1 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 first time are overwhelmed by all the possible bets. The goal here is to understand what these bets are and how the casino makes money. Probabilities and Expected Values Expected value is the expected return. We want to know what sort of payoff you can expect when you place a bet.. Some Simple Games Lets say you play a game where you roll a fair die (what does this mean? and get paid according to your roll: Roll Payout $ 4 $ 4 $ 3 $ 0 $ 0 $ 0 You have to pay $ to play this game. Is it worth it? What do you expect to happen in the long run? Here is how you might answer this: You roll a, of the time, and you get paid $4. You roll a, of the time, and you get paid $. You roll a 4, of the time, and you get paid $. You roll a 3, of the time, and you get paid $0. You roll a, of the time, and you get paid $0. You roll a, of the time, and you get paid $0. Sum these up to find the expected value: ( ( ( ( ( ( E(X = = 7.7 Thus, you expect to get $.7 back every time you play, making a cool $0.7 profit.
2 Another way to ask this very same question would be, how much is the fair price for this game? (The answer is, of course, $.7. Another way to answer this question is to use the following chart Roll Profit $ 3 $ 4 $ 0 3 $ - $ - $ - Computing expected value: E(X = ( 3 + ( + ( 0 + ( ( + Again, you see that you expect about a $0.7 profit... Notational notes ( ( + ( ( = 0.7 In the first computation, we were interested in the amount of money we would get back in a single game. Thus, in this case, X was this amount of money and E(X is the expected amount of money we get back. In the second computation, we were interested in the total profit we would make. In this case, X was the profit. Of course, it would have been nice to have used a different letter/variable for these things. If we did this and let M be the money from one game and P the profit, then we would have: P = M. Some Exercises for You Determine the expected value for the games.. Charge $ to play. Roll one die, with payouts as follows: Roll Payout $ $ 4 $ 3 $ 0 $ 0 $.0. Charge: $ to toss 3 coins. Toss the coins. If you get all heads or all tails, you receive $. If not, you get nothing. 3. Charge: $. Roll dice. If you roll odd numbers, like a 3 and a, you get $. If you roll even numbers, like 4 and, you get $. Otherwise, you get nothing. 4. Charge: $. Draw twice from a bag that has one $0 and 4 $ bills. You get to keep the bills.
3 Probabilities Versus Odds Lets explore this with a roll of a dice. If you roll a dice 00 times, you would expect to see the number one, 00 times: P (Roll a = Odds on the other hand are given as: (Chances For (Total Chances = = Odds(Roll a = (Chances For : (Total Chances = 00 : 00 = : Odds are usually written this way (with a colon.. Exercises If given odds, compute the probability. If given a probability, compute the odds.. Odds of an event are : 4. What is the probability?. Odds of an event are :. What is the probability? 3. Odds of an event are 3 :. What is the probability? 4. Odds of an event are 0 : 3. What is the probability?. Odds of an event are 3 : 0. What is the probability?. Probability of an event is. What are the odds? 3 7. Probability of an event is 3. What are the odds? 0 8. Probability of an event is 4. What are the odds? 3. Probability of an event is 4. What are the odds? 7 0. Probability of an event is 3%. What are the odds?. Probability of an event is 7. What are the odds? 3 Probability of the dice When throwing two dice and summing the numbers, the possible outcomes are through. To determine the probability of getting a number you make the observation that there are 3 different ways the two dice can be rolled. Question. Compute the probabilities for the sum of two rolled dice. 3
4 Solution: To determine the probability of rolling a number you count the number of ways to roll that number and divide by 3. 4 Craps Sum Combinations Probability , - = , -, 3- = 3 4-4, -3, 3-, 4- = 3 -, -4, 3-3, 4-, , -, 3-4, 4-3, -, - 3 = 8 -, 3-, 4-4, -3, , 4-, -4, -3 = , -, -4 = 3 = 3 8 -, In the game of craps there are a wide range of possible bets that one can make. There are single roll bets, line bets and more. The player places these bets by putting his money (gambling chips in the appropriate place on the craps table, see Figure. Figure : Craps Table Layout 4
