accuracy
how profitable today's game is compared to yesterday's.
|
|
[4]
|
“A Glossary of
Poker Terms” http://www.conjelco.com/pokglossary.html
|
|
[5]
|
“what is the
probability of getting a Royal Flush or Four of A Kind” http://www.sit.wisc.edu/~smwise/
|
|
[6]
|
http://www.cs.cornell.edu/cs100-sp99/ProgramDocs/P3_sol.htm
P3B Output:
P is the probability of
having exactly one pair in
an n-card hand.
n
P
----------------------
2
0.0588
3
0.1694
4
0.3042
5
0.4226
6
0.4855
7
0.4728
8
0.3923
9
0.2751
10
0.1599
11
0.0745
12
0.0262
13
0.0062
14
0.0007
|
|
[7]
|
“Mathematical Probability” http://www.ms.uky.edu:80/~viele/sta281/mathprob/mathprob.html
|
|
[8]
|
“C(13,5)*C(4,1)
= 1287*4 = 5148 ways of getting a flush. (including Straight flush 40)”
|
|
[9]
|
“The Chances Of
Winning The UK National Lottery” http://lottery.merseyworld.com/Info/Chances.html
|
|
[10]
|
“Link to nsaa” http://www.schoolnet.ca/vp-pv/amof/e_combI.htm
|
|
[11]
|
“We will
consider three types of hands, following the book. A) One pair, with three
different remaining cards, B) full house (3 of a kind, plus one pair), and C)
flush (all cards of the same suit)”
http://www.sscnet.ucla.edu:80/soc/faculty/campbell/210a_Fall1997/210a_notes_10_14_97.htm
|
|
[12]
|
“5 cards
selected at random from an ordinary deck.” http://www.math.iupui.edu/~momran/m118/chall.htm
|
|
[13]
|
“Ref. Book with
questions, Answers” http://www.cs.colostate.edu:80/~anderson/cs201/exercises-4.4.html
|
|
[14]
|
http://www.thewizardofodds.com/game/pokerodd.html
|
