Some Unusual Applications of Math
|
|
|
- Allen Hancock
- 5 years ago
- Views:
Transcription
1 Some Unusual Applications of Math Ron Gould Emory University Supported by Heilbrun Distinguished Emeritus Fellowship October 7, 2017
2 Game 1 - Three Card Game The Tools: A man has three cards, one red on both sides, one blue on both sides, and one blue on one side and red on the other side. Side 1 Side 2
3 Game 1 - Three Card Game The Tools: A man has three cards, one red on both sides, one blue on both sides, and one blue on one side and red on the other side. Side 1 Side 2
4 Game 1 - Three Card Game The Tools: A man has three cards, one red on both sides, one blue on both sides, and one blue on one side and red on the other side. Side 1 Side 2
5 Play The Game: He places the three cards into a hat and asks you to blindly select one card and only look at one side of the card. You show him that side of the card and he offers to bet even money he can tell you the color of the other side of the card.
6 Play The Game: He places the three cards into a hat and asks you to blindly select one card and only look at one side of the card. You show him that side of the card and he offers to bet even money he can tell you the color of the other side of the card. Is this a fair bet?
7 Common Thinking Suppose the side you show is red. Then, most people say this is a fair bet since: The card is either red-red or the card is red-blue and the man has a 50% chance of guessing correctly.
8 The Scam - I mean the real solution Despite the last argument there is a strategy to maximumize winning!
9 The Scam - I mean the real solution Despite the last argument there is a strategy to maximumize winning! The key is - 2 cards have the same color on both sides.
10 The Scam - I mean the real solution Despite the last argument there is a strategy to maximumize winning! The key is - 2 cards have the same color on both sides. The man will always guess the same color as that shown and be correct 2/3 of the time!
11 Guarini s Problem - Knights on a 3 3 chessboard Problem: Have the knights switch row positions using only legal moves.
12 Recall that knights only move in an L shape. Two questions are now clear: 1. Can this be done? 2. If it can be done, what is the minimum number of moves needed?
13 A Graph Model
14 A Graph Model
15 Tiling the chessboard with dominoes
16 Tiling the chessboard with dominoes
17 Holes in the Chessboard
18 With the lower left square removed, no tiling is possible, as only 63 squares remain and each dominoe covers exactly 2 squares.
19 With the lower left square removed, no tiling is possible, as only 63 squares remain and each dominoe covers exactly 2 squares. With both the upper left and lower right squares removed, still no tiling is possible, as each dominoe covers one white and one black square, and 30 white and 32 black squares remain!
20 Gomory s Theorem Theorem If you remove one black and one white square from anywhere on the 8 8 chessborad, then you can tile the remaining board with 31 dominoes.
21 Proof of Gomory s Theorem
22 The coin game - for 2 players One hundred randomly selected coins are placed in a row. The sum of the values of these coins is T which is odd. GAME: Player 1 selects one coin from either end of the row. Player 2 selects one coin from either end of the row. Play continues, alternating turns, until all coins have been selected. Problem: Show that there is a strategy so that Player 1 always wins! (i.e. get most of the money)
23 Sketch of proof
24 Sketch of proof 1. Think of the coins as partitioned into two groups.
25 Sketch of proof 1. Think of the coins as partitioned into two groups. 2. Count the value of the coins in one color to determine which color has the most (> T/2) WLOG say white.
26 Sketch of proof 1. Think of the coins as partitioned into two groups. 2. Count the value of the coins in one color to determine which color has the most (> T/2) WLOG say white. 3. Player 1 selects the end coin that is white.
27 Sketch of proof 1. Think of the coins as partitioned into two groups. 2. Count the value of the coins in one color to determine which color has the most (> T/2) WLOG say white. 3. Player 1 selects the end coin that is white. 4. Player 2 now has only red coins on either end to select from.
28 Sketch of proof 1. Think of the coins as partitioned into two groups. 2. Count the value of the coins in one color to determine which color has the most (> T/2) WLOG say white. 3. Player 1 selects the end coin that is white. 4. Player 2 now has only red coins on either end to select from. 5. Player 1 now selects the white coin uncovered by the choice Player 2 just made.
29 Sketch of proof 1. Think of the coins as partitioned into two groups. 2. Count the value of the coins in one color to determine which color has the most (> T/2) WLOG say white. 3. Player 1 selects the end coin that is white. 4. Player 2 now has only red coins on either end to select from. 5. Player 1 now selects the white coin uncovered by the choice Player 2 just made. 6. Player 2 again has only reds to choose from. This continues until the game ends with Player 1 having all the white coins and the win.