5 4. Single roll bets These bets are the easiest to understand. In a single roll bet the player is betting on a certain outcome in a single roll. 4.. Playing the field The most obvious single roll bet is perhaps playing the field. This bet is right in the middle of the table. On a roll of 3,4,,0 or, the player is paid even odds and on a roll of or the player is paid double odds. Thus, if $ is bet on the field and a 3,4,, 0 or is rolled the player is paid $ and keeps his original $. If a or is rolled, the player is paid $ and keeps his original $. Question. Compute the expected value of playing the field. Solution: Here is the expected value of one dollar bet on the field. E(X = = In other words, in the long run $ bet on the field will expect to pay the player $0.44. As we will see, this is better than some bets but it is not good enough. 4.. C and E These are the craps and yo bets. In the game of craps a roll of craps is a roll of a, 3 or. A roll of eleven is also called a yo. (At the craps table you will hear people calling for a lucky-yo, meaning they want an eleven rolled. A player can place a one-time bet on any of these numbers and the payoffs are printed on the craps table. Question 3. Fill in the table below. Roll Odds paid Actual Odds Probability Expected value of $ bet 3 30: 3: : 7: Yo : 7: : 3: Any Crap 7: 8: 0.88 Any 7 4: : Notice that the odds paid are printed on the table. So, for example, the odds paid for any seven is 4 to. Thus, if you put $ down and a seven is rolled this will pay you $4 plus your original bet (thus you will walk away having earned $4. Note that in this table we introduced the column Actual Odds. This is the odds that the casino should pay in order to be completely fair. In other words, if the casino paid these odds then the expected value of a dollar bet would be a dollar.
6 4. The pass line Line bets are the best example of a multi-roll bet. This means that the bet actually lasts several rolls the player does not generally win or lose immediately. 4.. The basic pass line bet When betting the pass line the player puts his bet right on the pass line, say $. Then, the first roll is called the come-out roll. This bet is immediately won if a 7 or is rolled (paid even odds and immediately lost of a craps (,3, is rolled. If any other number is rolled then a point is established and rolling continues until either that point is rolled again (good for the pass-line better or a 7 is rolled (good for the casino. Here is a typical sequence of rolls starting with the player putting $ on the pass line. Roll Description 3 Come out roll, craps. Player loses $ and places another $ on pass line. 7 Come out roll, lucky 7. Player is paid $ and keep his $ on the pass line. Come out roll, Yo. Player is paid $ and keep his $ on the pass line. 4 Come out roll, point of 4 established. No payouts Nothing happens, point is 4. Nothing happens, point is 4. Nothing happens, point is 4. 4 Point is made. Player is paid $ and keeps his $ on the pass line. Come out roll, point of is established. 3 Nothing happens, point is. 7 7-out. Player loses $ bet. Once a point has been established the player wants that point to be rolled and loses if a 7 is rolled. Just looking at probabilities, it is clear that the player is more likely to win if the point is or 8 and less likely to win if the point is 4 or 0. Lets compute some expected values. The key in computing the probabilities involved is first understanding that a pass line bet begins on the come out roll and ends when either the player is paid or when the player loses. Theoretically, this could take forever (nothing guarantees that the point or a 7 will ever be rolled. In any case, for the come-out roll, we can say the following: P (7 or = P (, 3 or = P (4,,, 8,, 0 = 3 (Player wins (Player craps out, loses (A point is established Once a point is established, we want to compute the probability of that point coming up again before a seven. This will, of course, depend on what that point is.
