Math Steven Noble. November 24th. Steven Noble Math 3790


 Iris Strickland
 4 years ago
 Views:
Transcription
1 Math 3790 Steven Noble November 24th
2 The Rules of Craps In the game of craps you roll two dice then, if the total is 7 or 11, you win, if the total is 2, 3, or 12, you lose, In the other cases (when the sum is 4, 5, 6, 8, 9 or 10), your sum is referred to as your point. You get the dice again. Now you keep rolling the dice until the sum is either 7, in which case you lose, or the sum is equal to your point, in which case you win.
3 Probability of winning Let us calculate the probability of winning at the game of craps.
4 Probability of winning Let us calculate the probability of winning at the game of craps. The probability of rolling a 7 with two dice is 6 36, since there are 36 possible outcomes for the two dice, and there are six desired outcomes, i.e., those that add up to seven, namely (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1). Similarly, the probability of rolling an 11 with two dice is 2 36, since the only ways to roll 11 are (5, 6) and (6, 5).
5 Probability of winning Let us calculate the probability of winning at the game of craps. The probability of rolling a 7 with two dice is 6 36, since there are 36 possible outcomes for the two dice, and there are six desired outcomes, i.e., those that add up to seven, namely (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1). Similarly, the probability of rolling an 11 with two dice is 2 36, since the only ways to roll 11 are (5, 6) and (6, 5). Therefore, the probability of winning on the first roll is = 8 36 = 2 9.
6 What if you don t win right away? Now, let s examine what happens when we roll a 4, 5, 6, 8, 9 or 10. Let s consider each case separately. First, consider the case when the point is 4.
7 What if you don t win right away? Now, let s examine what happens when we roll a 4, 5, 6, 8, 9 or 10. Let s consider each case separately. First, consider the case when the point is 4. We continue to roll the dice until the sum is either 4 (in which case we win), or roll a 7 (in which case we lose). We know that the game does not end until either of these two scenarios occur, so we want to determine the probability that the sum is 4 given that either the sum 4 or 7 has occurred. To elaborate, to find out the probability of winning the game when the point is 4, this simply the probability that we roll 4 before we roll 7. So, this is the same as saying, what is the probability that we roll a 4 given that we roll either a 4 or a 7.
8 Conditional Probability This brings us back to conditional probability. We say that P(A B) represents the probability that event A occurred given that event B occurred.
9 Conditional Probability This brings us back to conditional probability. We say that P(A B) represents the probability that event A occurred given that event B occurred. There is a nice formula for conditional probability: P(A B) = P(A B). P(B) (Here P(A B) is the probability that both A and B will occur.)
10 Returning to our example So in our example, let A be the event that the sum of the dice is 4, and let B be the event that the sum of the dice is either 4 or 7. We wish to find P(A B). Well, A B is simply A, namely the event that the sum is 4. Hence P(A B) = P(A) = 3 36, and P(B) = = 1 4, since there are three ways to roll a 3 with two dice, and six ways to roll a 7.
11 Returning to our example So in our example, let A be the event that the sum of the dice is 4, and let B be the event that the sum of the dice is either 4 or 7. We wish to find P(A B). Well, A B is simply A, namely the event that the sum is 4. Hence P(A B) = P(A) = 3 36, and P(B) = = 1 4, since there are three ways to roll a 3 with two dice, and six ways to roll a 7. Hence, P(Roll 4 Roll 4 or 7) = 3/36 9/36 = 1 3.
12 The rest of the cases Similarly, we find that P(Roll 5 Roll 5 or 7) = 4/36 10/36 = 2 5. P(Roll 6 Roll 6 or 7) = 5/36 11/36 = P(Roll 8 Roll 8 or 7) = 5/36 11/36 = P(Roll 9 Roll 9 or 7) = 4/36 10/36 = 2 5. P(Roll 10 Roll 10 or 7) = 3/36 9/36 = 1 3.
13 Putting it together Hence, the probability of winning when our first roll is 4 is P(Initial Roll Is 4) P(Roll 4 Roll 4 or 7) = = 1 36.
14 Putting it together Hence, the probability of winning when our first roll is 4 is P(Initial Roll Is 4) P(Roll 4 Roll 4 or 7) = = Similarly, we have P(Initial Roll Is 5) P(Roll 5 Roll 5 or 7) = = 2 45, P(Initial Roll Is 6) P(Roll 6 Roll 6 or 7) = 5 36 P(Initial Roll Is 8) P(Roll 8 Roll 8 or 7) = = , 5 11 = , P(Initial Roll Is 9) P(Roll 9 Roll 9 or 7) = = 2 45, P(Initial Roll Is 10) P(Roll 10 Roll 10 or 7) = = 1 36.