30 The Moral of the Story Never give a non-mathematician an even break!
31 The Paint Ball War The Game: Three friends, Matt (the mathematician) Phil (the physicist) and Ed (the engineer) decide to play paintball. Matt - limited experience: 50% shooter. Phil - plays more: 75% shooter. Ed - plays constantly: 100% shooter.
32 Some Rules - To Make the Game Fair Because of the ability differences, they decide to follow some old dueling traditions... so
33 Some Rules - To Make the Game Fair Because of the ability differences, they decide to follow some old dueling traditions... so Matt will get the first shot (50%).
34 Some Rules - To Make the Game Fair Because of the ability differences, they decide to follow some old dueling traditions... so Matt will get the first shot (50%). Phil will get the second shot (75%).
35 Some Rules - To Make the Game Fair Because of the ability differences, they decide to follow some old dueling traditions... so Matt will get the first shot (50%). Phil will get the second shot (75%). Ed will get the third shot (100%).
36 Some Rules - To Make the Game Fair Because of the ability differences, they decide to follow some old dueling traditions... so Matt will get the first shot (50%). Phil will get the second shot (75%). Ed will get the third shot (100%). Should it be needed, another round then follows among any survivors.
37 What strategy should Matt use?
38 What strategy should Matt use? If he shoots (and hits) Ed, then there is a 75% chance Phil shoots him.
39 What strategy should Matt use? If he shoots (and hits) Ed, then there is a 75% chance Phil shoots him. If he shoots Phil, then Ed immediately will shoot him!
40 What strategy should Matt use? If he shoots (and hits) Ed, then there is a 75% chance Phil shoots him. If he shoots Phil, then Ed immediately will shoot him! Surrender and go to a bar and wait for the winner to join him there later!
41 What strategy should Matt use? If he shoots (and hits) Ed, then there is a 75% chance Phil shoots him. If he shoots Phil, then Ed immediately will shoot him! Surrender and go to a bar and wait for the winner to join him there later! Matt s best move is to purposely miss!
42 Why is this a good move? Now Phil s shot Phil kills Ed Phil misses 3/4 1/4 1 L Matt kills Ed Matt wins Ed kills Phil Matt mises 1/2 1/2 Ed wins Figure : The strategy tree diagram. 1 Ed kills Matt P( Ed wins ) = = 1 8 =.125
43 Why is this a good move? p = prob Phil wins M misses Phil misses D... M misses 1/2 1/2 1/2 P kills E 3/4 1/2 1/4 Phil misses P kills M M kills P 1/4 3/4 M wins M kills P Phil wins M wins Figure : The strategy tree diagram. R R is the right subtree
44 Why is this a good move? Use the fact we are repeating turns. The subtree D contains this repeated pattern. (3/4 1/2 3/4) = x then then x 1/4 1/2 = y y 1/4 1/2 etc. So we get a geometric series with first term a = 3/4 1/2 3/4 = 9/32 and ratio r = 1/4 1/2 Hence, P(Phil wins) = a/(1 r) = 72/ Now P(Matt wins) = 1 P(Phil wins) P(Ed wins) = =.554.
45 The Moral of the Story Never give a non-mathematician and even break.
46 The 5 card trick - due to Fitch Chaney, 1950 Suppose you and a partner want to perform the following 5 card trick. A normal deck is shuffled and you are dealt 5 cards (without your partner seeing). Then you:
47 The 5 card trick - due to Fitch Chaney, 1950 Suppose you and a partner want to perform the following 5 card trick. A normal deck is shuffled and you are dealt 5 cards (without your partner seeing). Then you: J J
48 The Trick Now your partner enters the scene and identifies the face down card exactly!
49 How is this trick done?
50 How is this trick done? 1. We need to convey two pieces of information:
51 How is this trick done? 1. We need to convey two pieces of information: 2. The suit and the rank of the down card!
52 How the suit information is encrypted Given any 5 cards from a standard deck, the The Pigeon Hole Principle: There must be at least 2 cards of the same suit! Thus, we can use one card from that suit placed in a predetermined position to signal the suit. Say for now - the left most position. Now how do we handle the rank?
