Poker


1 pack, 52 cards, 4 suits.
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3 of a  kind  (but not 4 of a kind):

5 cards                                  Probability

A,A,A,no ace, no ace   (4/52)* (3/51)* (2/50)*(48/49)*(47/48)  = 47/270725

3 aces, 2  otherss in any order    (47/270725)* (5*4/2)  = 94/54145

3 of a kind                      (94/54145)*13=94/4165 ~ 0.022569 ~ 1/44.

approximation from a book: 1/47 ~ 0.0212766
many packs: (1/13)*(1/13)*(1/13)*(12/13)*(12/13)*10*13 ~ 0.0504184

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2 pairs (but not 3 of a kind):

5 cards                             Probability

A,A,9,9, no A,9            (4/52)* (3/51)*(4/52)* (3/51)*(44/48)=11/586092

2 aces, 2 nines, another  in any order     (11/586092)*(5*4/2)*3=55/97682

2 pairs      (55/97682)*(13*12/2) =165/3757 ~ 0.043918 ~ 1/23.

approximation from a book: 1/21 ~0.047619
many packs:(1/13)*(1/13)*(1/13)*(1/13)*(11/13)*30*13*6 ~ 0.0693253
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Probability of 4 of a kind

A,A,A,A, another card  (4/52)*(3/51)*(2/50)*(1/49)*(48/48)=1/ 270725
4 aces in any order 5*(1/ 270725)=1/ 54145
4 of a kind   13*(1/ 54145) =1/4165 ~ 0.000240096

many packs: (1/13)^3(12/13)*5=60/28561 ~  0.00210077
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Probability of a pair (but not 3 of a kind or 2 pairs)

A,A,no ace, no ace or pair with the third card, no ace or pair with third or fourth card:
     (4/52)*(3/51)*(48/50)*(44/49)*(40/48)=176/54145

two aces (but not 3 aces or two pairs):
10*(176/54145)=352/10829.

a pair:
(352/10829)*13=352/833 ~ 0.422569.

many decks:  (1/13)*(12/13)*(11/13)*(10/13)*10 =  13200/28561 ~  0.462169.
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