Bayesian Tutorial (Sheet Updated 20 March)

Size: px
Start display at page:

Download "Bayesian Tutorial (Sheet Updated 20 March)"

Transcription

1 Bayesian Tutorial (Sheet Updated 20 March) Practice Questions (for discussing in Class) Week starting 21 March What is the probability that the total of two dice will be greater than 8, given that the first die is a 6? 2. A patient s probability to have a liver disease increases if he is an alcoholic. Given that, 10% of the patients entering the clinic have liver disease, 5% of the clinic s patients are alcoholic and that, 7% of the patients diagnosed with liver disease are alcoholics, find the probability that an alcoholic patient is diagnosed with liver disease. 3. After tossing a fair coin three times: i. What is the probability of at least two tails? ii. What is the probability of exactly one tail? iii. Given that at least one tail is observed, what is the probability of observing at least two tails? 4. Given that 13% of patients who have lung cancer, and smoke; 3% of patients smoke and do not have lung cancer. 5% of patients have lung cancer and do not smoke. 79% of patients neither have lung cancer nor do they smoke. Draw an appropriate probability table and find the probability that a patient, picked at random, has lung cancer, given that he smokes. Also find the probability that a patient is a smoker given that he has lung cancer. From this, derive your inference. 5. Consider that 0.9% of the people have a genetic defect, 92% of the tests for gene are true positives, 9.8% of the tests are false positives. If a person gets a positive test result, what are the odds that they actually have the faulty gene? 6. The probability that it is Wednesday and that a student is absent is What is the probability that a student is absent given that today is Wednesday? 7. You go to see the doctor about an ingrowing toe-nail. The doctor selects you at random to have a blood test for swine flu, which for the purposes of this exercise we will say is currently suspected to affect 1 in 10,000 people in Australia. The test is 99% accurate, in the sense that the probability of a false positive is 1%. The probability of a false negative is zero. You test positive. What is the new probability that you have swine flu? Now imagine that you went to a friend s wedding in Mexico recently, and (for the purposes of this exercise) it is known that 1 in 200 people who visited Mexico recently come back with swine flu. Given the same test result as above, what should your revised estimate be for the probability you have the disease?

2 Poker Tournament Question Play this game in Groups, in YOUR OWN TIME! We are going to play a game of poker. Unlike usual games, we will discuss our cards and our bets as we play. The rules of the game Texas Hold em The aim of the game is to win the chips that are bet on each hand You do this by either o Having the best hand o Or forcing all the other players to throw in their hands instead of calling your bet i.e. they fold You will be dealt two cards that only you can see Once everyone has their cards, the person to the left of the dealer makes a bet based on their cards Everybody has the choice of: o Calling betting the same amount o Folding betting nothing and throwing in their cards o Raising Making the bet bigger, which means that the following players must bet the new higher amount if they want to call and that the players who have already bet have to make up the difference to stay in When everybody has bet the same amount or folded, three cards are dealt face up on the table. These cards are shared by all players There is another round of betting, as above, followed by one more shared card, another round of betting, and a final shared card. After a final round of betting, if more than one player is left in, the players show their hands and the player who can make the best hand using the middle cards plus 1 or 2 of his own cards wins. The order of the hands is as follows, with the best first: 1. Straight flush all cards the same suit and in unbroken order, e.g. 5,6,7,8,9 of spades 2. Four of a kind all cards have the same value, e.g. 5,5,5,5 one of each suit 3. Full house Three cards with one shared value plus two of another, e.g K,K,K,5,5 4. Flush All cards the same suit, e.g. 5 hearts. 5. Straight 5 cards in unbroken order, but of a mix of suits, e.g. 3,4 of spades, 5 of diamonds, 6 and 7 of hearts. 6. Three of a kind three cards of the same value 7. Two pairs e.g. 5,5,K,K 8. One pair e.g.7,7 9. No hand none of the above are made When two players have a hand of the same rank, the one that contains the highest valued card wins, except in the case of a full house, where the one with the highest ranking 3 of a kind wins. Round 1 Make sure you can answer these questions before you start the game! Supposing there s just me playing poker, no other players Let s say that I will bet whenever I get two fancy cards, but never any other time (fancy cards ace, Jack, Queen, King).