53 lexicographic ordering is the key! 3 unused cards - Use the relative ranks to have a low (L), medium (M) and high (H) card among the 3 remaining cards. The lexicographic orderings of these three produce: L M H (call this 1) L H M (call this 2) M L H (call this 3) M H L (call this 4) H L M (call this 5) H M L (call this 6)
54 The solution J J Q K A 2 3 J
55 The solution 8 10 J 5 Q 8 10 J 5 Q Q K A 2 3 J
56 The Moral of the Story Never give a non-mathematician an even break.
57 The Game of Penny Ante This game was invented by Walter Penny in The Game: Ask your opponent to select any pattern of heads (H) and tails (T) of length three they wish. Then you select a different pattern. Flip pennies repeatedly until one of the two patterns occurs on three consecutive tosses. The person with that pattern wins an even money bet.
58 Is this a fair game? Many people think the game is unfair because the second person only gets to choose their pattern after the first person has selected. This means they do not have all the choices the first person had.
59 Is this a fair game? Many people think the game is unfair because the second person only gets to choose their pattern after the first person has selected. This means they do not have all the choices the first person had. Those are the conditions - but that is not the advantage!
60 Suppose, for ease of argument, your opponent selected HHH.
61 Suppose, for ease of argument, your opponent selected HHH. Then you will select THH.
62 Case Analysis
63 Case Analysis... H H H... P = 1/8 that first three flips are heads
64 Case Analysis Now suppose this is the first HHH in the sequence.... H H H...
65 Case Analysis Now suppose this is the first HHH in the sequence.... T H H H... Then THH had to occur first! Thus, 7/8 of the time THH wins!
66 An analysis of all the other possibilities (more involved tree structures and infinite geometric series) shows that the following rule always wins with probability at least 2/3. Let the opponent select a pattern of three. Now move their first two choices to your last two choices and fill the first position with what ever choice does not form a palindrome!
67 The moral of the story Never give a non-mathematician an even break.
68 2 decks problem Suppose two standard decks of cards are individually shuffled and placed on a table. Now, the top card of each deck is turned over to determine if there is an EXACT MATCH. This process is continued throughout the remaining cards. Given an even money bet - would you bet there was an exact match before the cards run out?
69 The quick answer One view of this problem is that when the card from deck 1 is turned over, there is a 1 52 chance the card in deck 2 will match it. Thus, the expected number of matches when going through the entire deck would be: np = (52) 1 52 = 1. Thus, the actual length of the deck does not seem to matter.
70 Can we squeeze more out of this? That rough answer seems to say it would seem a wise bet to say there would be a match. BUT HOW WISE???.
71 What is really going on here? Another way to look at this problem, and one that brings far more to the table, is to think of the 2nd deck as a permutation of the first deck. Now, the question we are asking is how many permutations of the deck have at least one position unchanged (a fixed point)? This unchanged position gives us an exact match! A permutation with no fixed points is called a derangement.
72 Counting derangements - use IEP At this point, students would be familiar with De Morgan s Laws. So the step to the IEP is a natural one. Suppose we want to count how many permutations (of our 52 cards) have at least i matches with deck 1.
73 Counting derangements - use IEP At this point, students would be familiar with De Morgan s Laws. So the step to the IEP is a natural one. Suppose we want to count how many permutations (of our 52 cards) have at least i matches with deck 1. There are ( ) 52 i positions for the i matches to occur.
74 Counting derangements - use IEP At this point, students would be familiar with De Morgan s Laws. So the step to the IEP is a natural one. Suppose we want to count how many permutations (of our 52 cards) have at least i matches with deck 1. There are ( ) 52 i positions for the i matches to occur. Fill the other 52 i positions in (52 i)! possible ways.
75 Counting derangements - use IEP At this point, students would be familiar with De Morgan s Laws. So the step to the IEP is a natural one. Suppose we want to count how many permutations (of our 52 cards) have at least i matches with deck 1. There are ( ) 52 i positions for the i matches to occur. Fill the other 52 i positions in (52 i)! possible ways. Then, applying the IEP we obtain:
76 Counting derangements via IEP: D 52 = 52!. But we know from Calculus that: i=52 i=0 ( 1) i i! D n lim n n! = 1 e Hence, P(derangement).3679 and thus P(match).63. Thus, it is a very good even money bet!
77 The Game of NIM Game: Given t 2 piles of chips, two players proceed as follows: Player 1 removes some number of chips from exactly one pile. Player 2 then removes some number of chips from exactly one pile. Players alternate turns until no chips remain. The first player that cannot remove some chips loses.
78 Is there a strategy for optimal play? To find the strategy, it really helps to start at the end of the game. When no chips remain we call this the ZERO POSITION. Your goal as a player is to put your oponent in the zero position. How can we achieve this?