7 Lets do the computation for the point 4 (the calculation if the point was 0 is identical. The player will win if a 4 rolls before a seven. So, a winning sequence of rolls could look like 4 player rolls a 4 right away or 3,,, 8, 4 player rolls something other than a 4 or 7 and then finally a 4 So, the following are the probabilities of winning after a point of 4 has been established. P (4 on roll = P (4 on roll, no 7 =P (not a 4 or 7 P (4 = 3 4 P (4 on roll 3 =P (not a 4 or 7 P (4 on roll, no 7 = ( 3 4 Putting all this together, the probability of winning after a point of 4 has been established is a geometric series: P (4 before 7 =P (4 on roll + P ( 4 on roll, no 4 or 7 before roll + = ( ( ( = ( n 3 = ( 4 3 = 3 4 n=0 Similarly, we can compute the probability of winning other points. (And, included are the points of,3 and, which are not really points because they are craps. P ( before 7 =P ( before 7 = 7 P (3 before 7 =P ( before 7 = 4 P (4 before 7 =P (0 before 7 = 3 P ( before 7 =P ( before 7 = P ( before 7 =P (8 before 7 = We can now compute our probabilities of winning 7
8 Event Probability Win with a 7 or on come-out roll Lose with,3, on come-out roll ( Establish point 4 and win 3 = ( 3 Establish point 4 and lose 3 = ( 8 Establish point and win = ( 4 Establish point and lose 3 = ( Establish point and win 3 = ( 3 Establish point and lose 3 = ( Establish point 8 and win 3 = ( 3 Establish point 8 and lose 3 = ( Establish point and win = ( 4 Establish point and lose 3 = ( Establish point 0 and win 3 = ( 3 Establish point 0 and lose 3 = 8 Computing expected values of a $ bet on the pass lines gives 4.. Laying odds E(X = + = ( The pass line has one of the best expected values for your dollar in the casino. But, craps gives the player another method for increasing your odds. Once a point is established, the player is allowed to back his bet up with an odds bet, paid off at true odds. What this means is that the expected value of a $ odds bet is $. Playing Don t Pass In addition to the Pass line there is also the Don t Pass line where a player is betting with the casino. In other words, the Don t Pass player will win when all the other players are losing. This is true except that the Don t Pass line is actually the Don t Pass bar and the player does not win when on the come out roll when a craps of is rolled. So, what is the house advantage in this case? We compute expected value of $ placed on the Don t Pass line. Notice that our table is essentially the same as before. 8
9 Event Probability Lose with a 7 or on come-out roll Win with,3 on come-out roll Push with on come-out roll ( 3 Establish point 4 and lose 3 = ( 3 Establish point 4 and win 3 = ( 8 Establish point and lose = ( 4 Establish point and win 3 = ( Establish point and lose 3 = ( 3 Establish point and win 3 = ( Establish point 8 and lose 3 = ( 3 Establish point 8 and win 3 = ( Establish point and lose = ( 4 Establish point and win 3 = ( Establish point 0 and lose 3 = ( 3 Establish point 0 and win 3 = 8 Computing expected values of a $ bet on the pass lines gives E(X = + ( = Which is slightly better than playing the pass line. Crapless Craps There is a variation of craps (discovered by the author on a recent trip to a casino called Crappless Craps or No More Craps, Craps. What this means is that the craps and yo are all done away with on the come out roll. Instead, the,3,,and are valid points just like the 4,,,8, and 0. The odds bets for these are also paid off at true odds. This looks like a good thing because there are less ways to lose on the come-out roll. We now compute some probabilities and expected values.