15 Ok, now we ll really put it all together So all of these fractions above represent the probabilities of winning when the first roll is 4, 5, 6, 8, 9 and 10. Notice the symmetry of the numbers. So this is saying that the probability of winning by rolling 10 on your first roll is only 1 36, which is about three percent. To find the probability of winning at the game, we just add all these probabilities: = Since is approximately 49.3 percent, the probability of winning at this game is just under fifty percent.
16 The value of playing Let us say the casino lets you bet money on the Pass Line, or on the Don t Pass Line. If you bet 5 of the Pass Line, you win 5 dollars if the roller wins the game. If you bet 5 on the Don t Pass Line, you win 5 dollars if the roller loses the game. Since we ve just shown the probability of winning the game is less than fifty percent, it is statistically better to always bet on the Don t Pass Line  in other words, always bet that the roller will lose.
17 The value of playing Let us say the casino lets you bet money on the Pass Line, or on the Don t Pass Line. If you bet 5 of the Pass Line, you win 5 dollars if the roller wins the game. If you bet 5 on the Don t Pass Line, you win 5 dollars if the roller loses the game. Since we ve just shown the probability of winning the game is less than fifty percent, it is statistically better to always bet on the Don t Pass Line  in other words, always bet that the roller will lose. To find out the expected value of a decision you add up each each possible outcome times the probability of the outcome. So if you bet on Pass Line the expected value is (5 dollars) ( ) + ( 5 dollars) ( ) 251 = and the expected value of betting on Don t Pass Line is ( ) ( ) ( 5 dollars) + (5 dollars) =
18 Another Value So if you were to bet on the Don t Pass Line one thousand times, statistically you would expect to win about 70 dollars. But what if instead of winning 5 dollars you only win 4 dollars when you bet 5 dollars on Don t Pass Line. The the expected value is ( 5 dollars) ( ) + (4 dollars) ( ) = So no matter whether you choose to bet on Pass Line or Don t Pass Line you still have a negative expected value. However there is a third option of not playing that has an expected value of 0.
19 Monty Hall A (female) contestant on a game show is shown three doors by the (male) host. Behind one of these there is a car, and behind each of the other two there is a goat. She chooses one of the doors, hoping of course to get the car. Before the door is opened, the host (who knows what s behind each door) opens one of the remaining two doors to reveal a goat. At this point he offers her the chance to switch her choice if she so wishes. Should she switch, and if so, how does that change the probability of winning a car?
20 Monty Hall Solution Let us say that A is the event that the car is behind door A (the door she chooses); B is the event that the car is behind door B; and C is the event that the car is behind door C. Finally we will say that b is the event that the host opens door B.
21 Monty Hall Solution Let us say that A is the event that the car is behind door A (the door she chooses); B is the event that the car is behind door B; and C is the event that the car is behind door C. Finally we will say that b is the event that the host opens door B. What we want to know is P(A b) and P(C b). For this we enlist another formula. If A 1, A 2,..., A n are all the possible events that can happen, and X is some other event, then P(A 1 X ) = P(X A 1) P(X A 1 ) + P(X A 2) P(X A n ).
22 Applying this theorem So here our A i s are A, B, C and X is b. If the car is behind door A then there is a 0.5 chance that he ll open door C and a 0.5 chance that he ll open door B (this is because there are two scenarios and we don t have any extra knowledge to distinguish between the two). So P(b A) = 0.5. If the car is behind door B then the host must open door C so P(b B) = 0. And if the car is behind door C then the host must open door B so P(b C) = 1.
23 Applying this theorem So here our A i s are A, B, C and X is b. If the car is behind door A then there is a 0.5 chance that he ll open door C and a 0.5 chance that he ll open door B (this is because there are two scenarios and we don t have any extra knowledge to distinguish between the two). So P(b A) = 0.5. If the car is behind door B then the host must open door C so P(b B) = 0. And if the car is behind door C then the host must open door B so P(b C) = 1. Plugging into the formula we get that P(A b) = 1/2 1 = 1/3, P(C b) = 3/2 3/2 = 2/3.
24 Applying this theorem So here our A i s are A, B, C and X is b. If the car is behind door A then there is a 0.5 chance that he ll open door C and a 0.5 chance that he ll open door B (this is because there are two scenarios and we don t have any extra knowledge to distinguish between the two). So P(b A) = 0.5. If the car is behind door B then the host must open door C so P(b B) = 0. And if the car is behind door C then the host must open door B so P(b C) = 1. Plugging into the formula we get that P(A b) = 1/2 1 = 1/3, P(C b) = 3/2 3/2 = 2/3. So seeing as the car is more likely to be behind door C than A it makes more sense to switch than to stay where you are.
