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Royal Flush - Ace, King, Queen, Jack, 10 All of the Same Suit

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Royal Flush - Ace, King, Queen, Jack, 10 All of the Same Suit Counting Poker Hands and Finding Probabilities Counting Combinations by N. Rimmer Order doesn't matter n = the total number of objects you are choo sing from r = the number of objects you are choosing n! Cn, r = total number of ways to choose r differen t Cn, r = objects out of a total of n when order do esn't matter. r!() n− r ! Example: Total number of 5 card hands that can be dealt from a standard 52 card deck 10 2 52! 52! 525150⋅ ⋅ ⋅49 ⋅ 48 ⋅4746 ⋅ ⋅⋅⋅… 21 C = = = = 2,598,960 52,5 5!52()− 5! 5!47! ⋅ 5 ⋅4 ⋅⋅ 3 2 ⋅ 1 ⋅474645 ⋅ ⋅ ⋅⋅⋅… 21 On TI-83: To PRB Now type 5 Type 52 first Counting Poker Hands and Finding Probabilities 52 Card deck by N. Rimmer 26 Black Cards 26 Red Cards 13 Spades 13 Clubs 13 He arts 13 Dia monds These are called suits In each suit, you have the following 13 cards: A K Q J10 98 7 6 5 4 3 2 ace king queen jack rank: 10 This is called rank of the card suit: spades card: 10 of spades Counting Poker Hands and Finding Probabilities by N. Rimmer How many ways are there to get 5 cards in rank order ? AK Q J 10 9 8 7 6 5 K Q J 10 9 8 7 6 5 4 Q J 10 9 8 7 6 5 4 3 J 10 9 8 7 6 5 4 3 2 10 9 8 7 6 5 4 3 2 A This is called a straight. (Here we are not considering suits) Counting Poker Hands and Finding Probabilities by N. Rimmer Royal Flush - Ace, King, Queen, Jack, 10 all of the same suit One Royal Flush for each suit pick suit C P() Royal Flush = 4,1 C52,5 4 P() Royal Flush = 2,598,960 ≈ 0.0000015391 Odds( Royal Flush ) ≈ 1 in 649,740 Counting Poker Hands and Finding Probabilities by N. Rimmer Straight Flush - Five cards in rank order of the same suit pick suit pick straight royal C ⋅ C 4,1 10,1 − 4 P() Straight Flush = C52,5 36 P() Straight Flush = ≈ 0.0000138517 2,598,960 Odds( Straight Flush ) ≈1 in 72,193 Counting Poker Hands and Finding Probabilities by N. Rimmer 4 of a kind - Four cards of one rank and one other card pick the rank pick all suits pick other card C ⋅ C ⋅ C P()4 of akind = 13,1 4,4 48,1 C52,5 624 P()4 of akind = 2,598,960 ≈ 0.0002400960 Odds(4 of a kind ) ≈1 in 4,165 Counting Poker Hands and Finding Probabilities by N. Rimmer Full House - Three cards of one rank and two cards of another rank pick the rank pick the suits pick other rank pick other suits C ⋅ C ⋅ C 12,1 ⋅ C 4,2 P() Full House = 13,1 4,3 C52,5 3,744 P() Full House = ≈ 0.0014405762 2,598,960 Odds( Full House ) ≈1 in 694 Counting Poker Hands and Finding Probabilities by N. Rimmer Flush - Five cards of the same suit pick suit pick 5 ranks royal straight flushes C ⋅ C − 4 − 36 P() Flush = 4,1 13,5 C52,5 5,108 P() Flush = ≈ 0.0019654015 2,598,960 Odds( Flush ) ≈1 in 509 Counting Poker Hands and Finding Probabilities by N. Rimmer Straight - Five consecutive cards each of any suit pick straight pick each suit royal straight flushes C ⋅ C ⋅ C ⋅ C ⋅ C ⋅ C P() Straight = 10,1 4,1 4,1 4,1 4,1 4,1 −4 − 36 C52,5 10,200 P() Straight = ≈ 0.0039246468 2,598,960 Odds( Straight ) ≈1 in 255 Counting Poker Hands and Finding Probabilities by N. Rimmer 3 of a Kind – Three cards of one rank and two cards of different rank pick the rank pick the suits pick other's rank pick a suit pick a suit C ⋅ C ⋅ C ⋅ C ⋅ C P()3 of a kind = 13,1 4,3 12,2 4,1 4,1 C52,5 54,912 P()3 of a kind = ≈ 0.0211284514 2,598,960 Odds (3 of a kind) ≈ 1 in 47 Counting Poker Hands and Finding Probabilities by N. Rimmer 2 pair - 2 cards of one rank, 2 cards of another rank and another card of a third rank pick the rank of the pairs pick the suits pick the suits pick other card C ⋅ C ⋅ C ⋅ C P()2 pair = 13,2 4,2 4,2 44,1 C52,5 123,552 P()2 pair = ≈ 0.0475390156 2,598,960 Odds(2 pair ) ≈1 in 21 Counting Poker Hands and Finding Probabilities by N. Rimmer One pair - 2 cards of one rank, 3 other cards of different rank pick the rank pick the of the pair pick the suits other ranks pick the suits C ⋅ C ⋅ C ⋅⋅⋅ C C C P() One pair = 13,1 4,2 12,3 4,1 4,1 4,1 C52,5 1,098,240 P() One pair = ≈ 0.4225690276 2,598,960 Odds( One pair ) ≈ 2 in 5 Counting Poker Hands and Finding Probabilities by N. Rimmer No pair – 5 different ranks each of any suit pick 5 ranks pick each suit royal straight flushes flushes straights C 13,5 ⋅ CCCCC 4,1 ⋅ 4,1 ⋅ 4,1 ⋅ 4,1 ⋅ 4,1 −− 4 36 − 5,108 − 10,200 P() No pair = C52,5 1,302,540 P() No pair = ≈ 0.5011773940 2,598,960 Odds( No pair ) ≈1 in 2 Counting Poker Hands and Finding Probabilities by N. Rimmer.
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