# Probability and Casino Games

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Probability and Casino Games
Maximilian Mohr

Overview Gambler’s Fallacy Roulette Wheel Shooting Craps
Permutations and Combinations Poker Hands

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

Gambler’s Fallacy 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

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

Gambler’s Fallacy NOPE!!!

Gambler’s Fallacy The tenth toss has just as much of a chance to be heads as tales. It’s 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

Additionally 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

Roulette

Roulette 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

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

Craps

Craps Basics Craps is a game where gamblers wager of the outcome of rolls of dice. All possible rolls: 1 2 3 4 5 6 7 8 9 10 11 12

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

Pass Line Bet 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

Texas Hold’em

Texas Hold’em 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

Combinations & Permutations
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

Combination Lock?

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

Combination Formula ___n!___ r!(n-r)!
You divide out r! from the permutation formula because combinations of the same items in different orders don’t count as separate sets.

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

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

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

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

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

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

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

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

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

2 Pair 4.75%

Pair 42.3%

No Pair/High Card 50.1%

Homework 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 we’re both all in?

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