25 Sample problem 1 You are given one true or false question on a test. If you get the question right, your final mark is increased by one percent, but if you get the question wrong, your final mark is decreased by one percent. You can guess, or you can copy from your neighbour, but you know from experience that you you neighbour answers incorrectly 20% of the time. Besides, there is a 10% chance that the teacher would catch you cheating, and in this case, she would deduct five percent from your final mark. Moral issues aside what should you do: copy from your neighbour, or randomly guess on the question?
26 Sample problem 2 Two players alternately shoot themselves with a sixshooter, only one chamber of which contains a bullet. (This is called Russian Roulette). You have the first shot, so you decide the rules. Either both players take turns shooting the next chamber, or both players randomly spin the chamber before shooting. Naturally, you want to maximize your chances of surviving. Should you spin first and shoot, or shoot without spinning?
27 Sample problem 3 Alison, Bernon, and Chantel play the following game. They take turns (in the order A, B, C, A, B, C,...) rolling one die. Alison wins if she rolls 1, 2, or 3 on her turn. Bernon wins if he rolls 4 or 5 on his turn. Chantel wins if she rolls 6 on her turn. They keep repeating this until there is a winner. What is the probability that Alison wins the game? For example if Alison rolls a 5 then it is Bernon s turn. Say her rolls a 6. Then it s Chantel s turn. Say she rolls 6. Then Chantel wins.
Math Steven Noble. November 22nd. Steven Noble Math 3790
Math 3790 Steven Noble November 22nd Basic ideas of combinations and permutations Simple Addition. If there are a varieties of soup and b varieties of salad then there are a + b possible ways to order
More informationCasino Lab AP Statistics
Casino Lab AP Statistics Casino games are governed by the laws of probability (and those enacted by politicians, too). The same laws (probabilistic, not political) rule the entire known universe. If the
More informationSTATION 1: ROULETTE. Name of Guesser Tally of Wins Tally of Losses # of Wins #1 #2
Casino Lab 2017  ICM The House Always Wins! 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
More informationJunior Circle Meeting 5 Probability. May 2, ii. In an actual experiment, can one get a different number of heads when flipping a coin 100 times?
Junior Circle Meeting 5 Probability May 2, 2010 1. We have a standard coin with one side that we call heads (H) and one side that we call tails (T). a. Let s say that we flip this coin 100 times. i. How
More informationProbability Paradoxes
Probability Paradoxes Washington University Math Circle February 20, 2011 1 Introduction We re all familiar with the idea of probability, even if we haven t studied it. That is what makes probability so
More informationSection Summary. Finite Probability Probabilities of Complements and Unions of Events Probabilistic Reasoning
Section 7.1 Section Summary Finite Probability Probabilities of Complements and Unions of Events Probabilistic Reasoning Probability of an Event PierreSimon Laplace (17491827) We first study PierreSimon
More information4.2.4 What if both events happen?
4.2.4 What if both events happen? Unions, Intersections, and Complements In the mid 1600 s, a French nobleman, the Chevalier de Mere, was wondering why he was losing money on a bet that he thought was
More informationCSC/MTH 231 Discrete Structures II Spring, Homework 5
CSC/MTH 231 Discrete Structures II Spring, 2010 Homework 5 Name 1. A six sided die D (with sides numbered 1, 2, 3, 4, 5, 6) is thrown once. a. What is the probability that a 3 is thrown? b. What is the
More informationGuide. Odds. Understanding. The THE HOUSE ADVANTAGE
THE HOUSE ADVANTAGE A Guide The Odds to Understanding AMERICAN GAMING ASSOCIATION 1299 Pennsylvania Avenue, NW Suite 1175 Washington, DC 20004 2025522675 www.americangaming.org 2005 American Gaming Association.
More informationThe Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you $5 that if you give me $10, I ll give you $20.)
The Teachers Circle Mar. 2, 22 HOW TO GAMBLE IF YOU MUST (I ll bet you $ that if you give me $, I ll give you $2.) Instructor: Paul Zeitz (zeitzp@usfca.edu) Basic Laws and Definitions of Probability If
More informationOUTSIDE IOWA, CALL
WWW.1800BETSOFF.ORG OUTSIDE IOWA, CALL 18005224700 IOWA DEPARTMENT OF PUBLIC HEALTH, GAMBLING TREATMENT PROGRAM PROMOTING AND PROTECTING THE HEALTH OF IOWANS Printing is made possible with money from
More informationChapter 7 Homework Problems. 1. If a carefully made die is rolled once, it is reasonable to assign probability 1/6 to each of the six faces.