3 1. What is the probability of me making a bet, given that you don t know what cards I have? I need to play more hands than that, so I will bluff (bet on rubbish cards) exactly 20% of the time that I have rubbish cards. 2. Now what is the probability of me making a bet, given that you don t know what cards I have? Clue it is the conditional probability of me betting given that I do not have two fancy cards plus the answer to question 1. Now let s work out something more useful. I have made a bet and you want to know whether or not I am bluffing. 3. Use Bayes rule to work out the probability that I am bluffing. Then use the same rule to work out the probability that I have two fancy cards. Check that they sum to one to make sure you have it right. 4. Finally, if I also bet when I have any pair in my hand (5,5 for example), how does that affect the values calculated in 1,2 and 3 above? Once we ve answered these questions, we will try them out in a few hands. Round 2 Each player will say what cards they have and calculate the probability of being dealt those two cards. Hopefully a pattern will be quickly discovered! Players will be forced to bet a minimum amount at this point. We will then deal the three cards to the middle the so called Flop. Now each player will calculate the probability of them making a good hand from the last two unknown cards. Those with a low probability can fold. The others can bet. Again, another card will be dealt and probabilities will be calculated. People can bet or fold. The winner gets the chips. We will play a few more rounds, without people showing their cards until the end. Then we will look at the cards and calculate the probabilities that people were playing with.

4 Quick Review of Bayes Theorem and Bayesian Network Conditional Probability & Bayes Theorem: The conditional probability of an event B is the probability that the event will occur given the knowledge that an event A has already occurred. This probability is written P(B A), notation for the probability of B given A. P(B A) = P(A B) P(B) / P(A) Deriving Bayes theorem using conditional probability. P(A B) = P(A) * P(B A) P(B A) = P(B) * P(A B) Equating two of them, P(A) P(B A) = P(B) P(A B) This implies, P(B A) = P(A B) P(B) / P(A) Bayesian networks A Bayesian Network (BN) is a way of describing the relationships between causes & effects BNs used to support decision making and to find strategies to solve tasks under uncertainty BNs use Bayesian probability theory How are they different from others? Uncertainty is handled in a mathematically rigorous yet simple and efficient way. Network representation of problems. Most other methods do not include statistical information to make inferences. Combination of statistics and Bayesian network is powerful - Bayesian nets are a network-based framework for representing and analysing models involving uncertainty. Recall example Nodes: Random Variables Arcs: Casual or influential relationships Collection of nodes & arcs is termed graph or topology of BN (For each node, an associated Node Probability Table (NPT) give the the conditional probability of each possible outcome, given combinations of outcomes from parent nodes) BN present causal chains (i.e cause-effect relationships between parent and child nodes) Given evidence of past events, run the BN to see what the most likely future outcomes will be they are also robust to missing info.! Bayesian classification can help to predict information we do not know using information we do know and the likelihood of certain patterns in the data occurring (cf. learning!)

5 Example Solutions 1. What is the probability that the total of two dice will be greater than 8, given that the first die is a 6? Let A = first die is 6 Let B = total of two dice is greater than 8 We need to determine the conditional probability, P(B/A) i.e. the probability of an event (B) given that another event (A) has occurred. This can be computed by considering only outcomes for which the first die is a 6. Then, determine the proportion of these outcomes that total more than 8. All the possible outcomes for two dice can be calculated as below: There are 36 possible outcomes when a pair of dice is thrown. Consider that if one of the dice rolled is a 1, there are six possibilities for the other die. If one of the dice rolled a 2, the same is still true. And the same is true if one of the dice is a 3,4,5, or 6. If this is still confusing, look at the following (abbreviated) list of outcomes: [(1,1),(1,2),(1,3),(1,4),(1,5),(1,6); (2,1),(2,2),(2,3) (3,1),(3,2),3,3) (4,1) (5,1) (6,1). The total number of outcomes is 6 6 = 36 (or 6^2) Now, there are 6 outcomes for which the first die is a 6: (6,1),(6,2),(6,3),(6,4),(6,5),(6,6), and of these, there are four that total more than 8. The probability of a total greater than 8 given that the first die is 6, i.e. P(B/A) is therefore = 4/6 = 2/3. Alternatively, using Bayes Theorem: P (A) = 1/6 Favourable outcomes for A and B: (6, 3), (6, 4), (6, 5), (6, 6) P (A and B) = 4/ 36 P (B A) = P (A and B) / P (A) = 4/36 * 6/1 = 2/3 2. A patient s probability to have a liver disease increases if he is an alcoholic. Given that, 10% of the patients entering the clinic have liver disease, 5% of the clinic s patients are alcoholic and that, 7% of the patients diagnosed with liver disease are alcoholics, find the probability that an alcoholic patient is diagnosed with liver disease. Given: Probability of patients with Liver disease: P(A) = 0.10 Probability of patients who are alcoholic: P(B) = (7% patients diagnosed with liver disease (A) are alcoholics (B)) i.e. P(B A) = (Probability that alcoholic patient (B) is diagnosed with liver disease (A)), i.e. P(A B) = P(B A) P(A) / P(B) = 0.14

