# In the situations that we will encounter, we may generally calculate the probability of an event

Save this PDF as:

Size: px
Start display at page:

Download "In the situations that we will encounter, we may generally calculate the probability of an event"

## Transcription

1 What does it mean for something to be random? An event is called random if the process which produces the outcome is sufficiently complicated that we are unable to predict the precise result and are instead able to determine just a range of possible outcomes. It is quite easy to come up with examples of random processes in gambling (e.g. dealing from a shuffled deck of cards, rolling a pair of dice, spinning a roulette wheel), but there are plenty of examples we encounter daily where the mechanism that is generating the outcome is too complicated for us to predict with precision the outcome (e.g. how long it takes to make a commute, the weather forecast, the number of employees which are out sick in a given workday). It is precisely these types of situations that are the motivation for developing mathematical notation to make predictions about the future. A random variable is a mathematical construction which represents a random experiment or process. Usually we represent random variables by capital letters X, Y, Z,.... We will be thinking of random variables as roulette wheels which keep track of the possible outcomes and the frequency which those outcomes occur. Random variables have a range of possible values that they may take on (the numbers found on the edge of the wheel) and positive values representing the percentages that these values occur. The elementary outcomes are the individual possible values that the random variable may take on. An event is a subset of elementary outcomes and the field of events is the set of all possible events for this random variable. The empty set and the set of all elementary outcomes are always included in the field of events. The field of events is assumed to be closed under intersection, union and complementation. A probability measure is a function which associates to each event a number P [A] which will be a value in the interval from 0 to 1. This number reflects the confidence that the event A will occur where a value of 0 represents no chance that the event will occur and 1 that the event happens every time the experiment is conducted. The roulette representation of a random variable records the In general, we must have that if A and B are mutually exclusive events (the intersection of A and B is empty), then P [A or B] = P[A] + P[B]. by In the situations that we will encounter, we may generally calculate the probability of an event P [A] = the number of outcomes considered as part of A the total number of elementary outcomes A random variable for us will be represented as a wheel where the outside edge of this wheel 1

2 is labelled with the elementary outcomes and the portion of the arc of the circle occupied by any given elementary outcome is equal to the probability that the outcome occurs. We imagine that this wheel spins with a pointer lying somewhere on the outer edge (which we will not draw) and that when the random experiment is conducted we imagine that it is equivalent to giving this wheel a spin a recording the outcome that is lying next to the pointer. Assuming that there are not too many outcomes and the probability that they occur are within a range of pen and paper, this roulette wheel is a visual presentation of the information contained in the random variable. If the number of outcomes of the random variable are too large to draw or their probabilities too small (e.g. a random variable representing the outcomes of a lottery where there are roughly 14 million possiblilities), then we do not actually draw the wheel and instead must imagine that small fractions of an enormously large wheel could be drawn. Example: If our random variable represented the flipping of a coin then there are two possible elementary outcomes, { heads, tails }. There are four possible events which describes our field of events heads, tails, either heads or tails, neither heads nor tails. This random variable is visually represented by a roulette wheel with two equal sized regions. One half of the roulette wheel is labeled with heads and the other half is labelled with tails. Example: If our random variable represented rolling a 6 sided die, the elementary outcomes would be the set {1, 2, 3, 4, 5, 6}. Examples of events might be the roll was even or roll greater than 4 or more simply a roll of 3. The roulette wheel representing a roll of a die has six regions. Each sixth of the roulette wheel is labeled with one of the elementary outcomes of the rolling of a die. Odds are the not the opposite of evens 2

3 In gambling we also refer to the probability of an event through the odds. The odds of an event is the ratio of the probability of an event occurring to the event not occurring. Usually this is expressed in the form number to number or number : number (occassionally, and you need to be careful in this use, they are also expressed as number in number ). If the odds of an event are m to n then if m is bigger than n then it will occur more than 50 percent of the time, if m is less than n then the event will occur less than 50 percent of the time. We may also talk about the odds against an event happening. If the odds for an event happening are m to n then the odds against it happening are n to m. Example: Here is a mis-use of the vocabulary of odds: Sir, the possibility of successfully navigating an asteroid field is approximately 3,720 to 1! More precisely he should have said the odds against successfully navigating an asteroid field are approximately 3,720 to 1 Never tell me the odds. The advantage of using this notation is that when we are speaking, it is generally understood if the event is likely or unlikely to occur, so mixing up the order of the numbers is generally not too confusing. However, it is important to be precise when describing the rules of a game and the odds of 5:4 or 4:5 are not far off. Example: The OLG and most lottery boards state the odds of winning the lottery are 1 in number and they really mean that the probability of winning is 1/number. When lottery boards say they are using the notation of odds, they usually aren t. Example: The event of rolling a 3 on a die has probability 1/6. The odds that this event happens is 1 to 5 or 1 : 5 because the probability of the event happening is 1/6, the probability of the event not happening is 5/6, and so the odds are 1/6 5/6 = 1/5 which we then express as 1 : 5 or 1 to 5. Example: The odds of getting heads on the flip of a coin is 1 : 1. there are even odds on the event. In this case we say that How to go back and forth between odds and probability If p is the probability of an event A, then the odds of A occurring are p/(1 p) expressed in the form m to n where p/(1 p) = m/n Example: The probability of rolling two 1 s on a pair of dice is 1/36. The odds of this event occurring are = 1/35. So we say that the odds of rolling snake eyes is 1 to 35. 1/36 1 1/36 = 1/36 35/36 Example: If we bet the next number on a roulette spin will be a fixed number that we choose from 1-36 or 0 (European roulette wheel), then the probability that our choice is right is 1/37. The 3