Chapter 7 Homework Problems 1. If a carefully made die is rolled once, it is reasonable to assign probability 1/6 to each of the six faces. A. What is the probability of rolling a number less than 3. B.
More informationCounting and Probability
Counting and Probability Lecture 42 Section 9.1 Robb T. Koether HampdenSydney College Wed, Apr 9, 2014 Robb T. Koether (HampdenSydney College) Counting and Probability Wed, Apr 9, 2014 1 / 17 1 Probability
More informationStatistics 1040 Summer 2009 Exam III
Statistics 1040 Summer 2009 Exam III 1. For the following basic probability questions. Give the RULE used in the appropriate blank (BEFORE the question), for each of the following situations, using one
More informationStatistics Intermediate Probability
Session 6 oscardavid.barrerarodriguez@sciencespo.fr April 3, 2018 and Sampling from a Population Outline 1 The Monty Hall Paradox Some Concepts: Event Algebra Axioms and Things About that are True Counting
More informationTable Games Rules. MargaritavilleBossierCity.com FIN CITY GAMBLING PROBLEM? CALL
Table Games Rules MargaritavilleBossierCity.com 1 855 FIN CITY facebook.com/margaritavillebossiercity twitter.com/mville_bc GAMBLING PROBLEM? CALL 8005224700. Blackjack Hands down, Blackjack is the most
More informationIf event A is more likely than event B, then the probability of event A is higher than the probability of event B.
Unit, Lesson. Making Decisions Probabilities have a wide range of applications, including determining whether a situation is fair or not. A situation is fair if each outcome is equally likely. In this
More information2. The value of the middle term in a ranked data set is called: A) the mean B) the standard deviation C) the mode D) the median
1. An outlier is a value that is: A) very small or very large relative to the majority of the values in a data set B) either 100 units smaller or 100 units larger relative to the majority of the values
More informationCS 361: Probability & Statistics
February 7, 2018 CS 361: Probability & Statistics Independence & conditional probability Recall the definition for independence So we can suppose events are independent and compute probabilities Or we
More informationProbability and the Monty Hall Problem Rong Huang January 10, 2016
Probability and the Monty Hall Problem Rong Huang January 10, 2016 Warmup: There is a sequence of number: 1, 2, 4, 8, 16, 32, 64, How does this sequence work? How do you get the next number from the previous
More informationSTAT 311 (Spring 2016) Worksheet: W3W: Independence due: Mon. 2/1
Name: Group 1. For all groups. It is important that you understand the difference between independence and disjoint events. For each of the following situations, provide and example that is not in the
More information, x {1, 2, k}, where k > 0. (a) Write down P(X = 2). (1) (b) Show that k = 3. (4) Find E(X). (2) (Total 7 marks)
1. The probability distribution of a discrete random variable X is given by 2 x P(X = x) = 14, x {1, 2, k}, where k > 0. Write down P(X = 2). (1) Show that k = 3. Find E(X). (Total 7 marks) 2. In a game
More informationCh Probability Outcomes & Trials
Learning Intentions: Ch. 10.2 Probability Outcomes & Trials Define the basic terms & concepts of probability. Find experimental probabilities. Calculate theoretical probabilities. Vocabulary: Trial: realworld
More information2 A fair coin is flipped 8 times. What is the probability of getting more heads than tails? A. 1 2 B E. NOTA
For all questions, answer E. "NOTA" means none of the above answers is correct. Calculator use NO calculators will be permitted on any test other than the Statistics topic test. The word "deck" refers
More informationIntroduction to Probability
6.04/8.06J Mathematics for omputer Science Srini Devadas and Eric Lehman pril 4, 005 Lecture Notes Introduction to Probability Probability is the last topic in this course and perhaps the most important.