6 3. After tossing a fair coin three times: i. What is the probability of at least two tails? Let H represent an outcome of heads and T represent an outcome of Tails For three tosses of the coin all the possible outcomes are H-H-H T-H-H H-T-H H-H-T T-H-T T-T-H H-T-T T-T-T (or, 2^3 since, each toss leads to two outcomes) The above eight possible outcomes are the sample space. The outcomes that have at least two tails in them are T-H-T, T-T-H, H-T-T, and T-T-T. Therefore, there are four of the eight outcomes that have two or more tails in them. This means that the probability of throwing at least two tails in three tosses is 4 out of 8, which reduces to: 50 percent. OR: P (HTT U THT U TTH U TTT) = 1/8 + 1/8 + 1/8 + 1/8 = 4/8 (since events are mutually exclusive) ii. What is the probability of exactly one tail? P (HHT U HTH U THH) = 1/8 + 1/8 + 1/8 = 3/8 iii. Given that at least one tail is observed, what is the probability of observing at least two tails? A1 = event that at least one tail is observed (T 1) A2 = event of observing at least two tails (T 2) Intuitive Answer: It would be pretty quick to just list all possible outcomes. There are 8 ways you can flip 3 coins. Only one of them is eliminated by the condition (HHH). How many of the remaining results have at least two tails? (4/7) Using Bayes Theorem: P (A1 ) = 7/8 P (A2) = 4/8 P (A2 A1) = P(A1 A2) / P(A1) = (4/8) / (7/8) = 4/7 (To find P(T 1 T 2), we should just list the possible sequences of coin tosses that would allow this (4 out of the 8 tosses, as the required conditions eliminate four remaining sequences: H-H-H,T-H-H, H-T-H, H-H-T)

7 4. Given that 13% of patients who have lung cancer, and smoke; 3% of patients smoke and do not have lung cancer. 5% of patients have lung cancer and do not smoke. 79% of patients neither have lung cancer nor do they smoke. Draw an appropriate probability table and find the probability that a patient, picked at random, has lung cancer, given that he smokes. Also find the probability that a patient is a smoker given that he has lung cancer. From this, derive your inference. S A B 0.79 Disease Status (Joint-probabilities Table!) Smoker (B) Nonsmoker (~B) Lung No Lung Cancer (A) Cancer (~A) A = lung cancer B = smoker P(A B) = 0.13/0.16 = which is 81.25% P(B A) = 0.13/0.18 = which is 72.22% Inference is P(A B) and P(B A) are not equal. Q5. Consider that 0.9% of the people have a genetic defect, 92% of the tests for gene are true positives, 9.8% of the tests are false positives. If a person gets a positive test result, what are the odds that they actually have the faulty gene? Let P(A) = probability of having the faulty gene = (Hence, P(~A) = 0.991) B = positive test result P(A B) = Probability of having the gene given a positive test result. P(B A) = Probability of a positive test result given that the person has the gene = 0.92 P(B ~A) = Probability of a positive test if the person does not have the gene = P(B) = P(B A) + P(B ~A) = P(B A) * P(A) + P(B ~A)*P(~A) = 0.92 * * 0.991= P(A B) = P(B A) * P(A) / P(B) = (0.92 * 0.009) / ( ) = , which is 7.86% probability of having the faulty gene. Q6. The probability that it is Wednesday and that a student is absent is What is the probability that a student is absent given that today is Wednesday? P(Absent Wednesday) = P(Wednesday and Absent) / P(Wednesday) = 0.04/0.2 = 0.2 which is 20%. (assuming there are five working days in a student s week!)

8 Q7 You go to see the doctor about an ingrowing toe-nail. The doctor selects you at random to have a blood test for swine flu, which for the purposes of this exercise we will say is currently suspected to affect 1 in 10,000 people in Australia. The test is 99% accurate, in the sense that the probability of a false positive is 1%. The probability of a false negative is zero. You test positive. What is the new probability that you have swine flu? Now imagine that you went to a friend s wedding in Mexico recently, and (for the purposes of this exercise) it is known that 1 in 200 people who visited Mexico recently come back with swine flu. Given the same test result as above, what should your revised estimate be for the probability you have the disease? Let P(D) be the probability one has swine flu. Let P(T) be the probability of a positive test. We wish to know P(D T). Bayes theorem says: P T D P(D) P D T = P T which in this case can be rewritten as: P T D P(D) P D T = P T D P D + P T ~ D P(~ D) where P(~ D) means the probability of not having swine flu. We have P(D) = (the a priori probability one has swine flu). P(~ D) = P(T D) = 1 (if one has swine flu the test is always positive). P(T ~ D) = 0.01 (1% chance of a false positive, i.e. test wrongly indicates the condition is present). Plugging these numbers in we get: P D T = That is, even though the test was positive one s chance of having swine flu is only 1%. However, if one went to Mexico recently then his starting P(D) is In this case P D T = and you should be a lot more worried! Aside: Recap definitions of True/False Positives/Negatives (Confusion Matrix) There is quite a bit of terminological confusion in this area. Many people find it useful to come back to a confusion matrix to think about this. In a classification / screening test, you can have four different situations: Condition: A Not A Test says A True positive False positive Test says Not A False negative True negative