4 odds that we win this bed is then 1/37 36/37 = 1/36 and so we say they are 1 to 36. Example: The probability of dealing out two cards and both of them are Aces is 6/1326 = 1/221. To find the odds we compute the fraction 1/ /221 = 1/ /221 = 1/220 and then we convert this to odds notation by 1 : 220 or instead say 1 to 220. If the odds of an event are m to n then the probability of the event occurring is because p/(1 p) = m/n so pn = m mp and pn + pm = m so p = m m+n. m m+n. This is Example: If you look on betting websites they will often provide you with the odds of winning a given bet. For instance at the website the odds of the Field bet are posted at 5 : 4. Technically, what is being posted are the odds against winning the bet. The probability of winning the bet is 4/(4 + 5) = 4/9. Odds on a payout We also use the notation of odds to talk about payouts. When we say a bet pays 35 to 1 odds this means that the bet pays \$35 for every \$1 that was bet. When you bet that \$1 and you win, you also get it back so you will actually be returned \$36. If you lose that bet, then the \$1 is gone. At a gambling table the bets are often made by using chips. In those cases when the odds are listed where the second number is not 1 then it is most often the case that the bets must be made in multiples of that number. For example, on the Horn bet in craps the payout is 27 : 4 on a 2 or 12. The bettor must place 4 chips on this bet and when the bet wins by getting the 2 or 12 he/she is returned 31 chips in total, 27 more than the 4 that were bet. For casino bets the rules are set in the game so that the payout of a given bet favors the casino. If a bet is fair and the odds of winning the bet were m : n and there is only one type of payout (sometimes there are different payouts for different outcomes and the situation is a lot more complicated), then the payout should be n : m. In a casino, since the bet is not fair, the payout will be somewhat less than n : m. How much less will indicate the size of the cut that the casino is taking. There is no set rule about how casinos determine what the payoff odds are. They set them so that the bets are close enough to a fair game that they will get people taking those bets, but if they can get away with giving a payout which is very unfavorable to a bettor they will set the odds as they like. On some level gamblers are still customers in a casino and if another casino down the street is offering better odds then a casino will lose those customers. 4

5 For instance, in roulette we found that the odds of winning the bet on a single number is 1 : 36. The payout odds on this bet are 35 : 1. The house is taking a cut on this bet by making the payout odds 35 to 1 when the odds of winning the bet are 1 to 36. Similarly, the odds of winning the corner bet in roulette (betting that one of 4 numbers will come up in the next roulette spin) is 1 : 8 but the payout of this bet is 7 : 1. What we call the house advantage (see below) on these two bets is the same, but that is not easy to see just by looking at the odds alone. Everyone expects to win, only the house does for sure When the elementary outcomes of a random variable are all numbers then it makes sense to talk about the average value or the expected value of the random variable. For a random variable X with elementary outcomes {x 1, x 2, x 3,..., x n }, the expected value (denoted E(X)) is defined to be E(X) = x 1 P (X = x 1 ) + x 2 P (X = x 2 ) + x 3 P (X = x 3 ) + + x n P (X = x n ). This quantity has some intuitive meaning as sort of the average value that occurs if the experiment is repeated many times. Depending on what the random variable represents, this might not give us much intuition about the outcome of the experiment, but when the random variable represents a bet the expected value does give us some intuition about the bet. When our random variable represents a gambling bet the expected value has a direct interpretation because in this case the elementary outcomes are the amount of money which is won or lost for each of the events in the bet. In this case, the expected value represents the amount of money that we should expect to win or lose on average per game that is played assuming that the bet is played many times. This does not mean that if we play the game many times that we should 5