More informationLISTING THE WAYS. getting a total of 7 spots? possible ways for 2 dice to fall: then you win. But if you roll. 1 q 1 w 1 e 1 r 1 t 1 y
LISTING THE WAYS A pair of dice are to be thrown getting a total of 7 spots? There are What is the chance of possible ways for 2 dice to fall: 1 q 1 w 1 e 1 r 1 t 1 y 2 q 2 w 2 e 2 r 2 t 2 y 3 q 3 w 3
More informationStatistics Laboratory 7
Pass the Pigs TM Statistics 104  Laboratory 7 On last weeks lab we looked at probabilities associated with outcomes of the game Pass the Pigs TM. This week we will look at random variables associated
More informationSECONDARY 2 Honors ~ Lesson 9.2 Worksheet Intro to Probability
SECONDARY 2 Honors ~ Lesson 9.2 Worksheet Intro to Probability Name Period Write all probabilities as fractions in reduced form! Use the given information to complete problems 13. Five students have the
More informationMath 3201 Midterm Chapter 3
Math 3201 Midterm Chapter 3 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which expression correctly describes the experimental probability P(B), where
More informationSIC BO ON THE MULTI TERMINALS
How to play SIC BO ON THE MULTI TERMINALS LET S PLAY SIC BO Sic Bo is a Chinese dice game with a history dating back centuries. Originally played using painted bricks, modern Sic Bo has evolved into the
More informationChapter 3: Probability (Part 1)
Chapter 3: Probability (Part 1) 3.1: Basic Concepts of Probability and Counting Types of Probability There are at least three different types of probability Subjective Probability is found through people
More informationProbability. A Mathematical Model of Randomness
Probability A Mathematical Model of Randomness 1 Probability as Long Run Frequency In the eighteenth century, Compte De Buffon threw 2048 heads in 4040 coin tosses. Frequency = 2048 =.507 = 50.7% 4040
More informationMATHEMATICS E102, FALL 2005 SETS, COUNTING, AND PROBABILITY Outline #1 (Probability, Intuition, and Axioms)
MATHEMATICS E102, FALL 2005 SETS, COUNTING, AND PROBABILITY Outline #1 (Probability, Intuition, and Axioms) Last modified: September 19, 2005 Reference: EP(Elementary Probability, by Stirzaker), Chapter
More informationDISCUSSION #8 FRIDAY MAY 25 TH Sophie Engle (Teacher Assistant) ECS20: Discrete Mathematics
DISCUSSION #8 FRIDAY MAY 25 TH 2007 Sophie Engle (Teacher Assistant) ECS20: Discrete Mathematics 2 Homework 8 Hints and Examples 3 Section 5.4 Binomial Coefficients Binomial Theorem 4 Example: j j n n
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 13
CS 70 Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 13 Introduction to Discrete Probability In the last note we considered the probabilistic experiment where we flipped a
More information3.2 Measures of Central Tendency
Math 166 Lecture Notes  S. Nite 9/22/2012 Page 1 of 5 3.2 Measures of Central Tendency Mean The average, or mean, of the n numbers x = x 1 + x 2 +... + x n n x1,x2,...,xn is x (read x bar ), where Example
More information6. a) Determine the probability distribution. b) Determine the expected sum of two dice. c) Repeat parts a) and b) for the sum of
d) generating a random number between 1 and 20 with a calculator e) guessing a person s age f) cutting a card from a wellshuffled deck g) rolling a number with two dice 3. Given the following probability
More informationProbability of Independent and Dependent Events. CCM2 Unit 6: Probability
Probability of Independent and Dependent Events CCM2 Unit 6: Probability Independent and Dependent Events Independent Events: two events are said to be independent when one event has no affect on the probability
More informationProbability Homework Pack 1
Dice 2 Probability Homework Pack 1 Probability Investigation: SKUNK In the game of SKUNK, we will roll 2 regular 6sided dice. Players receive an amount of points equal to the total of the two dice, unless
More informationMULTI TERMINAL TABLE GAMES ROULETTE BACCARAT
Getting started with MULTI TERMINAL TABLE GAMES ROULETTE BACCARAT LET S PLAY Multi Terminal Table Games are the newest and most exciting addition to our promise of offering you unforgettable gaming experiences.
More information[Independent Probability, Conditional Probability, Tree Diagrams]
Name: Year 1 Review 119 Topic: Probability Day 2 Use your formula booklet! Page 5 Lesson 118: Probability Day 1 [Independent Probability, Conditional Probability, Tree Diagrams] Read and Highlight Station
More information02. Probability: Intuition  Ambiguity  Absurdity  Puzzles
University of Rhode Island DigitalCommons@URI Nonequilibrium Statistical Physics Physics Course Materials 10192015 02. Probability: Intuition  Ambiguity  Absurdity  Puzzles Gerhard Müller University
More informationName: Probability, Part 1 March 4, 2013
1) Assuming all sections are equal in size, what is the probability of the spinner below stopping on a blue section? Write the probability as a fraction. 2) A bag contains 3 red marbles, 4 blue marbles,
More informationExam III Review Problems
c Kathryn Bollinger and Benjamin Aurispa, November 10, 2011 1 Exam III Review Problems Fall 2011 Note: Not every topic is covered in this review. Please also take a look at the previous WeekinReviews
More informationPresentation by Toy Designers: Max Ashley
A new game for your toy company Presentation by Toy Designers: Shawntee Max Ashley As game designers, we believe that the new game for your company should: Be equally likely, giving each player an equal
More informationCS1802 Week 9: Probability, Expectation, Entropy
CS02 Discrete Structures Recitation Fall 207 October 30  November 3, 207 CS02 Week 9: Probability, Expectation, Entropy Simple Probabilities i. What is the probability that if a die is rolled five times,
More informationEx 1: A coin is flipped. Heads, you win $1. Tails, you lose $1. What is the expected value of this game?