9 In this table, true positive, false negative, false positive and true negative are events (or their probability). What you have is therefore probably a true positive rate and a false negative rate. The distinction matters because it emphasizes that both numbers have a numerator and a denominator. Where things get a bit confusing is that you can find several definitions of false positive rate and false negative rate, with different denominators. For example, Wikipedia provides the following definitions (they seem pretty standard): True positive rate (or sensitivity): TPR=TP/(TP+FN) False positive rate: FPR=FP/(FP+TN) True negative rate (or specificity): TNR=TN/(FP+TN) In all cases, the denominator is the column total. This also gives a cue to their interpretation: The true positive rate is the probability that the test says A when the real value is indeed A (i.e., it is a conditional probability, conditioned on A being true). This does not tell you how likely you are to be correct when calling A (i.e., the probability of a true positive, conditioned on the test result being A ). Assuming the false negative rate is defined in the same way, we then have we then have FNR=1 TPR We cannot however directly derive the false positive rate from either the true positive or false negative rates because they provide no information on the specificity, i.e., how the test behaves when not A is the correct answer. There are however other definitions in the literature! NOTE: 1) True +ve and false -ve make 100% 2) False +ve and true -ve make 100% 3) There is no relation between true positives and false positives.

Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314

Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314 Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space

More information

Texas Hold em. From highest to lowest, the possible five card hands in poker are ranked as follows:

Texas Hold em. From highest to lowest, the possible five card hands in poker are ranked as follows: Texas Hold em Poker is one of the most popular card games, especially among betting games. While poker is played in a multitude of variations, Texas Hold em is the version played most often at casinos

More information

Probability --QUESTIONS-- Principles of Math 12 - Probability Practice Exam 1 www.math12.com

Probability --QUESTIONS-- Principles of Math 12 - Probability Practice Exam 1 www.math12.com Probability --QUESTIONS-- Principles of Math - Probability Practice Exam www.math.com Principles of Math : Probability Practice Exam Use this sheet to record your answers:... 4... 4... 4.. 6. 4.. 6. 7..

More information

Chapter 6. 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? Ans: 4/52.

Chapter 6. 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? Ans: 4/52. Chapter 6 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? 4/52. 2. What is the probability that a randomly selected integer chosen from the first 100 positive

More information

Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions.

Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions. Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

More information

Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to

More information

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE

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

More information

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

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.

More information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,

More information

Conditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Conditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Conditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of conditional probability and independence

More information

Pattern matching probabilities and paradoxes A new variation on Penney s coin game

Pattern matching probabilities and paradoxes A new variation on Penney s coin game Osaka Keidai Ronshu, Vol. 63 No. 4 November 2012 Pattern matching probabilities and paradoxes A new variation on Penney s coin game Yutaka Nishiyama Abstract This paper gives an outline of an interesting

More information

A Few Basics of Probability

A Few Basics of Probability A Few Basics of Probability Philosophy 57 Spring, 2004 1 Introduction This handout distinguishes between inductive and deductive logic, and then introduces probability, a concept essential to the study

More information

PROBABILITY. The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE

PROBABILITY. The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE PROBABILITY 53 Chapter 3 PROBABILITY The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE 3. Introduction In earlier Classes, we have studied the probability as

More information

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

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

More information

AP Stats - Probability Review

AP Stats - Probability Review AP Stats - Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose

More information

6.3 Conditional Probability and Independence

6.3 Conditional Probability and Independence 222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

More information

Ready, Set, Go! Math Games for Serious Minds

Ready, Set, Go! Math Games for Serious Minds Math Games with Cards and Dice presented at NAGC November, 2013 Ready, Set, Go! Math Games for Serious Minds Rande McCreight Lincoln Public Schools Lincoln, Nebraska Math Games with Cards Close to 20 -

More information

Chapter 4 - Practice Problems 1

Chapter 4 - Practice Problems 1 Chapter 4 - Practice Problems SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. ) Compare the relative frequency formula

