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Probability and Casino Games Maximilian Mohr

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Gamblers Fallacy Roulette Wheel Shooting Craps Permutations and Combinations Poker Hands Overview

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The error of thinking that the probability of a particular outcome increases with the successive number of opposite outcomes Gamblers Fallacy

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Coin Toss If you toss a coin ten times, and the first nine results are heads, you might think that the probability of the fifth coin being heads is very low Gamblers Fallacy

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Probability of one toss heads = ½ Probability of 10 tosses being heads = 1/(2^10) = 1/1024 So the probability of the 10 th toss being heads = 1/1024, right? Gamblers Fallacy

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NOPE!!! Gamblers Fallacy

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The tenth toss has just as much of a chance to be heads as tales. Its 1/2 At the point of your contemplation, the coin has already flipped 9 times, and each individual coin flip is not affected by the previous flip The universe is not keeping a memory of what has happened and what should come next Gamblers Fallacy

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The probability of getting 10 heads in a row OR 9 heads then 1 tail both have the same odds. Each is an exact combination of 10 outcomes H H H H H H H H H H H H H H H H H H H T There are two possibilities for each outcome, and compounded ten times, both sets have a probability of 1/1024 Additionally

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Roulette

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Roulette is a game in which a small ball is spun around a rotating wheel with 38 different pockets In American roulette, the pockets are numbered 00, 0, 1-36 Gamblers bet on the outcome of the pockets into which the ball will fall Each number is also colored either red or black Roulette

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Single bet – a bet on one exact outcome Split – a bet placed on two adjoining squares Betting on three numbers in a horizontal line Square – marker placed at the intersection of four numbers Types of Inside Roulette Bets

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Manque (1-18) Passe (19-36) Red or Black Bets Even or Odd Bets Column Bet – All 12 numbers in any vertical line Types of Outside Roulette Bets

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Craps

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Craps is a game where gamblers wager of the outcome of rolls of dice. All possible rolls: Craps Basics

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The most basic bet, a bet for the shooter to win If the roll is 7 or 11, shooter wins Probability of winning this way? Pass Line Bet

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If the roll is 2, 3, or 12, the shooter loses Probability of losing? Pass Line Bet

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If the roll is any other value, the player establishes a point. If the player rolls that point again before a 7, he/she wins If the player rolls a 7 before that point, he/she loses. Example: Rolling a 4 Pass Line Bet

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Texas Holdem

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The basics Two cards dealt to each player, 5 cards turned over in succession from the deck Player with the best hand using his cards and the flop is the winner Texas Holdem

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Which applies to poker hands?? Permutation: a specific set of items in a particular order Combination: a specific set of items in no particular order Combinations & Permutations

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Combination Lock?

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Permutation Formula Where n is the number of items to choose from, and r is how many youre picking out

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___n!___ r!(n-r)! You divide out r! from the permutation formula because combinations of the same items in different orders dont count as separate sets. Combination Formula

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How many possible poker hands in all? Out of 52 possible cards, choose a combination of 5

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52 choose 5 = 2,598,960 How Many Hands?

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Getting 10-J-Q-K-A all of the same suit Royal Flush

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Discounting the possibility of getting a royal flush 5 cards in numerical order, all in the same suit % Straight Flush

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4 of one value of any card, the same value from each suit 0.024% 4 of a Kind

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3 cards of a kind + a pair 0.144% Full House

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5 cards of any value, all from the same suit Discounting straight flushes 0.197% Flush

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5 cards in rank order from any suit Discount royal flush and straight flush 0.392% Straight

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3 of the same rank, other two cards can be anything 2.11% 3 of a kind

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4.75% 2 Pair

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42.3% Pair

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50.1% No Pair/High Card

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If I am playing a pass line bet in craps, and I roll a 9, how much more likely am I to lose than win? What percent chance do I have of beating a player with a regular flush in poker, if were both all in? Homework

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ons-permutations.html ons-permutations.html +fallacy +fallacy Sources

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