AFM Unit 7 Day 5 Notes Expected Value and Fairness Name Date Expected Value: the weighted average of possible values of a random variable, with weights given by their respective theoretical probabilities.
More informationMath 147 Lecture Notes: Lecture 21
Math 147 Lecture Notes: Lecture 21 Walter Carlip March, 2018 The Probability of an Event is greater or less, according to the number of Chances by which it may happen, compared with the whole number of
More informationMath 152: Applicable Mathematics and Computing
Math 152: Applicable Mathematics and Computing May 8, 2017 May 8, 2017 1 / 15 Extensive Form: Overview We have been studying the strategic form of a game: we considered only a player s overall strategy,
More informationMath 106 Lecture 3 Probability  Basic Terms Combinatorics and Probability  1 Odds, Payoffs Rolling a die (virtually)
Math 106 Lecture 3 Probability  Basic Terms Combinatorics and Probability  1 Odds, Payoffs Rolling a die (virtually) m j winter, 00 1 Description We roll a sixsided die and look to see whether the face
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Math 1342 Practice Test 2 Ch 4 & 5 Name 1) Nanette must pass through three doors as she walks from her company's foyer to her office. Each of these doors may be locked or unlocked. 1) List the outcomes
More informationDiscrete Random Variables Day 1
Discrete Random Variables Day 1 What is a Random Variable? Every probability problem is equivalent to drawing something from a bag (perhaps more than once) Like Flipping a coin 3 times is equivalent to
More informationChapter 6: Probability and Simulation. The study of randomness
Chapter 6: Probability and Simulation The study of randomness Introduction Probability is the study of chance. 6.1 focuses on simulation since actual observations are often not feasible. When we produce
More informationContents of this Document [ntc2]
Contents of this Document [ntc2] 2. Probability: Intuition  Ambiguity  Absurdity  Puzzles Regular versus random schedules [nln40] Pick the winning die [nex2] Educated guess [nex4] Coincident birthdays
More informationUnit 9: Probability Assignments
Unit 9: Probability Assignments #1: Basic Probability In each of exercises 1 & 2, find the probability that the spinner shown would land on (a) red, (b) yellow, (c) blue. 1. 2. Y B B Y B R Y Y B R 3. Suppose
More informationGeometry: Shapes, Symmetry, Area and Number PROBLEMS & INVESTIGATIONS
Overhead 0 Geometry: Shapes, Symmetry, Area and Number Session 5 PROBLEMS & INVESTIGATIONS Overview Using transparent pattern blocks on the overhead, the teacher introduces a new game called Caterpillar
More informationThreePrisoners Puzzle. The rest of the course. The Monty Hall Puzzle. The SecondAce Puzzle
The rest of the course ThreePrisoners Puzzle Subtleties involved with maximizing expected utility: Finding the right state space: The wrong state space leads to intuitively incorrect answers when conditioning
More informationBouncy Dice Explosion
Bouncy Dice Explosion The Big Idea This week you re going to toss bouncy rubber dice to see what numbers you roll. You ll also play War to see who s the high roller. Finally, you ll move onto a giant human
More informationExpectation Variance Discrete Structures
Expectation Variance 1 Markov Inequality Y random variable, Y(s) 0, then P( Y x) E(Y)/x Andrei Andreyevich Markov 18561922 2 Chebyshev Inequality Y random variable, then P( YE(Y) x) V(Y)/x 2 Pafnuty
More informationProbability with Set Operations. MATH 107: Finite Mathematics University of Louisville. March 17, Complicated Probability, 17th century style
Probability with Set Operations MATH 107: Finite Mathematics University of Louisville March 17, 2014 Complicated Probability, 17th century style 2 / 14 Antoine Gombaud, Chevalier de Méré, was fond of gambling
More information1. More on Binomial Distributions
Math 25Introductory Statistics Lecture 9/27/06. More on Binomial Distributions When we toss a coin four times, and we compute the probability of getting three heads, for example, we need to know how many
More informationMTH 103 H Final Exam. 1. I study and I pass the course is an example of a. (a) conjunction (b) disjunction. (c) conditional (d) connective
MTH 103 H Final Exam Name: 1. I study and I pass the course is an example of a (a) conjunction (b) disjunction (c) conditional (d) connective 2. Which of the following is equivalent to (p q)? (a) p q (b)
More information1. A factory makes calculators. Over a long period, 2 % of them are found to be faulty. A random sample of 100 calculators is tested.