6 expect to win (the expected value) times (the number of bets), instead what it means is that the more times we play (the amount won or lost) divided by (the number of bets) should get closer and closer to the expected value. This will be made more precise when we study the weak law of large numbers. If the expected value of a bet is positive, then the bet favors the player. If the expected value of a given bet is negative then the bet favors the entity the bet is made against (e.g. the house, the bookie, the opponent). If the expected value of the bet is 0 then we say it is a fair bet. When the expected value is negative, the house advantage is negative of the expected value expressed as a percentage of a \$1 bet. Example: The field bet is made on a single roll of a pair of dice. On a \$1 bet, if the dice add up to 2 or 12 the player wins \$2. If the dice add up to 3, 4, 9, 10, 11 the player wins \$1, otherwise the player loses the \$1 bet. The probability that the roll will be 2 or 12 is 2/36. The probability that the roll is 3, 4, 9, 10, 11 is 14/36. The probability that the roll is 5,6,7 or 8 is 20/36. The expected value of this bet is E(field bet) = = What this says is that the bet favors the house and the house advantage is approximately 5.6%. Example: the expected value of a coin toss doesn t make sense because the elementary outcomes of a coin toss are heads or tails. Since these are not numerical values the expected value doesn t represent anything. However, if we put a bet on the outcome of a flip of a coin such as If the outcome is heads I owe you \$100, and if the outcome is tails you owe me \$100. In this case the expected value of the bet is E(\$100 on the flip of a coin) = = 0. This is what we would call a fair bet. The expected value is an important single number that in a way summarizes something about the bet (or random variable) that is being considered, but a single number cannot tell us everything about a bet. Consider the following example: Example: Say that you need to raise \$1,000,000 by a deadline and choose to do it by betting. The positive outcomes of the following three bets will solve your problem by raising the \$1,000,000. Some of the outcomes in the bets could cause you to be in debt for the rest of your life. Say that you are given the choice of the following: a bet where you are given a 1 in 11 chance of losing \$10,000,000 but a 10 in 11 chance of winning the \$1,000,000. 6

7 a bet where you have a 1 in 2 chance of losing \$1,000,000 and a 1 in 2 chance of winning \$1,000,000. a 999,999 in 1,000,000 chance of losing \$1 and a 1 in a 1,000,000 chance of winning \$1,000,000. In all three of these bets the expected value is 0, but the outcomes are very different. A lot of information about these bets is lost by considering only the expected values of the bets. In each case, there is some positive probability of achieving the \$1,000,000 outcome. In the first two bets you don t care what happens in the long run, you probably would not be able to make that bet more than once. Probability when you know something We know that the probability of any single number coming up on roulette is 1/37 (assuming a European roulette wheel) so the probability of seeing the number 14 (in particular) on the next roll is 1/37. Imagine a roulette wheel is rigged so that although we don t know the next spins value, we do know that the next spin is red. In this case, since 14 is red it is still possible that the result is 14, but the information that we are given has now changed the probability that 14 will be the next spin because it is a red value. Since there are now 18 red values, one of them is to come up we can determine that the probability of getting a 14 given that the next spin is red is 1/18. This is called the conditional probability of the event that a 14 comes up given that the next value is red. There are other situations where we are given some information about the outcome, for example what are the chances that it will rain in the next 24 hours given that it is cloudy now. Other times if we don t have the outcome we like, we repeat the experiment again until we have an outcome that falls into an event that we do like. For example, in the game of craps we might roll a pair of dice until their is either a 5 or a 7 showing, this would be a situation where we would ask the probability of an event given that the roll is either a 5 or a 7. Both of these are situations are examples we would like to determine the probability of an outcome given another outcome. If we consider random variable as a wheel then the conditional probability can be represented by forbidding certain parts of the wheel of coming up or spinning the wheel again if the forbidden results happen to come up. The resulting wheel is then crippled by blackening in or excluding certain results from arising. Consider the example mentioned above where we know that the next spin of the wheel is red. We would represent this on our roulette wheel by blackening in the possible outcomes that are forbidden as in the image below: 7

8 We imagine that if we spin this roulette wheel that we will respin if the outcome lies in one of the the blackened in areas. In this case we are left with 18 spots where the roulette wheel is allowed to stop and those 18 spots occupy still one 37 th of the total portion of the wheel. What we do to that wheel is take out the blackened in areas and stretch the remaining parts so that they take up a proportional amount of the resulting wheel. The wheel above then represents the conditional random variable of the spin of the roulette wheel given the that the spin is red. 8

### The overall size of these chance errors is measured by their RMS HALF THE NUMBER OF TOSSES NUMBER OF HEADS MINUS 0 400 800 1200 1600 NUMBER OF TOSSES

INTRODUCTION TO CHANCE VARIABILITY WHAT DOES THE LAW OF AVERAGES SAY? 4 coins were tossed 1600 times each, and the chance error number of heads half the number of tosses was plotted against the number

