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**Probability and Casino Games**

Maximilian Mohr

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**Overview Gambler’s Fallacy Roulette Wheel Shooting Craps**

Permutations and Combinations Poker Hands

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

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**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

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**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?

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Gambler’s Fallacy NOPE!!!

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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

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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

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Roulette

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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

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**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

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**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

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Craps

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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

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**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?

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**Pass Line Bet If the roll is 2, 3, or 12, the shooter loses**

Probability of losing?

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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

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Texas Hold’em

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**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

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**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

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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 you’re picking out

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**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.

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

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

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**Straight Flush Discounting the possibility of getting a royal flush**

5 cards in numerical order, all in the same suit %

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**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%

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

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**Flush 5 cards of any value, all from the same suit**

Discounting straight flushes 0.197%

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**Straight 5 cards in rank order from any suit**

Discount royal flush and straight flush 0.392%

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**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%

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2 Pair 4.75%

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Pair 42.3%

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No Pair/High Card 50.1%

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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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Sources

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1 Counting Rules. 2 The probability of a specific event or outcome is a fraction. In the numerator we have the number of ways the specific event can occur.

1 Counting Rules. 2 The probability of a specific event or outcome is a fraction. In the numerator we have the number of ways the specific event can occur.

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