10 Event Probability Win with 7 on come-out roll ( Establish point and win 3 7 = ( Establish point and lose 3 7 = ( 4 Establish point 3 and win 8 4 = ( 7 Establish point 3 and lose = ( 4 Establish point 4 and win 3 = ( 3 Establish point 4 and lose 3 = ( 8 Establish point and win = ( 4 Establish point and lose 3 = ( Establish point and win 3 = ( 3 Establish point and lose 3 = ( Establish point 8 and win 3 = ( 3 Establish point 8 and lose 3 = ( Establish point and win = ( 4 Establish point and lose 3 = ( Establish point 0 and win 3 = ( 3 Establish point 0 and lose 3 = ( 8 Establish point and win 8 4 = ( 7 Establish point and lose = ( 4 Establish point and win 3 7 = Establish point and lose 3 Computing expected value of a $ bet in on this pass line. E(X = + = ( 7 = 4 ( Amazingly, this makes it a worse bet than regular craps, by a substantial amount. Also, the casino will not allow a player to play Don t (why not?. 7 How fast will you lose money? Here are some estimate to help compute this. Over an hour, we can expect about: : dice rolls (observed, info from Internet : points to be established : craps on come out roll 4: 7 or on the come out roll 0
11 7. Pass line player Basically, this means that if you are betting on the pass line then every hour you will place around 7 bets. If you are making $00 bets then this means you will have placed $00 on the table 7 times and on you will receive $8. back from these bets. In other words, in an hour you will have lost only about $ Field player Over an hour, the field player will make about bets and, if playing $00 a bet then the field player should receive $4.44 from every bet. This means that this field player will lose about $ How to minimize your losses? There are clearly some bad bets, such as the field (and, the field is not even the worst bet. So, first thing to do is to not play the bad bets. When playing the pass-line, the house edge is only about.4%. If you play with odds and place as many odds behind your bet then you can expect to decrease the house advantage. The more odds you place on your pass-line bet the more you decrease the advantage that the house has. 7.4 Pass-line with odds Here we assume a player places a pass-line bet of $ and then, when a point is established, places an odds bets behind the pass-line. Most casino s limit the amount of this odds bet and 0 is a high amount. What this means is that if your pass-line bet is $ then your odds bet can be at most $0. Here are the payoffs on such a bet if the number is hit. Point Bet Payout So, if the point is 4 and that point is hit then the player would have $ on the pass-line and $0 for odds. The player would be paid $ for his pass-line bet and $0 for his odds bet. How do you compute the house advantage now? 7. Methods to win? I found these on the Internet:
12 As one Internet site said, There are NO winning systems for Craps that work! It is instructive to see if you can figure out what is wrong with these systems. 7.. The Craps Count Method The -Count method for finding hot shooters and reducing a player s overall risk at the craps table starts with a point number (the point numbers are 4,,, 8,, or 0 and ends with a point number. Let s take it step by step. A new shooter has just received the dice and is on his come-out roll. If he rolls any of the point numbers on his come-out roll, that constitutes the -count. If he rolls a, 3, 7,, or on the come-out, the count has not started yet. The second roll after the -count is automatically the -count, regardless of what number is rolled, and this applies to the 3-count and 4-count as well. The next roll after the 4-count is the -Count only if the number rolled is again one of the point numbers above. If not, it is the 4-count and holding until the -Count is achieved. When the -Count is achieved, you then begin to place money at risk. You can make a come bet or make a place bet. If you usually make three come bets you can actually make your first come bet after the three count. You would then make your second come bet after the four count. You don t take odds on your come bets until after the five count. At this time you would also make your third come bet. 7.. The Martingale or Rothstein System Bet unit on the Pass Line. If you lose bet 3 units (double your bet plus. If you lose again bet 7 units (double your bet plus and so on. Sooner or later you must win and you will be up. If you win then start betting again from the unit up The Patience or Watching System Wait until the dice pass four times and then put $4 on them losing on the fifth roll. If you lose, put double your bet ($8 on them losing on the next roll and so on until you win The Hot and Cold System This system says you will win when the dice are hot and win when the dice are cold. You bet on the dice to win when the shooter makes a pass. You continue to bet on the dice to win until a losing through occurs and then you switch your bets on to Don t Pass. You simply switch when the dice are having a hot run and back again when the dice have a cold run.