### Unit 19: Probability Models

Unit 19: Probability Models Summary of Video Probability is the language of uncertainty. Using statistics, we can better predict the outcomes of random phenomena over the long term from the very complex,

### Expected Value and the Game of Craps

Expected Value and the Game of Craps Blake Thornton Craps is a gambling game found in most casinos based on rolling two six sided dice. Most players who walk into a casino and try to play craps for the

### Section 7C: The Law of Large Numbers

Section 7C: The Law of Large Numbers Example. You flip a coin 00 times. Suppose the coin is fair. How many times would you expect to get heads? tails? One would expect a fair coin to come up heads half

### Probability and Expected Value

Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are

### 36 Odds, Expected Value, and Conditional Probability

36 Odds, Expected Value, and Conditional Probability What s the difference between probabilities and odds? To answer this question, let s consider a game that involves rolling a die. If one gets the face

### Basic Probability. Probability: The part of Mathematics devoted to quantify uncertainty

AMS 5 PROBABILITY Basic Probability Probability: The part of Mathematics devoted to quantify uncertainty Frequency Theory Bayesian Theory Game: Playing Backgammon. The chance of getting (6,6) is 1/36.

### We { can see that if U = 2, 3, 7, 11, or 12 then the round is decided on the first cast, U = V, and W if U = 7, 11 X = L if U = 2, 3, 12.

How to Play Craps: Craps is a dice game that is played at most casinos. We will describe here the most common rules of the game with the intention of understanding the game well enough to analyze the probability

### Introductory Probability. MATH 107: Finite Mathematics University of Louisville. March 5, 2014

Introductory Probability MATH 07: Finite Mathematics University of Louisville March 5, 204 What is probability? Counting and probability 2 / 3 Probability in our daily lives We see chances, odds, and probabilities

### AMS 5 CHANCE VARIABILITY

AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and

### MA 1125 Lecture 14 - Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.

MA 5 Lecture 4 - Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the

### Week 5: Expected value and Betting systems

Week 5: Expected value and Betting systems Random variable A random variable represents a measurement in a random experiment. We usually denote random variable with capital letter X, Y,. If S is the sample

### V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay \$ to play. A penny and a nickel are flipped. You win \$ if either

### Expected Value. 24 February 2014. Expected Value 24 February 2014 1/19

Expected Value 24 February 2014 Expected Value 24 February 2014 1/19 This week we discuss the notion of expected value and how it applies to probability situations, including the various New Mexico Lottery

### Elementary Statistics and Inference. Elementary Statistics and Inference. 16 The Law of Averages (cont.) 22S:025 or 7P:025.

Elementary Statistics and Inference 22S:025 or 7P:025 Lecture 20 1 Elementary Statistics and Inference 22S:025 or 7P:025 Chapter 16 (cont.) 2 D. Making a Box Model Key Questions regarding box What numbers

### Probability: The Study of Randomness Randomness and Probability Models. IPS Chapters 4 Sections 4.1 4.2

Probability: The Study of Randomness Randomness and Probability Models IPS Chapters 4 Sections 4.1 4.2 Chapter 4 Overview Key Concepts Random Experiment/Process Sample Space Events Probability Models Probability

### Introduction to Discrete Probability. Terminology. Probability definition. 22c:19, section 6.x Hantao Zhang

Introduction to Discrete Probability 22c:19, section 6.x Hantao Zhang 1 Terminology Experiment A repeatable procedure that yields one of a given set of outcomes Rolling a die, for example Sample space

### You can place bets on the Roulette table until the dealer announces, No more bets.

Roulette Roulette is one of the oldest and most famous casino games. Every Roulette table has its own set of distinctive chips that can only be used at that particular table. These chips are purchased

### Statistics and Random Variables. Math 425 Introduction to Probability Lecture 14. Finite valued Random Variables. Expectation defined

Expectation Statistics and Random Variables Math 425 Introduction to Probability Lecture 4 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 9, 2009 When a large

### Ch. 13.3: More about Probability

Ch. 13.3: More about Probability Complementary Probabilities Given any event, E, of some sample space, U, of a random experiment, we can always talk about the complement, E, of that event: this is the

You are about to learn the very best method there is to beat an even-money bet ever devised. This works on almost any game that pays you an equal amount of your wager every time you win. Casino games are

### 6.042/18.062J Mathematics for Computer Science. Expected Value I

6.42/8.62J Mathematics for Computer Science Srini Devadas and Eric Lehman May 3, 25 Lecture otes Expected Value I The expectation or expected value of a random variable is a single number that tells you

### Probability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes

Probability Basic Concepts: Probability experiment: process that leads to welldefined results, called outcomes Outcome: result of a single trial of a probability experiment (a datum) Sample space: all

### Gaming the Law of Large Numbers

Gaming the Law of Large Numbers Thomas Hoffman and Bart Snapp July 3, 2012 Many of us view mathematics as a rich and wonderfully elaborate game. In turn, games can be used to illustrate mathematical ideas.

