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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