1. A factory makes calculators. Over a long period, 2 % of them are found to be faulty. A random sample of 0 calculators is tested. Write down the expected number of faulty calculators in the sample. Find
More informationSection 6.5 Conditional Probability
Section 6.5 Conditional Probability Example 1: An urn contains 5 green marbles and 7 black marbles. Two marbles are drawn in succession and without replacement from the urn. a) What is the probability
More informationFor question 1 n = 5, we let the random variable (Y) represent the number out of 5 who get a heart attack, p =.3, q =.7 5
1 Math 321 Lab #4 Note: answers may vary slightly due to rounding. 1. Big Grack s used car dealership reports that the probabilities of selling 1,2,3,4, and 5 cars in one week are 0.256, 0.239, 0.259,
More informationRestricted Choice In Bridge and Other Related Puzzles
Restricted Choice In Bridge and Other Related Puzzles P. Tobias, 9/4/2015 Before seeing how the principle of Restricted Choice can help us play suit combinations better let s look at the best way (in order
More informationModule 5: Probability and Randomness Practice exercises
Module 5: Probability and Randomness Practice exercises PART 1: Introduction to probability EXAMPLE 1: Classify each of the following statements as an example of exact (theoretical) probability, relative
More informationMath 1342 Exam 2 Review
Math 1342 Exam 2 Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) If a sportscaster makes an educated guess as to how well a team will do this
More informationMathacle. Name: Date:
Quiz Probability 1.) A telemarketer knows from past experience that when she makes a call, the probability that someone will answer the phone is 0.20. What is probability that the next two phone calls
More informationBasic Probability Concepts
6.1 Basic Probability Concepts How likely is rain tomorrow? What are the chances that you will pass your driving test on the first attempt? What are the odds that the flight will be on time when you go
More informationMath : Probabilities
20 20. Probability EPProgram  Strisuksa School  Roiet Math : Probabilities Dr.Wattana Toutip  Department of Mathematics Khon Kaen University 200 :Wattana Toutip wattou@kku.ac.th http://home.kku.ac.th/wattou
More informationChapter 3: PROBABILITY
Chapter 3 Math 3201 1 3.1 Exploring Probability: P(event) = Chapter 3: PROBABILITY number of outcomes favourable to the event total number of outcomes in the sample space An event is any collection of
More informationQuant Interview Guide
Quant Interview Guide This guide is the property of CanaryWharfian.co.uk, and any replication or reselling of this guide without the consent of the owners of the aforementioned site is strictly prohibited.
More informationMake better decisions. Learn the rules of the game before you play.
BLACKJACK BLACKJACK Blackjack, also known as 21, is a popular casino card game in which players compare their hand of cards with that of the dealer. To win at Blackjack, a player must create a hand with
More informationBouncy Dice Explosion
The Big Idea Bouncy Dice Explosion This week you re going to toss bouncy rubber dice to see what numbers you roll. You ll also play War to see who s the high roller. Finally, you ll move onto a giant human
More informationCraps Wizard App Quick Start Guide
Craps Wizard App Quick Start Guide Most Control Throw Dice Shooters will have what they need to start using this App at home. But if you are just starting out, you need to do a lot more steps that are
More informationProbability (Devore Chapter Two)
Probability (Devore Chapter Two) 101635101 Probability Winter 20112012 Contents 1 Axiomatic Probability 2 1.1 Outcomes and Events............................... 2 1.2 Rules of Probability................................
More informationFoundations to Algebra In Class: Investigating Probability
Foundations to Algebra In Class: Investigating Probability Name Date How can I use probability to make predictions? Have you ever tried to predict which football team will win a big game? If so, you probably
More informationMath 146 Statistics for the Health Sciences Additional Exercises on Chapter 3
Math 46 Statistics for the Health Sciences Additional Exercises on Chapter 3 Student Name: Find the indicated probability. ) If you flip a coin three times, the possible outcomes are HHH HHT HTH HTT THH
More informationProbability. March 06, J. Boulton MDM 4U1. P(A) = n(a) n(s) Introductory Probability
Most people think they understand odds and probability. Do you? Decision 1: Pick a card Decision 2: Switch or don't Outcomes: Make a tree diagram Do you think you understand probability? Probability Write
More informationChapter 6: Probability and Simulation. The study of randomness
Chapter 6: Probability and Simulation The study of randomness 6.1 Randomness Probability describes the pattern of chance outcomes. Probability is the basis of inference Meaning, the pattern of chance outcomes
More informationGrade 7/8 Math Circles February 25/26, Probability
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Probability Grade 7/8 Math Circles February 25/26, 2014 Probability Centre for Education in Mathematics and Computing Probability is the study of how likely
More informationNorth Seattle Community College Winter ELEMENTARY STATISTICS 2617 MATH Section 05, Practice Questions for Test 2 Chapter 3 and 4
North Seattle Community College Winter 2012 ELEMENTARY STATISTICS 2617 MATH 109  Section 05, Practice Questions for Test 2 Chapter 3 and 4 1. Classify each statement as an example of empirical probability,
More informationBell Work. WarmUp Exercises. Two sixsided dice are rolled. Find the probability of each sum or 7
WarmUp Exercises Two sixsided dice are rolled. Find the probability of each sum. 1. 7 Bell Work 2. 5 or 7 3. You toss a coin 3 times. What is the probability of getting 3 heads? WarmUp Notes Exercises
More informationINDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2
INDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2 WARM UP Students in a mathematics class pick a card from a standard deck of 52 cards, record the suit, and return the card to the deck. The results
More informationHazard: The Scientist s Analysis of the Game.
