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 13 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 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 P() Flush = 4,1 13,5 −4 − 36 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

P() Straight = 10,200 ≈ 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

P()3 of a kind = 54,912 ≈ 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 ⋅ CCCCC ⋅ ⋅ ⋅ ⋅ −− − − P() No pair = 13,5 4,1 4,1 4,1 4,1 4,1 4 36 5,108 10,200 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