# MTH 201 Project-Topic #1: A Poker Game Tom Andary Essay

MTH 201 Project-Topic #1: A Poker GameTom Andary
Section 4
There are many types of combinations desirable to possess in a poker game. Some are more desirable than others. What makes any given combination of cards powerful in the game of poker? The probability that it will occur. Specifically, the less likely a card combination is to occur, the more powerful it is. For starters, it is necessary to state that there are 2,598,960 possible combinations, or hands, from a standard 52 card poker deck. Also note that 1,302,540 of these hands do not contain any useful combination in the game of poker. That leaves 1,296,420 hands that could win you a game of poker.
To get a pair, you are trying for 2 cards of a specific denomination out of 4 that exist, with three cards that don’t amount to anything special from the remaining 48. For drawing the pair, you have combination of (4,2) (for drawing 2 cards from a four card set) times 13 (the number of 4 card sets), multiplied by the combination of (48,3). Then subtract the possibility of getting a full house, 2 pair, and 3 of a kind, to get 1,098,240 different ways of drawing a pair in a game of poker. This carries a probability of .4225.

To get two pair, you are attempting to pull two cards each from two four-card sets, plus one random card not from either of the previous sets. First, you multiply the combination of (4,2) by 13, just as you would for 1 pair. Multiply the result by 12 times the combination of (4,2) and the combination of (44,1). Divide this answer by 2, to account for the fact that reverse pairs cannot exist side by side. (i.e. one pair can not be the 2 of hearts/2 of diamonds, and the other pair be the 2 of diamonds/2 of hearts.) The final answer is 123,552 possibilities, being a probability of .0475.

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Three of a kind is having three of one denomination, and two random cards which are not a pair. Take the combination of (4,3) times 13, multiplied by the combination of (48,2) to get 54,912. Then subtract the possibility of having a full house to get 54,912. Divide that by the magic number of possible poker hands to get your probability of .0211.

A straight is cards forming a consecutive order, but not a straight flush or royal flush. Take the 4 suits of cards, and raise them to the 5th power, representing the 5 cards in your hand. Multiply the result by ten, then subtract the number of possible straight and royal flushes. This gives you 10,200 possibilities, divided by 2,598,960 for a probability of .0039.

A Flush is having 5 cards from the same suit that are not in consecutive order so as to be a straight or royal flush. Take the combination of (13,5) (for 5 cards out of a 13 card suit) and multiply it by 4 (for the 4 suits) and subtract the possibility of a straight or royal flush to get 5,108. Divide 5,108 by the number of possible poker hands to get the probability of .0019.

A full house is three of a kind and a pair together. Take the combination of (4,3) times 13 and multiply the answer by the combination of (4,2) times 12 (because you can’t have a pair of a kind you have 3 taken out of already.) to get 3744. Divide that by the magic number of poker hands to get your probability of .0014.

Noticing that the less probable a hand is, the easier it is to calculate? It keeps getting that way right down to the grand old royal flush?
Four of a kind is pretty self explanatory. You posses all four of a given denomination of card. Take 13 times the combination of (4,4) and multiply the answer by the combination of (48,1) to get 624. Divide 624 by 2,598,960 to find the probability of .00024.

A straight flush is having 5 cards of the same suit in consecutive order, not including the royal flush. With (A,2,3,4,5) being the lowest straight flush, and (9,10,J,Q,K) being the highest, that totals 9 possibilities for each suit, 36 in all. Divide 36 by the now enormous-looking 2,598,960 for your highly unlikely probability of .000013.

Now for the easiest to calculate, but hardest to obtain hand in the game of poker. The royal flush is possessing the 10, Jack, Queen, King, and Ace of the same suit. There are only 4 possibilities of this happening. Divide this by the number of possible poker hands to get the infinitesimally small probability of .0000015. That’s 649,740 to 1 odds on getting a royal flush on your first deal!
Simply enough, the value of a hand of poker is based on two things:
1. The probability of the hand occurring off the first deal.

2. If two players have the same kind of hand, the one with the highest cards wins.
Here’s how the poker hands rank in order from worst to best:
? 1 Pair
? 2 Pair
? Three of a kind
? Straight
? Flush
? Full House
? Four of a Kind
? Straight Flush
? Royal Flush

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