Probability and Casino Games

Maximilian Mohr

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

Overview

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

Gambler’s Fallacy

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

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

Roulette

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

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

Craps

Craps is a game where gamblers wager of the outcome of rolls of dice.

All possible rolls:

Craps Basics

1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

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

Pass Line Bet

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

Texas Hold’em

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

Combination Lock?

Permutation Formula

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

___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.

Combination Formula

How many possible poker hands in all?

Out of 52 possible cards, choose a combination of 5

52 choose 5 = 2,598,960

How Many Hands?

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

Royal Flush

Discounting the possibility of getting a royal flush

5 cards in numerical order, all in the same suit

0.00139%

Straight Flush

4 of one value of any card, the same value from each suit

0.024%

4 of a Kind

3 cards of a kind + a pair 0.144%

Full House

5 cards of any value, all from the same suit

Discounting straight flushes 0.197%

Flush

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

Straight

3 of the same rank, other two cards can be anything

2.11%

3 of a kind

4.75%

2 Pair

42.3%

Pair

50.1%

No Pair/High Card

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?

Homework

http://www.mathsisfun.com/combinatorics/combinations-permutations.html

http://en.wikipedia.org/wiki/Poker_probability http://dictionary.reference.com/browse/mont

e+carlo+fallacy

http://en.wikipedia.org/wiki/Craps

Sources