Math and Probability in Poker

Published by southlax21
October 20, 2009, Category: Mathematics

A brief history of the game of poker and its mathematical connotations.

“I believe in poker the way I believe in the American Dream. Poker is good for you. It enriches the soul, sharpens the intellect, heals the spirit, and – when played well, nourishes the wallet.” These are the feelings of Lou Krieger, professional poker player and author on poker. Poker is a great game to play because it teaches many important skills like patience, planning, and probabilities. Believe it or not, the first idea for the King in a deck of cards was based off of real kings who ruled during the classical and post classical era. The King of spades represented King David from the bible, the King of hearts represented Charlemagne of France, the King of diamonds represented Julius Caesar, and the King of clubs was Alexander the Great. Other connections can be made in a more religious sense are that the Ace is a reminder that there is one god. The 2 signifies the two parts of the Bible, and the 3 stands for the Father, the Son, and the Holy Ghost. 4 is representing Four Apostles, 5 is for the five glorified virgins. 6 for the six days it took God to create the earth, and 7 for the day God rested, 8 for family of Noah that God saved after the flood, and 9 for the lepers Jesus healed. 10 is for the Ten Commandments, and the Jack as a reminder of Satan, the first angel, but god kicked out of heaven for his evil ways. The Queen for the Virgin Mary and the King represents that Jesus is the King of Kings. All these religious allusions may seem a little far-fetched, and in reality they are! These ideas were thought up by a quick thinking sailor in the British Royal Navy in the 1700’s when he was caught playing cards instead of going to mass. He quickly explained all these connections to God and his commanding officer not only let him off the hook, he asked to borrow the cards to pray to. That sailor must have been a terrific poker player, being able to think that quickly under pressure! It is time for the probabilities of each of the hands to be addressed. To find the probability of a hand, one must find the answer to 52nCr5. NCr is the symbol for computations, or “blank choose blank”. 52nCr5 is used because there are fifty-two possible cards to make a hand out of, and you can use five of them.

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That number, 2,598,960, is the total amount of possible five card poker hands. Now that we have a base number, which will be referred to as h, we can continue to figure out the probability of the hands. The formula for probability will be shown as (n/h), where n is the amount of possible hands of a combination. The most difficult hand to get in poker is the royal flush. There are only 4 different ways to have a royal flush, so it is shown as (4nCr1). To find the probability of a royal flush, you need to divide the amount of possible hands for that set by the amount of possible hands, h. (4/h) ≈ .00000154%. Another way to represent that figure into more manageable terms is to find the odds of the hand. The formula to find the odds is (1/P) – 1:1. The P in this formula stands in for the previously calculated probability. The odds of a royal flush are represented ((1/0.00000154%) – 1): 1, or 649,349:1. The next strongest hand is the straight flush. There are 40 different straight flushes that can be made, but four of them are royal flushes, so there are then 36 possible straight flushes possible. First the probability of a straight flush must be found, so the formula (36/h) is used. (36/h) ≈ 0.0000139%. The odds of having a straight flush are found by using the odds formula, (1/P) – 1:1, which is found to be 71,941:1. The next hand is a four of a kind. There are 13 different ranks to have a four of a kind with, and then there are 48 cards left over in the deck, so to find the number of different four of a kinds you do (13nCr1)×(48nCr1) = 624. The probability of a four of a kind is (624/h) ≈ .00024%, and the odds are (1/P) – 1:1 = 4166:1. Some professional poker players never see any of these hands in their life, but as it’s represented by the odds and probabilities, it is a huge achievement. The next group of hands are common and are seen usually once a game in poker, if not more. The full house, a three of a kind and one pair, has one triple out of thirteen different values, so it is (13nCr1), and only three of the four cards are needed for a triple, so you do (13nCr1) × (4nCr3). To find the probability of the pair, there can only be a pair for twelve values, since one was needed for the three of a kind, so you calculate (12nCr1) × (4nCr2), because only two cards are needed for a pair. After calculating (13nCr1) × (4nCr3) × (12nCr1) × (4nCr2) = 3,744, which is the number of different full houses that can be dealt, the probability is found to be (3,744h) ≈ .00144%. The odds of a full house are (1/P) – 1:1 = 693:1. The next hand is the flush. The amount of different flushes is figured out by taking (13nCr5), for the five cards of any of the thirteen, multiplied by (4nCr1) each of the suits, and then subtract 40, for the 40 different straight flushes. That equation is (13nCr5) × (4nCr1) – 40 = 5,108. Then the probability is found using (n/h) ≈ 0.00197%. The odds formula is then used, (1/P) – 1:1, and the odds of getting a flush come out to be 507:1. The hand that follows a flush is a straight. The frequency of a straight is computed by (10nCr1), because there are 10 possible sequences for a straight, multiplied by (4nCr1)^5, because of the five cards in four possible suits, and then just subtract 40 to show the 40 possible straight flushes. ((10nCr1) × (4nCr1)^5) – 40 = 10,200. The probability is found by (10,200/h) ≈ 0.00392%. The odds of getting a straight, represented by the formula (1/P) – 1:1 = 254:1. The calculations for finding the frequency, probability, and odds of the more common hands become very tedious because there are more possibilities that can be dealt. Three of a kind is the hand after a straight, and its frequency is found by using (13nCr1), because any of the 13 ranks can be used, and multiplying it by (4nCr3), because only three of the four cards in the rank are used, and multiplying that by (12nCr2), because two more cards are needed from the remaining twelve ranks, and multiplying that by (4nCr1)², because the remaining two cards can have any of the four suits. The equation in its entirety is (13nCr1) × (4nCr3) × (12nCr2) × (4nCr1)² = 54,912. After computing the probability of getting three of a kind by using the probability equation, (n/h) ≈ 0.0211%, and the odds formula is used, the answer comes out to be 46:1. To find the frequency of getting two pairs, the formula starts with (13nCr2), because two ranks must be used, and it is multiplied by (4nCr2)², because only two cards of each rank used are needed for the pairs. Then that is multiplied by (11nCr1) because the final card can be of any rank, and finally it is all multiplied by (4nCr1) because this can happen within any suit. The whole equation equals 123,552. The probability is found by solving the equation (n/h) ≈ 0.0475%. Then calculate the odds the formula for odds is used, (1/P) – 1:1, and the odds of getting a two pair come out to be 20:1. The next hand is having one pair. The frequency of getting one pair is (13nCr1), for any rank can be used, multiplied by (4nCr2), for two cards are needed from the chosen rank, multiplied by (12nCr3), for the remaining three cards can come from any of the 12 ranks left over, multiplied by (4nCr1)³ because the remaining three cards can have any of the four suits. That equation is shown as (13nCr1) × (4nCr2) × (12nCr3) × (4nCr1)³ = 1,098,240. The probability formula is used and the probability of getting one pair comes out to be 0.423%. The formula for calculating odds is utilized and it equals 1.4:1. Poker, Texas Hold’em in particular, employs many different styles of betting and bluffing. This is what makes the game so appealing to so many card players in the world. Although having good bluffs and tricks is helpful, “Most of the money you’ll win comes not from the brilliance of your own play, but from the ineptitude of your opponents.” That was a quote by Lou Krieger and he could not have said it better. Poker, like any other game, takes practice, patience, and a little luck and is a great game to play with friends, and in some cases, a good investment. It brings people together in social environments and teaches money management. Poker is a rapidly growing past time, and there are plenty of opportunities to play with friends or online. Just remember one thing when going into a game of poker, “Trust everyone, but always cut the cards.”- Benny Binion.