79 Base two representations The key is to think of each pile of chips as a number and to look at the base 2 representation of this number. Say we have three piles of 8, 5 and 6 chips.
80 Base 2 view 8: : :
81 Base 2 view 8: : :
82 Base 2 view 8: : : A: (not a zero position)
83 What to do? Player 1 would like to put the other player into a zero position. We think of a zero position as So to get our parity counts above to a zero position we must remove chips from one pile so that all the colums have an even number of ones: This can always be done!
84 One such move Looking at our example: 8: : : A: (not a zero position) Removing 5 chips from the pile of 8 will produce what we need!
85 3: : : A: (zero position)
86 Player 2 is now in a hopeless position!
87 Player 2 is now in a hopeless position! Any move made by Player 2 produces a none zero position!
88 Player 2 is now in a hopeless position! Any move made by Player 2 produces a none zero position! Player 1 will then repeat the strategy of moving to a zero position!
89 Player 2 is now in a hopeless position! Any move made by Player 2 produces a none zero position! Player 1 will then repeat the strategy of moving to a zero position! Eventually player 1 will reduce to the ultimate zero position - no chips, and win the game
Senior Math Circles February 10, 2010 Game Theory II
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Senior Math Circles February 10, 2010 Game Theory II Take-Away Games Last Wednesday, you looked at take-away
Domino Games. Variation - This came can also be played by multiplying each side of a domino.
Domino Games Domino War This is a game for two people. 1. Place all the dominoes face down. 2. Each person places their hand on a domino. 3. At the same time, flip the domino over and whisper the sum of
GAMES AND STRATEGY BEGINNERS 12/03/2017
GAMES AND STRATEGY BEGINNERS 12/03/2017 1. TAKE AWAY GAMES Below you will find 5 different Take Away Games, each of which you may have played last year. Play each game with your partner. Find the winning
Dependence. Math Circle. October 15, 2016
Dependence Math Circle October 15, 2016 1 Warm up games 1. Flip a coin and take it if the side of coin facing the table is a head. Otherwise, you will need to pay one. Will you play the game? Why? 2. If
EE 126 Fall 2006 Midterm #1 Thursday October 6, 7 8:30pm DO NOT TURN THIS PAGE OVER UNTIL YOU ARE TOLD TO DO SO
EE 16 Fall 006 Midterm #1 Thursday October 6, 7 8:30pm DO NOT TURN THIS PAGE OVER UNTIL YOU ARE TOLD TO DO SO You have 90 minutes to complete the quiz. Write your solutions in the exam booklet. We will
Probability. 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
Random 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,
Chapter 1. Probability
Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.
Quant 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.
18.S34 (FALL, 2007) PROBLEMS ON PROBABILITY
18.S34 (FALL, 2007) PROBLEMS ON PROBABILITY 1. Three closed boxes lie on a table. One box (you don t know which) contains a $1000 bill. The others are empty. After paying an entry fee, you play the following
Random 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,
Probability --QUESTIONS-- Principles of Math 12 - Probability Practice Exam 1
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..
Math 4610, Problems to be Worked in Class
Math 4610, Problems to be Worked in Class Bring this handout to class always! You will need it. If you wish to use an expanded version of this handout with space to write solutions, you can download one
Name. Is the game fair or not? Prove your answer with math. If the game is fair, play it 36 times and record the results.
Homework 5.1C You must complete table. Use math to decide if the game is fair or not. If Period the game is not fair, change the point system to make it fair. Game 1 Circle one: Fair or Not 2 six sided
The 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
Chapter 1. Probability
Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.
Probability MAT230. Fall Discrete Mathematics. MAT230 (Discrete Math) Probability Fall / 37
Probability MAT230 Discrete Mathematics Fall 2018 MAT230 (Discrete Math) Probability Fall 2018 1 / 37 Outline 1 Discrete Probability 2 Sum and Product Rules for Probability 3 Expected Value MAT230 (Discrete
Section The Multiplication Principle and Permutations
Section 2.1 - The Multiplication Principle and Permutations Example 1: A yogurt shop has 4 flavors (chocolate, vanilla, strawberry, and blueberry) and three sizes (small, medium, and large). How many different
2. The Extensive Form of a Game
2. The Extensive Form of a Game In the extensive form, games are sequential, interactive processes which moves from one position to another in response to the wills of the players or the whims of chance.