### Chapter 16. Law of averages. Chance. Example 1: rolling two dice Sum of draws. Setting up a. Example 2: American roulette. Summary.

Overview Box Part V Variability The Averages Box We will look at various chance : Tossing coins, rolling, playing Sampling voters We will use something called s to analyze these. Box s help to translate

### Basic Probability Theory II

RECAP Basic Probability heory II Dr. om Ilvento FREC 408 We said the approach to establishing probabilities for events is to Define the experiment List the sample points Assign probabilities to the sample

### Betting systems: how not to lose your money gambling

Betting systems: how not to lose your money gambling G. Berkolaiko Department of Mathematics Texas A&M University 28 April 2007 / Mini Fair, Math Awareness Month 2007 Gambling and Games of Chance Simple

### Elementary Statistics and Inference. Elementary Statistics and Inference. 17 Expected Value and Standard Error. 22S:025 or 7P:025.

Elementary Statistics and Inference S:05 or 7P:05 Lecture Elementary Statistics and Inference S:05 or 7P:05 Chapter 7 A. The Expected Value In a chance process (probability experiment) the outcomes of

### Comparing & Contrasting. - mathematically. Ways of comparing mathematical and scientific quantities...

Comparing & Contrasting - mathematically Ways of comparing mathematical and scientific quantities... Comparison by Division: using RATIOS, PERCENTAGES, ODDS, and PROPORTIONS Definition of Probability:

### Orange High School. Year 7, 2015. Mathematics Assignment 2

Full name: Class teacher: Due date: Orange High School Year 7, 05 Mathematics Assignment Instructions All work must be your own. You are encouraged to use the internet but you need to rewrite your findings

### 6.3 Probabilities with Large Numbers

6.3 Probabilities with Large Numbers In general, we can t perfectly predict any single outcome when there are numerous things that could happen. But, when we repeatedly observe many observations, we expect

### 13.0 Central Limit Theorem

13.0 Central Limit Theorem Discuss Midterm/Answer Questions Box Models Expected Value and Standard Error Central Limit Theorem 1 13.1 Box Models A Box Model describes a process in terms of making repeated

### (SEE IF YOU KNOW THE TRUTH ABOUT GAMBLING)

(SEE IF YOU KNOW THE TRUTH ABOUT GAMBLING) Casinos loosen the slot machines at the entrance to attract players. FACT: This is an urban myth. All modern slot machines are state-of-the-art and controlled

### PLACE BETS (E) win each time a number is thrown and lose if the dice ODDS AND LAYS HARDWAYS (F) BUY & LAY BETS (G&H)

craps PASS LINE BET (A) must be rolled again before a 7 to win. If the Point is and the shooter continues to throw the dice until a Point is established and a 7 is rolled before the Point. DON T PASS LINE

### Section 6.2 Definition of Probability

Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability that it will

### Standard 12: The student will explain and evaluate the financial impact and consequences of gambling.

STUDENT MODULE 12.1 GAMBLING PAGE 1 Standard 12: The student will explain and evaluate the financial impact and consequences of gambling. Risky Business Simone, Paula, and Randy meet in the library every

### Week 2: Conditional Probability and Bayes formula

Week 2: Conditional Probability and Bayes formula We ask the following question: suppose we know that a certain event B has occurred. How does this impact the probability of some other A. This question

### Law of Large Numbers. Alexandra Barbato and Craig O Connell. Honors 391A Mathematical Gems Jenia Tevelev

Law of Large Numbers Alexandra Barbato and Craig O Connell Honors 391A Mathematical Gems Jenia Tevelev Jacob Bernoulli Life of Jacob Bernoulli Born into a family of important citizens in Basel, Switzerland

### The Casino Lab STATION 1: CRAPS

The Casino Lab Casinos rely on the laws of probability and expected values of random variables to guarantee them profits on a daily basis. Some individuals will walk away very wealthy, while others will

### STRIKE FORCE ROULETTE

STRIKE FORCE ROULETTE Cycles, cycles, cycles... You cannot get away from them in the game of Roulette. Red, black, red, black... Red, red, red, red. Black, black, black... Red, red, black, black... 1st

### Lecture 13. Understanding Probability and Long-Term Expectations

Lecture 13 Understanding Probability and Long-Term Expectations Thinking Challenge What s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing).

### calculating probabilities

4 calculating probabilities Taking Chances What s the probability he s remembered I m allergic to non-precious metals? Life is full of uncertainty. Sometimes it can be impossible to say what will happen

### The study of probability has increased in popularity over the years because of its wide range of practical applications.