13 7.. The Place Betting System A big time gamblers system which has even deceived some Craps operators. You place the maximum limit (although it can be less on all of the six place bets: 4,,, 8, and 0. If the limit is $00 then this makes a total of $,00. As soon as one of these bets is won you collect on it and call off the remaining five place number bets. Yes, most casinos allow you to call off place number bets at any time. The theory is that the bettor has six numbers against the house s one, the 7. There are 4 ways to make the place numbers and only ways to make a 7 so the odds are worked out as 4 to in favor of the gambler with the system. 7.. The Right and Wrong Way System In this system you place your bet (say $0 on the Don t Pass line before the come out throw. The shooter is then supposed to throw a point on the come out roll, either 4,,, 8, or 0. You then put $0 on the points place number. The idea is if the shooter doesn t throw the point you have still won on the Don t Pass bet and you are still even. You ve lost $0 on the place number but won $0 on Don t Pass. If the shooter makes his point you lose $0 on the Don t Pass but your up by between $48 and $0 depending on which place number it is. 8 More Probability and Expected Value Challenges Question 4. Suppose you roll a die and receive the number of dollars that you roll. What is the fair value to play this game? Question. If you flip a coin twice, what the probability that it will come up heads each time? What if you flip the coin 3 times? n times? Question. Suppose we play a game in which you flip a coin until a heads comes up for the first time. What is the probability that the first heads will come up on the first flip? Second flip? Third flip? What about the n th flip? Question 7. Play a game where you flip a coin until you get a heads. If the first heads comes up on the k th flip, then you win k dollars. So if you get H on the first try, you win $. If you get T and then H, you win $4. If you get TTTTH, you win $3. What is the least amount of money a player can win in this game? What is the most? How much would you be willing to pay to play this game? What is a fair amount to pay to play this game? 3
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
SIXTEEN PLAYERS, TWO DICE, ONE SHOOTER Craps is the most exciting game at Potawatomi Bingo Casino. One shooter of the dice determines the outcome for up to 15 other players. Craps also features a variety
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
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
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 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
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
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
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,
REWARD System For Even Money Bet in Roulette By Izak Matatya By even money betting we mean betting on Red or Black, High or Low, Even or Odd, because they pay 1 to 1. With the exception of the green zeros,
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
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
(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
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
Baccarat was made famous in the United States when a tuxedoed Agent 007 played at the same tables with his arch rivals in many James Bond films. You don t have to wear a tux or worry about spies when playing
After mining all the available NBA data for the last 15 years, we found the keys to a successful basketball betting system: If you follow this system exactly, you can expect to hit 90% of your NBA bets.
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
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
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
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
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
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
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
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
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
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
The New Mexico Lottery 26 February 2014 Lotteries 26 February 2014 1/27 Today we will discuss the various New Mexico Lottery games and look at odds of winning and the expected value of playing the various
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
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.
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
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
Expectation Statistics and Random Variables Math 425 Introduction to Probability Lecture 4 Kenneth Harris email@example.com Department of Mathematics University of Michigan February 9, 2009 When a large
Mathematical Expectation Properties of Mathematical Expectation I The concept of mathematical expectation arose in connection with games of chance. In its simplest form, mathematical expectation is the
Introduction and Overview Probability and Statistics is a topic that is quickly growing, has become a major part of our educational program, and has a substantial role in the NCTM Standards. While covering
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
Inside the pokies - player guide 3nd Edition - May 2009 References 1, 2, 3 Productivity Commission 1999, Australia s Gambling Industries, Report No. 10, AusInfo, Canberra. 4 Victorian Department of Justice,
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
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
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,
6.042/8.062J Mathematics for Comuter Science December 2, 2006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Random Walks Gambler s Ruin Today we re going to talk about one-dimensional random walks. In
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
Welcome to the casino secret to profitable options trading. I am very excited that we re going to get the chance to discuss this topic today. That s because, in this video course, I am going to teach you
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
Chapter 5. Discrete Probability Distributions Chapter Problem: Did Mendel s result from plant hybridization experiments contradicts his theory? 1. Mendel s theory says that when there are two inheritable
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;
Pascal s wager So far we have discussed a number of arguments for or against the existence of God. In the reading for today, Pascal asks not Does God exist? but Should we believe in God? What is distinctive
Björn Bäckström, Lars Olsén and Simon Renström Copyright 2011 ClaroBet AB. Updated: May. 12, 11 You may not distribute or copy any of the text or use it in any other way except for personal use, without
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
Combinatorics: The Fine Art of Counting Week 7 Lecture Notes Discrete Probability Continued Note Binomial coefficients are written horizontally. The symbol ~ is used to mean approximately equal. The Bernoulli
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
A P S T A T S A Fabulous PROBABILITY C A S I N O L A B AP Statistics Casino Lab 1 AP STATISTICS CASINO LAB: INSTRUCTIONS The purpose of this lab is to allow you to explore the rules of probability in the
Rules of core casino games in Great Britain June 2011 Contents 1 Introduction 3 2 American Roulette 4 3 Blackjack 5 4 Punto Banco 7 5 Three Card Poker 9 6 Dice/Craps 11 2 1 Introduction 1.1 This document
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
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).