More information

Thursday, October 18, 2001 Page: 1 STAT 305. Solutions

Thursday, October 18, 2001 Page: 1 STAT 305. Solutions Thursday, October 18, 2001 Page: 1 1. Page 226 numbers 2 3. STAT 305 Solutions S has eight states Notice that the first two letters in state n +1 must match the last two letters in state n because they

More information

Probability definitions

Probability definitions Probability definitions 1. Probability of an event = chance that the event will occur. 2. Experiment = any action or process that generates observations. In some contexts, we speak of a data-generating

More information

Probabilistic Strategies: Solutions

Probabilistic Strategies: Solutions Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6-sided dice. What s the probability of rolling at least one 6? There is a 1

More information

Probability and Expected Value

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

More information

Lab 11. Simulations. The Concept

Lab 11. Simulations. The Concept Lab 11 Simulations In this lab you ll learn how to create simulations to provide approximate answers to probability questions. We ll make use of a particular kind of structure, called a box model, that

More information

Unit 19: Probability Models

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,

More information

Definition and Calculus of Probability

Definition and Calculus of Probability In experiments with multivariate outcome variable, knowledge of the value of one variable may help predict another. For now, the word prediction will mean update the probabilities of events regarding the

More information

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

More information

Session 8 Probability

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

More information

Bayesian Updating with Discrete Priors Class 11, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Bayesian Updating with Discrete Priors Class 11, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals Bayesian Updating with Discrete Priors Class 11, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1. Be able to apply Bayes theorem to compute probabilities. 2. Be able to identify

More information

Decision Making Under Uncertainty. Professor Peter Cramton Economics 300

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

More information

Fundamentals of Probability

Fundamentals of Probability Fundamentals of Probability Introduction Probability is the likelihood that an event will occur under a set of given conditions. The probability of an event occurring has a value between 0 and 1. An impossible

More information

Curriculum Design for Mathematic Lesson Probability

Curriculum Design for Mathematic Lesson Probability Curriculum Design for Mathematic Lesson Probability This curriculum design is for the 8th grade students who are going to learn Probability and trying to show the easiest way for them to go into this class.

More information

Week 2: Conditional Probability and Bayes formula

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

More information

Probabilities of Poker Hands with Variations

Probabilities of Poker Hands with Variations Probabilities of Poker Hands with Variations Jeff Duda Acknowledgements: Brian Alspach and Yiu Poon for providing a means to check my numbers Poker is one of the many games involving the use of a 52-card

More information

Ch. 13.2: Mathematical Expectation

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

More information

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

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

More information

Combinatorics 3 poker hands and Some general probability

Combinatorics 3 poker hands and Some general probability Combinatorics 3 poker hands and Some general probability Play cards 13 ranks Heart 4 Suits Spade Diamond Club Total: 4X13=52 cards You pick one card from a shuffled deck. What is the probability that it

More information

2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways.

2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways. Math 142 September 27, 2011 1. How many ways can 9 people be arranged in order? 9! = 362,880 ways 2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways. 3. The letters in MATH are

More information

(b) You draw two balls from an urn and track the colors. When you start, it contains three blue balls and one red ball.

(b) You draw two balls from an urn and track the colors. When you start, it contains three blue balls and one red ball. Examples for Chapter 3 Probability Math 1040-1 Section 3.1 1. Draw a tree diagram for each of the following situations. State the size of the sample space. (a) You flip a coin three times. (b) You draw

More information

Champion Poker Texas Hold em

Champion Poker Texas Hold em Champion Poker Texas Hold em Procedures & Training For the State of Washington 4054 Dean Martin Drive, Las Vegas, Nevada 89103 1 Procedures & Training Guidelines for Champion Poker PLAYING THE GAME Champion

More information

Source. http://en.wikipedia.org/wiki/poker

Source. http://en.wikipedia.org/wiki/poker AI of poker game 1 C H U N F U N G L E E 1 0 5 4 3 0 4 6 1 C S E 3 5 2 A R T I F I C I A L I N T E L L I G E N C E P R O. A N I T A W A S I L E W S K A Source CAWSEY, ALISON. THE ESSENCE OF ARTIFICAL INTELLIGENCE.

More information

Current California Math Standards Balanced Equations

Current California Math Standards Balanced Equations Balanced Equations Current California Math Standards Balanced Equations Grade Three Number Sense 1.0 Students understand the place value of whole numbers: 1.1 Count, read, and write whole numbers to 10,000.

