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---------------------------- - 2362 - How to Calculate Odds for Poker Hands.
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- The logic of calculations in poker are calculations of “ combinations“. For a Full House you need a combination of 3 of a kind and 2 of a kind in 5 card draw The calculations us the rule of “product“. That is what multiplication is. It uses the rule of product.
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- Whenever you have a choice, and, followed by another choice, and, followed by another choice you get the “combination” of the number of ways the result can turn out by multiplying the choices together. The rule of product is the function of “ and’s”. This happens and this happens and this happens, what are the number of ways the result can happen.
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- (There is another rule called the rule of “sums”. We do not us that in poker, but, it is the function of “or’s. This happens or this happens you “add’ the two together to get the number of ways the results could happen.)
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- Poker uses “combinations” which uses the “rule of product“.
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- For example, with a deck of cards. You first have a choice of numbers, 1 through 13, that is ace through king. Then, and followed by, a choice of suits, 1 of 4: clubs, hearts, spades, diamonds. The number of combinations is their product 13 * 4 = 52. Therefore poker uses a deck of 52 cards.
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- The rule of combinations requires that repetitions not be allowed and the order of choices does not matter. This is true in poker. Each card is unique, no repetitions. You can form a Straight in your 5 card hand regardless of the order you received them. So, the rule of combinations applies.
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- To give you the bigger picture there could be choices where order does matter and repetitions are allowed. ZIP codes for example, 95405. This is the rule of “arrangements” and it has different formulas. Poker is the rule of combinations, not arrangements.
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- How many combinations of 5 card hands can you get from a 52 card deck?
- ----------- 52 cards choose 5
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------------ This is the rule of “n” choose “k”
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------------ “n” cards choose “k” for your hand.
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----------- The formula for combinations is: -----------n choose k = n! / k! * ( n - k)!
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------------ “n!” is n “factorial”
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------------ “n!” is 52 * 51 * 50 * 49 * ……….. 5* 4* 3* 2* 1
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------------ “k”, k factorial is 5* 4* 3* 2* 1
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------------ ”(n-k)” is 47!
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------------ Plug these values of “52 choose 5” into the formula for combinations:
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-------------- 52! / 5! * 47! = 52 * 51 * 50 * 49 * 48 * 47! / 5* 4* 3* 2* 1 * 47!
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----------- = 52 * 51 10 * 49 * 2 = 2,598,960
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- Therefore, there are over 2 ½ million possible poker hands
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- 52 cards choose 5 = 2,598,960 combinations.
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- What are the odds of drawing a Full House? That is 3 of a kind and 2 of a kind. The rule for 52 cards choose 5 being a Full House is the product rule of the 1st card drawn times the 2nd card drawn times the 3rd card drawn, times the 4th card drawn, times the 5th card drawn with the result being 3 numbers the same and 2 numbers the same.
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- Your first card has 13 choices of numbers, Ace through King, and that is going to be paired, 13 choose 1.
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- Your second card has now 12 choices of numbers since the 3 of a kind can not be the same numbers as the pair, 12 choose 1.
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- Next you have 4 suits to choose 3 that are the same number as the second card.
------------ 4 choose 3 = n! / k! * ( n - k )! = 4! / 3! * (1)!
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------------ 4 choose 3 = 4 combinations.
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- Next you have 4 suits to choose 2 that match the number of your first card.
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------------ 4 choose 2 = 4! / 2! * (2)! = 4 * 3 * 2 * 1 / 2 * 2 = 6
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------------ 4 choose 2 = 6 combinations.:
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- Now you use the product rule to multiple them all together.
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------------ 13 * 12 * 4 * 6 = 3,744 combinations to get a Full House.
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- There are 2,598,960 possible poker hands. The are 3,744 combinations that will be a Full House So your odds are 0.144% of getting a Full House. That is a little over 1 chance in 1,000 of getting a Full House on a 5 card draw. Good Luck.
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We use the same logic to determine the odds of drawing a Straight. A Straight is a consecutive sequence of 5 numbers. Let the Ace be a one or a 14. Low or High. There are 10 possible Straights.