6.7. Probability. The study of probability has increased in popularity over the years because of its wide range of practical applications. In probability, each repetition of an experiment is called a trial,

### 3.2 Roulette and Markov Chains

238 CHAPTER 3. DISCRETE DYNAMICAL SYSTEMS WITH MANY VARIABLES 3.2 Roulette and Markov Chains In this section we will be discussing an application of systems of recursion equations called Markov Chains.

### Unit 18: Introduction to Probability

Unit 18: Introduction to Probability Summary of Video There are lots of times in everyday life when we want to predict something in the future. Rather than just guessing, probability is the mathematical

### TABLE OF CONTENTS. ROULETTE FREE System #1 ------------------------- 2 ROULETTE FREE System #2 ------------------------- 4 ------------------------- 5

IMPORTANT: This document contains 100% FREE gambling systems designed specifically for ROULETTE, and any casino game that involves even money bets such as BLACKJACK, CRAPS & POKER. Please note although

### TOPIC P2: SAMPLE SPACE AND ASSIGNING PROBABILITIES SPOTLIGHT: THE CASINO GAME OF ROULETTE. Topic P2: Sample Space and Assigning Probabilities

TOPIC P2: SAMPLE SPACE AND ASSIGNING PROBABILITIES SPOTLIGHT: THE CASINO GAME OF ROULETTE Roulette is one of the most popular casino games. The name roulette is derived from the French word meaning small

### Slide 1 Math 1520, Lecture 23. This lecture covers mean, median, mode, odds, and expected value.

Slide 1 Math 1520, Lecture 23 This lecture covers mean, median, mode, odds, and expected value. Slide 2 Mean, Median and Mode Mean, Median and mode are 3 concepts used to get a sense of the central tendencies

### Probability Models.S1 Introduction to Probability

Probability Models.S1 Introduction to Probability Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard The stochastic chapters of this book involve random variability. Decisions are

### 14.4. Expected Value Objectives. Expected Value

. Expected Value Objectives. Understand the meaning of expected value. 2. Calculate the expected value of lotteries and games of chance.. Use expected value to solve applied problems. Life and Health Insurers

### MAT 1000. Mathematics in Today's World

MAT 1000 Mathematics in Today's World We talked about Cryptography Last Time We will talk about probability. Today There are four rules that govern probabilities. One good way to analyze simple probabilities

### COME OUT POINT The number (4, 5, 6, 8, 9 or 10) thrown by the shooter on the Come Out roll.

CRAPS A lively Craps game is the ultimate when it comes to fun and excitement. In this fast-paced game, there are many ways to bet and just as many ways to win! It s as simple as placing a bet on the Pass

### Probability, statistics and football Franka Miriam Bru ckler Paris, 2015.

Probability, statistics and football Franka Miriam Bru ckler Paris, 2015 Please read this before starting! Although each activity can be performed by one person only, it is suggested that you work in groups

### How to Play. Player vs. Dealer

How to Play You receive five cards to make your best four-card poker hand. A four-card Straight is a Straight, a four-card Flush is a Flush, etc. Player vs. Dealer Make equal bets on the Ante and Super

### MAS113 Introduction to Probability and Statistics

MAS113 Introduction to Probability and Statistics 1 Introduction 1.1 Studying probability theory There are (at least) two ways to think about the study of probability theory: 1. Probability theory is a

### The game of roulette is played by throwing a small ball onto a rotating wheel with thirty seven numbered sectors.

LIVE ROULETTE The game of roulette is played by throwing a small ball onto a rotating wheel with thirty seven numbered sectors. The ball stops on one of these sectors. The aim of roulette is to predict

### If, under a given assumption, the of a particular observed is extremely. , we conclude that the is probably not

4.1 REVIEW AND PREVIEW RARE EVENT RULE FOR INFERENTIAL STATISTICS If, under a given assumption, the of a particular observed is extremely, we conclude that the is probably not. 4.2 BASIC CONCEPTS OF PROBABILITY

### Contemporary Mathematics- MAT 130. Probability. a) What is the probability of obtaining a number less than 4?

Contemporary Mathematics- MAT 30 Solve the following problems:. A fair die is tossed. What is the probability of obtaining a number less than 4? What is the probability of obtaining a number less than

### PROBABILITY 14.3. section. The Probability of an Event

4.3 Probability (4-3) 727 4.3 PROBABILITY In this section In the two preceding sections we were concerned with counting the number of different outcomes to an experiment. We now use those counting techniques

### Ch. 13.2: Mathematical Expectation

Ch. 13.2: Mathematical Expectation Random Variables Very often, we are interested in sample spaces in which the outcomes are distinct real numbers. For example, in the experiment of rolling two dice, we

### Statistical Inference. Prof. Kate Calder. If the coin is fair (chance of heads = chance of tails) then

Probability Statistical Inference Question: How often would this method give the correct answer if I used it many times? Answer: Use laws of probability. 1 Example: Tossing a coin If the coin is fair (chance

### Math/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability

Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock

### The Normal Approximation to Probability Histograms. Dice: Throw a single die twice. The Probability Histogram: Area = Probability. Where are we going?