More information

Betting systems: how not to lose your money gambling

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

More information

Discrete Structures for Computer Science

Discrete Structures for Computer Science Discrete Structures for Computer Science Adam J. Lee adamlee@cs.pitt.edu 6111 Sennott Square Lecture #20: Bayes Theorem November 5, 2013 How can we incorporate prior knowledge? Sometimes we want to know

More information

Math 141. Lecture 2: More Probability! Albyn Jones 1. jones@reed.edu www.people.reed.edu/ jones/courses/141. 1 Library 304. Albyn Jones Math 141

Math 141. Lecture 2: More Probability! Albyn Jones 1. jones@reed.edu www.people.reed.edu/ jones/courses/141. 1 Library 304. Albyn Jones Math 141 Math 141 Lecture 2: More Probability! Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Outline Law of total probability Bayes Theorem the Multiplication Rule, again Recall

More information

CS 341 Software Design Homework 5 Identifying Classes, UML Diagrams Due: Oct. 22, 11:30 PM

CS 341 Software Design Homework 5 Identifying Classes, UML Diagrams Due: Oct. 22, 11:30 PM CS 341 Software Design Homework 5 Identifying Classes, UML Diagrams Due: Oct. 22, 11:30 PM Objectives To gain experience doing object-oriented design To gain experience developing UML diagrams A Word about

More information

Random variables, probability distributions, binomial random variable

Random variables, probability distributions, binomial random variable Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that

More information

The temporary new rules and amendments authorize casino licensees to. offer a supplemental wager in the game of three card poker known as the three

The temporary new rules and amendments authorize casino licensees to. offer a supplemental wager in the game of three card poker known as the three Progressive Wager and Envy Bonus In Three Card Poker Accounting And Internal Controls Gaming Equipment Rules Of The Game Temporary Amendments: N.J.A.C. 19:45-1.20; 19:46-1.10A; and 19:47-20.1, 20.6, 20.10

More information

Stat 20: Intro to Probability and Statistics

Stat 20: Intro to Probability and Statistics 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

More information

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

Standard 12: The student will explain and evaluate the financial impact and consequences of gambling. TEACHER GUIDE 12.1 GAMBLING PAGE 1 Standard 12: The student will explain and evaluate the financial impact and consequences of gambling. Risky Business Priority Academic Student Skills Personal Financial

More information

Basic Probability Theory II

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

More information

Contemporary Mathematics Online Math 1030 Sample Exam I Chapters 12-14 No Time Limit No Scratch Paper Calculator Allowed: Scientific

Contemporary Mathematics Online Math 1030 Sample Exam I Chapters 12-14 No Time Limit No Scratch Paper Calculator Allowed: Scientific Contemporary Mathematics Online Math 1030 Sample Exam I Chapters 12-14 No Time Limit No Scratch Paper Calculator Allowed: Scientific Name: The point value of each problem is in the left-hand margin. You

More information

Minimax Strategies. Minimax Strategies. Zero Sum Games. Why Zero Sum Games? An Example. An Example

Minimax Strategies. Minimax Strategies. Zero Sum Games. Why Zero Sum Games? An Example. An Example Everyone who has studied a game like poker knows the importance of mixing strategies With a bad hand, you often fold But you must bluff sometimes Lectures in Microeconomics-Charles W Upton Zero Sum Games

More information

Worldwide Casino Consulting Inc.

Worldwide Casino Consulting Inc. Card Count Exercises George Joseph The first step in the study of card counting is the recognition of those groups of cards known as Plus, Minus & Zero. It is important to understand that the House has

More information

MATH 140 Lab 4: Probability and the Standard Normal Distribution

MATH 140 Lab 4: Probability and the Standard Normal Distribution MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes

More information

Know it all. Table Gaming Guide

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

More information

Math 3C Homework 3 Solutions

Math 3C Homework 3 Solutions Math 3C Homework 3 s Ilhwan Jo and Akemi Kashiwada ilhwanjo@math.ucla.edu, akashiwada@ucla.edu Assignment: Section 2.3 Problems 2, 7, 8, 9,, 3, 5, 8, 2, 22, 29, 3, 32 2. You draw three cards from a standard

More information

STATISTICS HIGHER SECONDARY - SECOND YEAR. Untouchability is a sin Untouchability is a crime Untouchability is inhuman

STATISTICS HIGHER SECONDARY - SECOND YEAR. Untouchability is a sin Untouchability is a crime Untouchability is inhuman STATISTICS HIGHER SECONDARY - SECOND YEAR Untouchability is a sin Untouchability is a crime Untouchability is inhuman TAMILNADU TEXTBOOK CORPORATION College Road, Chennai- 600 006 i Government of Tamilnadu

More information

Conditional Probability

Conditional Probability 6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman April 21, 2005 Lecture Notes Conditional Probability Suppose that we pick a random person in the world. Everyone has an equal

More information

Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University

Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University 1 Chapter 1 Probability 1.1 Basic Concepts In the study of statistics, we consider experiments

More information

Gaming the Law of Large Numbers

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.