Lake Forest College Lake Forest College Publications FirstYear Writing Contest Spring 2003 Hazard: The Scientist s Analysis of the Game. Kaloian Petkov Follow this and additional works at: https://publications.lakeforest.edu/firstyear_writing_contest
More informationRandom Variables. A Random Variable is a rule that assigns a number to each outcome of an experiment.
Random Variables When we perform an experiment, we are often interested in recording various pieces of numerical data for each trial. For example, when a patient visits the doctor s office, their height,
More informationGrade 8 Math Assignment: Probability
Grade 8 Math Assignment: Probability Part 1: Rock, Paper, Scissors  The Study of Chance Purpose An introduction of the basic information on probability and statistics Materials: Two sets of hands Paper
More informationAlgebra 2 Notes Section 10.1: Apply the Counting Principle and Permutations
Algebra 2 Notes Section 10.1: Apply the Counting Principle and Permutations Objective(s): Vocabulary: I. Fundamental Counting Principle: Two Events: Three or more Events: II. Permutation: (top of p. 684)
More information1. The chance of getting a flush in a 5card poker hand is about 2 in 1000.
CS 70 Discrete Mathematics for CS Spring 2008 David Wagner Note 15 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette wheels. Today
More informationor More Events Activities D2.1 Open and Shut Case D2.2 Fruit Machines D2.3 Birthdays Notes for Solutions (1 page)
D2 Probability of Two or More Events Activities Activities D2.1 Open and Shut Case D2.2 Fruit Machines D2.3 Birthdays Notes for Solutions (1 page) ACTIVITY D2.1 Open and Shut Case In a Game Show in America,
More informationProbability and Statistics  Grade 5
Probability and Statistics  Grade 5. If you were to draw a single card from a deck of 52 cards, what is the probability of getting a card with a prime number on it? (Answer as a reduced fraction.) 2.
More informationA Probability Work Sheet
A Probability Work Sheet October 19, 2006 Introduction: Rolling a Die Suppose Geoff is given a fair sixsided die, which he rolls. What are the chances he rolls a six? In order to solve this problem, we
More informationGrades 46 Teacher Math Workshop SAGE Conference Session for MAME Winnipeg MB Canada October 19
Grades 46 Teacher Math Workshop SAGE Conference Session for MAME Winnipeg MB Canada October 19 Contents of this handout copyright protected by Box Cars And OneEyed Jacks Inc, No Sweat Education Inc.,
More informationFraction Race. Skills: Fractions to sixths (proper fractions) [Can be adapted for improper fractions]
Skills: Fractions to sixths (proper fractions) [Can be adapted for improper fractions] Materials: Dice (2 different colored dice, if possible) *It is important to provide students with fractional manipulatives
More informationRandom Variables. Outcome X (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5. (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) }
Random Variables When we perform an experiment, we are often interested in recording various pieces of numerical data for each trial. For example, when a patient visits the doctor s office, their height,
More informationMonty Hall Problem & Birthday Paradox
Monty Hall Problem & Birthday Paradox Hanqiu Peng Abstract There are many situations that our intuitions lead us to the wrong direction, especially when we are solving some probability problems. In this
More informationPart 1: I can express probability as a fraction, decimal, and percent
Name: Pattern: Part 1: I can express probability as a fraction, decimal, and percent For #1 to #4, state the probability of each outcome. Write each answer as a) a fraction b) a decimal c) a percent Example:
More information1 of 5 7/16/2009 6:57 AM Virtual Laboratories > 13. Games of Chance > 1 2 3 4 5 6 7 8 9 10 11 3. Simple Dice Games In this section, we will analyze several simple games played with dicepoker dice, chuckaluck,
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Study Guide for Test III (MATH 1630) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the number of subsets of the set. 1) {x x is an even
More information