The Normal Approximation to Probability Histograms Where are we going? Probability histograms The normal approximation to binomial histograms The normal approximation to probability histograms of sums

### Session 8 Probability

Key Terms for This Session Session 8 Probability Previously Introduced frequency New in This Session binomial experiment binomial probability model experimental probability mathematical probability outcome

### Bonus Maths 2: Variable Bet Sizing in the Simplest Possible Game of Poker (JB)

Bonus Maths 2: Variable Bet Sizing in the Simplest Possible Game of Poker (JB) I recently decided to read Part Three of The Mathematics of Poker (TMOP) more carefully than I did the first time around.

### This Method will show you exactly how you can profit from this specific online casino and beat them at their own game.

This Method will show you exactly how you can profit from this specific online casino and beat them at their own game. It s NOT complicated, and you DON T need a degree in mathematics or statistics to

### Chapter 16: law of averages

Chapter 16: law of averages Context................................................................... 2 Law of averages 3 Coin tossing experiment......................................................

### Stat 134 Fall 2011: Gambler s ruin

Stat 134 Fall 2011: Gambler s ruin Michael Lugo Setember 12, 2011 In class today I talked about the roblem of gambler s ruin but there wasn t enough time to do it roerly. I fear I may have confused some

### PROBABILITY NOTIONS. Summary. 1. Random experiment

PROBABILITY NOTIONS Summary 1. Random experiment... 1 2. Sample space... 2 3. Event... 2 4. Probability calculation... 3 4.1. Fundamental sample space... 3 4.2. Calculation of probability... 3 4.3. Non

### AP Statistics 7!3! 6!

Lesson 6-4 Introduction to Binomial Distributions Factorials 3!= Definition: n! = n( n 1)( n 2)...(3)(2)(1), n 0 Note: 0! = 1 (by definition) Ex. #1 Evaluate: a) 5! b) 3!(4!) c) 7!3! 6! d) 22! 21! 20!

### Statistics. Head First. A Brain-Friendly Guide. Dawn Griffiths

A Brain-Friendly Guide Head First Statistics Discover easy cures for chart failure Improve your season average with the standard deviation Make statistical concepts stick to your brain Beat the odds at

### Introduction to the Rebate on Loss Analyzer Contact: JimKilby@usa.net 702-436-7954

Introduction to the Rebate on Loss Analyzer Contact: JimKilby@usa.net 702-436-7954 One of the hottest marketing tools used to attract the premium table game customer is the "Rebate on Loss." The rebate

### Practical Probability:

Practical Probability: Casino Odds and Sucker Bets Tom Davis tomrdavis@earthlink.net April 2, 2011 Abstract Gambling casinos are there to make money, so in almost every instance, the games you can bet

### Chapter 5 Section 2 day 1 2014f.notebook. November 17, 2014. Honors Statistics

Chapter 5 Section 2 day 1 2014f.notebook November 17, 2014 Honors Statistics Monday November 17, 2014 1 1. Welcome to class Daily Agenda 2. Please find folder and take your seat. 3. Review Homework C5#3

### Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

### MONT 107N Understanding Randomness Solutions For Final Examination May 11, 2010

MONT 07N Understanding Randomness Solutions For Final Examination May, 00 Short Answer (a) (0) How are the EV and SE for the sum of n draws with replacement from a box computed? Solution: The EV is n times

### COMMON CORE STATE STANDARDS FOR

COMMON CORE STATE STANDARDS FOR Mathematics (CCSSM) High School Statistics and Probability Mathematics High School Statistics and Probability Decisions or predictions are often based on data numbers in

### THE WINNING ROULETTE SYSTEM by http://www.webgoldminer.com/

THE WINNING ROULETTE SYSTEM by http://www.webgoldminer.com/ Is it possible to earn money from online gambling? Are there any 100% sure winning roulette systems? Are there actually people who make a living

### HONORS STATISTICS. Mrs. Garrett Block 2 & 3

HONORS STATISTICS Mrs. Garrett Block 2 & 3 Tuesday December 4, 2012 1 Daily Agenda 1. Welcome to class 2. Please find folder and take your seat. 3. Review OTL C7#1 4. Notes and practice 7.2 day 1 5. Folders