More information

REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.

REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k. REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game

More information

Sue Fine Linn Maskell

Sue Fine Linn Maskell FUN + GAMES = MATHS Sue Fine Linn Maskell Teachers are often concerned that there isn t enough time to play games in maths classes. But actually there is time to play games and we need to make sure that

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level S2 of challenge: B/C S2 Mathematical goals Starting points Materials required Time needed Evaluating probability statements To help learners to: discuss and clarify some common misconceptions about

More information

Chapter 4 - Practice Problems 2

Chapter 4 - Practice Problems 2 Chapter - Practice Problems 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated probability. 1) If you flip a coin three times, the

More information

Poker. 10,Jack,Queen,King,Ace. 10, Jack, Queen, King, Ace of the same suit Five consecutive ranks of the same suit that is not a 5,6,7,8,9

Poker. 10,Jack,Queen,King,Ace. 10, Jack, Queen, King, Ace of the same suit Five consecutive ranks of the same suit that is not a 5,6,7,8,9 Poker Poker is an ideal setting to study probabilities. Computing the probabilities of different will require a variety of approaches. We will not concern ourselves with betting strategies, however. Our

More information

Chapter 13 & 14 - Probability PART

Chapter 13 & 14 - Probability PART Chapter 13 & 14 - Probability PART IV : PROBABILITY Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14 - Probability 1 / 91 Why Should We Learn Probability Theory? Dr. Joseph

More information

How to Play. Player vs. Dealer

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

More information

STA 371G: Statistics and Modeling

STA 371G: Statistics and Modeling STA 371G: Statistics and Modeling Decision Making Under Uncertainty: Probability, Betting Odds and Bayes Theorem Mingyuan Zhou McCombs School of Business The University of Texas at Austin http://mingyuanzhou.github.io/sta371g

More information

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

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

More information

6th Grade Lesson Plan: Probably Probability

6th Grade Lesson Plan: Probably Probability 6th Grade Lesson Plan: Probably Probability Overview This series of lessons was designed to meet the needs of gifted children for extension beyond the standard curriculum with the greatest ease of use

More information

Probability. a number between 0 and 1 that indicates how likely it is that a specific event or set of events will occur.

Probability. a number between 0 and 1 that indicates how likely it is that a specific event or set of events will occur. Probability Probability Simple experiment Sample space Sample point, or elementary event Event, or event class Mutually exclusive outcomes Independent events a number between 0 and 1 that indicates how

More information

Introduction to Probability

Introduction to Probability 3 Introduction to Probability Given a fair coin, what can we expect to be the frequency of tails in a sequence of 10 coin tosses? Tossing a coin is an example of a chance experiment, namely a process which

More information

Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80)

Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) CS 30, Winter 2016 by Prasad Jayanti 1. (10 points) Here is the famous Monty Hall Puzzle. Suppose you are on a game show, and you

More information

Section 6-5 Sample Spaces and Probability

Section 6-5 Sample Spaces and Probability 492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)

More information

Chapter 4 Lecture Notes

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,

More information

Expected Value and the Game of Craps

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

More information

Playing around with Risks

Playing around with Risks Playing around with Risks Jurgen Cleuren April 19th 2012 2011 CTG, Inc. Introduction Projects are done in a probabilistic environment Incomplete information Parameters change over time What is true in

More information

5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1

5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1 MATH 13150: Freshman Seminar Unit 8 1. Prime numbers 1.1. Primes. A number bigger than 1 is called prime if its only divisors are 1 and itself. For example, 3 is prime because the only numbers dividing

More information

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

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,

More information

PROBABILITY. Chapter. 0009T_c04_133-192.qxd 06/03/03 19:53 Page 133

PROBABILITY. Chapter. 0009T_c04_133-192.qxd 06/03/03 19:53 Page 133 0009T_c04_133-192.qxd 06/03/03 19:53 Page 133 Chapter 4 PROBABILITY Please stand up in front of the class and give your oral report on describing data using statistical methods. Does this request to speak

More information

For two disjoint subsets A and B of Ω, say that A and B are disjoint events. For disjoint events A and B we take an axiom P(A B) = P(A) + P(B)

For two disjoint subsets A and B of Ω, say that A and B are disjoint events. For disjoint events A and B we take an axiom P(A B) = P(A) + P(B) Basic probability A probability space or event space is a set Ω together with a probability measure P on it. This means that to each subset A Ω we associate the probability P(A) = probability of A with