Chapter 3 Prediction Markets, Fair Games and Martingales Prediction markets...... are speculative markets created for the purpose of making predictions. The current market prices can then be interpreted
EECS 401 Due on Feb 2, 2007 PROBLEM 1 (25 points) Joe and Helen each know that the a priori probability that her mother will be home on any given night is 0.6. However, Helen can determine her mother s
The Sure Roulette Method Win a minimum of 10 betting units per hour Roulette is the oldest of all the casino games. It has been Europe's favorite gambling game since the latter part of the 18th century.
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
Third Grade Math Games Unit 1 Lesson Less than You! 1.3 Addition Top-It 1.4 Name That Number 1.6 Beat the Calculator (Addition) 1.8 Buyer & Vendor Game 1.9 Tic-Tac-Toe Addition 1.11 Unit 2 What s My Rule?
If you like a flutter on the horses or any other sport then I would strongly recommend Betfair to place those bets. I find it amazing the number of people still using high street bookmakers which offer
2urbo Blackjack Type of Game The game of 2urbo Blackjack utilizes a player-dealer position and is a California game. The player-dealer shall collect all losing wagers, pay all winning wagers, and may not
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
Chapter 16. Risk and Uncertainty Part A 2009, Kwan Choi Expected Value X i = outcome i, p i = probability of X i EV = pix For instance, suppose a person has an idle fund, $100, for one month, and is considering
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:
. 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
January 2012, Number 64 ALL-WAYS TM NEWSLETTER Help from the Collective Public How Much to Wager How to Apportion Wager Dollars Inside This Newsletter Announcements Winter: A Good Time to Explore, Learn
Week 4: Gambler s ruin and bold play Random walk and Gambler s ruin. Imagine a walker moving along a line. At every unit of time, he makes a step left or right of exactly one of unit. So we can think that
Learn How to Use The Roulette Layout To Calculate Winning Payoffs For All Straight-up Winning Bets Understand that every square on every street on every roulette layout has a value depending on the bet
Forex Trading What Finally Worked For Me If you find typographical errors in this book they are here for a purpose. Some people actually enjoy looking for them and we strive to please as many people as
In conditions of laissez-faire the avoidance of wide fluctuations in employment may, therefore, prove impossible without a far-reaching change in the psychology of investment markets such as there is no
The Easy Roulette System Please note that all information is provided as is and no guarantees are given whatsoever as to the amount of profit you will make if you use this system. Neither the seller of
Stat 20: Intro to Probability and Statistics Lecture 16: More Box Models Tessa L. Childers-Day UC Berkeley 22 July 2014 By the end of this lecture... You will be able to: Determine what we expect the sum
How-To-Make-Money-Easily The Hidden Secrets Of Making Money Online Every Day! www.how-to-make-money-easily.com 2008 Congratulations!! You re about to discover a slightly strange, rather unkown method to
Greg Fletcher Sharpshooter Roulette Video Guide Russell Hunter Publishing, Inc. Sharpshooter Roulette Video Guide 2015 Greg Fletcher and Russell Hunter Publishing. All Rights Reserved All rights reserved.
Random Fibonacci-type Sequences in Online Gambling Adam Biello, CJ Cacciatore, Logan Thomas Department of Mathematics CSUMS Advisor: Alfa Heryudono Department of Mathematics University of Massachusetts
Your consent to our cookies if you continue to use this website.