### The number (4, 5, 6, 8, 9 or 10) thrown by the shooter on the Come Out roll.

CRAPS A lively Craps game is the ultimate when it comes to fun and excitement. In this fast-paced game, there are many ways to bet and just as many ways to win! It s as simple as placing a bet on the Pass

### \$2 4 40 + ( \$1) = 40

THE EXPECTED VALUE FOR THE SUM OF THE DRAWS In the game of Keno there are 80 balls, numbered 1 through 80. On each play, the casino chooses 20 balls at random without replacement. Suppose you bet on the

### Beating Roulette? An analysis with probability and statistics.

The Mathematician s Wastebasket Volume 1, Issue 4 Stephen Devereaux April 28, 2013 Beating Roulette? An analysis with probability and statistics. Every time I watch the film 21, I feel like I ve made the

### Decision Making Under Uncertainty. Professor Peter Cramton Economics 300

Decision Making Under Uncertainty Professor Peter Cramton Economics 300 Uncertainty Consumers and firms are usually uncertain about the payoffs from their choices Example 1: A farmer chooses to cultivate

### Probability. 4.1 Sample Spaces

Probability 4.1 Sample Spaces For a random experiment E, the set of all possible outcomes of E is called the sample space and is denoted by the letter S. For the coin-toss experiment, S would be the results

### 7.S.8 Interpret data to provide the basis for predictions and to establish

7 th Grade Probability Unit 7.S.8 Interpret data to provide the basis for predictions and to establish experimental probabilities. 7.S.10 Predict the outcome of experiment 7.S.11 Design and conduct an

### Know it all. Table Gaming Guide

Know it all. Table Gaming Guide Winners wanted. Have fun winning at all of your favorite games: Blackjack, Craps, Mini Baccarat, Roulette and the newest slots. Add in seven mouthwatering dining options

### Chapter 7 Probability. Example of a random circumstance. Random Circumstance. What does probability mean?? Goals in this chapter

Homework (due Wed, Oct 27) Chapter 7: #17, 27, 28 Announcements: Midterm exams keys on web. (For a few hours the answer to MC#1 was incorrect on Version A.) No grade disputes now. Will have a chance to

### ECE 316 Probability Theory and Random Processes

ECE 316 Probability Theory and Random Processes Chapter 4 Solutions (Part 2) Xinxin Fan Problems 20. A gambling book recommends the following winning strategy for the game of roulette. It recommends that

### Lesson 1: Experimental and Theoretical Probability

Lesson 1: Experimental and Theoretical Probability Probability is the study of randomness. For instance, weather is random. In probability, the goal is to determine the chances of certain events happening.

### Chance and Uncertainty: Probability Theory

Chance and Uncertainty: Probability Theory Formally, we begin with a set of elementary events, precisely one of which will eventually occur. Each elementary event has associated with it a probability,

### How to Beat Online Roulette!

Martin J. Silverthorne How to Beat Online Roulette! Silverthorne Publications, Inc. How to Beat Online Roulette! COPYRIGHT 2015 Silverthorne Publications Inc. All rights reserved. Except for brief passages

### The Mathematics of Gambling

The Mathematics of Gambling with Related Applications Madhu Advani Stanford University April 12, 2014 Madhu Advani (Stanford University) Mathematics of Gambling April 12, 2014 1 / 23 Gambling Gambling:

### Formula for Theoretical Probability

Notes Name: Date: Period: Probability I. Probability A. Vocabulary is the chance/ likelihood of some event occurring. Ex) The probability of rolling a for a six-faced die is 6. It is read as in 6 or out

### FOUR CARD POKER EXPERIENCEEVERYTHING. 24/7 ACTION SilverReefCasino.com (866) 383-0777

FOUR CARD POKER Four Card Poker is a game that rewards skill, patience and nerve. It is similar to Three Card poker but with one major difference. In Three Card Poker, the play wager must equal the ante;

### Responsible Gambling Education Unit: Mathematics A & B

The Queensland Responsible Gambling Strategy Responsible Gambling Education Unit: Mathematics A & B Outline of the Unit This document is a guide for teachers to the Responsible Gambling Education Unit:

### Table of Contents. Casino Help... 1 Introduction... 1. Using This Help Guide... 1 Getting Started... 2

Table of Contents Casino Help... 1 Introduction... 1 Introduction... 1 Using This Help Guide... 1 Getting Started... 2 Getting Started... 2 Web Casino... 3 Game Client... 4 To Play the Games... 4 Common

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Practice Test Chapter 9 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the odds. ) Two dice are rolled. What are the odds against a sum

### Solutions: Problems for Chapter 3. Solutions: Problems for Chapter 3

Problem A: You are dealt five cards from a standard deck. Are you more likely to be dealt two pairs or three of a kind? experiment: choose 5 cards at random from a standard deck Ω = {5-combinations of