More information

Vieta s Formulas and the Identity Theorem

Vieta s Formulas and the Identity Theorem Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion

More information

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)

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

More information

E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

More information

Prime Time: Homework Examples from ACE

Prime Time: Homework Examples from ACE Prime Time: Homework Examples from ACE Investigation 1: Building on Factors and Multiples, ACE #8, 28 Investigation 2: Common Multiples and Common Factors, ACE #11, 16, 17, 28 Investigation 3: Factorizations:

More information

Homework 3 Solution, due July 16

Homework 3 Solution, due July 16 Homework 3 Solution, due July 16 Problems from old actuarial exams are marked by a star. Problem 1*. Upon arrival at a hospital emergency room, patients are categorized according to their condition as

More information

What Is Probability?

What Is Probability? 1 What Is Probability? The idea: Uncertainty can often be "quantified" i.e., we can talk about degrees of certainty or uncertainty. This is the idea of probability: a higher probability expresses a higher

More information

1/3 1/3 1/3 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0 1 2 3 4 5 6 7 8 0.6 0.6 0.6 0.6 0.6 0.6 0.6

1/3 1/3 1/3 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0 1 2 3 4 5 6 7 8 0.6 0.6 0.6 0.6 0.6 0.6 0.6 HOMEWORK 4: SOLUTIONS. 2. A Markov chain with state space {, 2, 3} has transition probability matrix /3 /3 /3 P = 0 /2 /2 0 0 Show that state 3 is absorbing and, starting from state, find the expected

More information

Math Games For Skills and Concepts

Math Games For Skills and Concepts Math Games p.1 Math Games For Skills and Concepts Original material 2001-2006, John Golden, GVSU permission granted for educational use Other material copyright: Investigations in Number, Data and Space,

More information

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 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

More information

Comparison of frequentist and Bayesian inference. Class 20, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Comparison of frequentist and Bayesian inference. Class 20, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Comparison of frequentist and Bayesian inference. Class 20, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Be able to explain the difference between the p-value and a posterior

More information

Analysis of poker strategies in heads-up poker

Analysis of poker strategies in heads-up poker BMI paper Analysis of poker strategies in heads-up poker Author: Korik Alons Supervisor: Dr. S. Bhulai VU University Amsterdam Faculty of Sciences Study Business Mathematics and Informatics De Boelelaan

More information

PROBABILITY SECOND EDITION

PROBABILITY SECOND EDITION PROBABILITY SECOND EDITION Table of Contents How to Use This Series........................................... v Foreword..................................................... vi Basics 1. Probability All

More information

UNIT 7A 118 CHAPTER 7: PROBABILITY: LIVING WITH THE ODDS

UNIT 7A 118 CHAPTER 7: PROBABILITY: LIVING WITH THE ODDS 11 CHAPTER 7: PROBABILITY: LIVING WITH THE ODDS UNIT 7A TIME OUT TO THINK Pg. 17. Birth orders of BBG, BGB, and GBB are the outcomes that produce the event of two boys in a family of. We can represent

More information

Coin Flip Questions. Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT.

Coin Flip Questions. Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT. Coin Flip Questions Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT. 1 How many ways can you get exactly 1 head? 2 How many ways can you get exactly 2 heads?

More information

Slots... 1. seven card stud...22

Slots... 1. seven card stud...22 GAMING GUIDE table of contents Slots... 1 Blackjack...3 Lucky Ladies...5 Craps...7 Roulette... 13 Three Card Poker... 15 Four Card Poker... 17 Texas Hold em Bonus Poker... 18 omaha Poker... 21 seven card

More information

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

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

More information

Fighting an Almost Perfect Crime

Fighting an Almost Perfect Crime Fighting an Almost Perfect Crime Online Poker Fraud Detection Philipp Schosteritsch, MSc Consultant for Software Development and Technical Advisor on Cyber Security & Privacy Foundation Graz, 2014 Contents

More information

Lecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett

Lecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.

More information

1. General...3. 2. Black Jack...5. 3. Double Deck Black Jack...13. 4. Free Bet Black Jack...20. 5. Craps...28. 6. Craps Free Craps...

1. General...3. 2. Black Jack...5. 3. Double Deck Black Jack...13. 4. Free Bet Black Jack...20. 5. Craps...28. 6. Craps Free Craps... Table of Contents Sec Title Page # 1. General...3 2. Black Jack...5 3. Double Deck Black Jack...13 4. Free Bet Black Jack...20 5. Craps...28 6. Craps Free Craps...36 7. Roulette...43 8. Poker...49 9. 3-Card

More information