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---------- 1 ----------1 2 3 4 5 6 7 8 9 10 11 12 13 14
---------- 2 --------- 2 3 4 5 6
---------- 3 ---------- 3 4 5 6 7
---------- 4 ---------- 4 5 6 7 8
---------- 5 ---------- 5 6 7 8 9
---------- 6 ---------- 6 7 8 9 10
---------- 7 ---------- 7 8 9 10 11 (Jack)
---------- 8 ---------- 8 9 10 11 12 (Queen)
---------- 9 ---------- 9 10 11 12 13 (King)
---------- 10 ---------- 10 11 12 13 14 (Ace)
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- So, there are 10 possible 5 card Straights.
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- Those are the numbers but each card has to choose a suit, 4 choose 1.
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------------------ 4 choose 1 = 4! /1! * 3! = 4
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------------------ Product rule 10 * 4* 4* 4 * 4 * 4 =10,240 combinations.
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- What are the odds of drawing a Straight = 10,240 / 2,598,960 = 0.394%
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- The odds are 3 times better than a Full House, so a Full House beats a Straight.
Good Luck.
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- Does a Flush beat a Straight? What are the odds of drawing a Flush? All 5 the same suit.
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----------- To choose the first card it is 4 choose 1, that is the 1 to decide the suit for the Flush.
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------------ For the numbers you need 13 choose 5 of the same suit
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-------------- 13! / 5! - 8! = 13 * 12 *11 * 10 * 9 * 8! / 5 * 4 * 3 * 2 * 8! = 5,148 combinations.
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- So, there are half as many combinations to give us a Flush as to give us a Straight, 5,148 versus 10,240. So, a Flush beats a Straight.
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- The odds of drawing a Flush are 5,148 / 2,598,960 = 0.198% chance, Good Luck.
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- What are the odds of drawing a single Pair?
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- The first card is 13 choose 1 = 13
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- Out of the 4 suits choose 2 = 4 * 3 * 2 / 2 * 2 = 6 combinations.
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- To get a single pair the other cards can’t match so those odds are 12 choose 3
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--------- 12 choose 3 = 12! /3! * ( 12-3)! = 12 * 11 * 10 * 9! / 3 * 2 * 9! = 13 * 6 * 220 for the numbers = 17,160 combinations.
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- Now choose the suit for the last 3 cards, 4 choose 1, 4 choose 1, 4 choose 1 = 4 * 4 * 4 = 64 combinations.
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----------17,160 * 64 = 1,098,240
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- There are over 1 million ways to draw a single pair.
- The odds are 1,098,240 / 2,598,960 = 42.3%
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- The probability of drawing a single pair is 42%, still less than a 50 : 50 chance.
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OK, here is the last one: What are the odds of getting all “ garbage”?
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- The odds of not getting a single pair are 13 choose 5:
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----------- 13! / 5! * (13-5)! = 13 * 12 * 11 * 10 * 9 * 8! / % * 4 * 3 * 2 * 8! = 1,287 for the numbers.
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---------- 4 choose 1, 4 choose 1, 4 choose 1, 4 choose 1, 4 choose 1 for each of the 5 cards to get suits.
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------------ 1,287 * 1024 = 1,317,888
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- Now this is over counted for the number of Straights in the no pair possibilities. So we need to subtract the number of Straights. 1,317,888 - 10,240 = 1,307,648
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- And we need to subtract the number of Flushes in the no pair possibilities. 1,307,648 - 5,148 = 1,302,500
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- However, a Straight Flush has been counted twice, once in the Straights and once in the Flushes. Add 10 * 4 = 40. 1,302,500 + 40 = 1,302,540
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- Therefore, the odds of getting “garbage are 1,302,540 / 2,598,960 = 50,1%
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- You always knew this. There is over 50% chance of getting all” garbage” in a 5 card draw of Poker.
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- There is another way to look at the above problem. For the numbers there are (13 choose 5 for the numbers - 10 Straights) * For the suits there are ( 4*4*4*4*4 - 4 for that Straight Flush) = (1,287 - 10 ) * ( 1,024 - 4) = (1,277) * ( 1,020) = 1,302,540, the same answer for the combinations of 5 cards that are Garbage. Good Luck.
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- Yes, best of luck. Now you know your chances of beating the odds.
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- May 13, 2019. 1213
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--------------------- Monday, May 13, 2019 -------------------------
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