Discrete probability and the laws of chance
Chapter 8 Discrete probability and the laws of chance 8.1 Multiple Events and Combined Probabilities 1 Determine the probability of each of the following events assuming that the die has equal probability
Outcome X (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5 {(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)}
Section 8: Random Variables and probability distributions of discrete random variables In the previous sections we saw that when we have numerical data, we can calculate descriptive statistics such as
Poker Hand Rankings Highest to Lowest A Poker Hand s Rank determines the winner of the pot!
POKER GAMING GUIDE Poker Hand Rankings Highest to Lowest A Poker Hand s Rank determines the winner of the pot! ROYAL FLUSH Ace, King, Queen, Jack, and 10 of the same suit. STRAIGHT FLUSH Five cards of
EECS 203 Spring 2016 Lecture 15 Page 1 of 6
EECS 203 Spring 2016 Lecture 15 Page 1 of 6 Counting We ve been working on counting for the last two lectures. We re going to continue on counting and probability for about 1.5 more lectures (including
CS1802 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,
The next several lectures will be concerned with probability theory. We will aim to make sense of statements such as the following:
CS 70 Discrete Mathematics for CS Fall 2004 Rao Lecture 14 Introduction to Probability The next several lectures will be concerned with probability theory. We will aim to make sense of statements such
The tenure game. The tenure game. Winning strategies for the tenure game. Winning condition for the tenure game
The tenure game The tenure game is played by two players Alice and Bob. Initially, finitely many tokens are placed at positions that are nonzero natural numbers. Then Alice and Bob alternate in their moves
MTH 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)
JIGSAW ACTIVITY, TASK # Make sure your answer in written in the correct order. Highest powers of x should come first, down to the lowest powers.
JIGSAW ACTIVITY, TASK #1 Your job is to multiply and find all the terms in ( 1) Recall that this means ( + 1)( + 1)( + 1)( + 1) Start by multiplying: ( + 1)( + 1) x x x x. x. + 4 x x. Write your answer
Sept. 26, 2012
Mathematical Games Marin Math Circle linda@marinmathcircle.org Sept. 26, 2012 Some of these games are from the book Mathematical Circles: Russian Experience by D. Fomin, S. Genkin, and I. Itenberg. Thanks
Grade 6 Math Circles Fall Oct 14/15 Probability
1 Faculty of Mathematics Waterloo, Ontario Centre for Education in Mathematics and Computing Grade 6 Math Circles Fall 2014 - Oct 14/15 Probability Probability is the likelihood of an event occurring.
Probability of Independent and Dependent Events
706 Practice A Probability of In and ependent Events ecide whether each set of events is or. Explain your answer.. A student spins a spinner and rolls a number cube.. A student picks a raffle ticket from
Week in Review #5 ( , 3.1)
Math 166 Week-in-Review - S. Nite 10/6/2012 Page 1 of 5 Week in Review #5 (2.3-2.4, 3.1) n( E) In general, the probability of an event is P ( E) =. n( S) Distinguishable Permutations Given a set of n objects
LAMC Junior Circle February 3, Oleg Gleizer. Warm-up
LAMC Junior Circle February 3, 2013 Oleg Gleizer oleg1140@gmail.com Warm-up Problem 1 Compute the following. 2 3 ( 4) + 6 2 Problem 2 Can the value of a fraction increase, if we add one to the numerator
Chapter 4 Number Theory
Chapter 4 Number Theory Throughout the study of numbers, students Á should identify classes of numbers and examine their properties. For example, integers that are divisible by 2 are called even numbers
I.M.O. Winter Training Camp 2008: Invariants and Monovariants
I.M.. Winter Training Camp 2008: Invariants and Monovariants n math contests, you will often find yourself trying to analyze a process of some sort. For example, consider the following two problems. Sample
Yale University Department of Computer Science
LUX ETVERITAS Yale University Department of Computer Science Secret Bit Transmission Using a Random Deal of Cards Michael J. Fischer Michael S. Paterson Charles Rackoff YALEU/DCS/TR-792 May 1990 This work
Beeches Holiday Lets Games Manual
Beeches Holiday Lets Games Manual www.beechesholidaylets.co.uk Page 1 Contents Shut the box... 3 Yahtzee Instructions... 5 Overview... 5 Game Play... 5 Upper Section... 5 Lower Section... 5 Combinations...
Probability (Devore Chapter Two)
Probability (Devore Chapter Two) 1016-351-01 Probability Winter 2011-2012 Contents 1 Axiomatic Probability 2 1.1 Outcomes and Events............................... 2 1.2 Rules of Probability................................
Lesson 10: Using Simulation to Estimate a Probability
Lesson 10: Using Simulation to Estimate a Probability Classwork In previous lessons, you estimated probabilities of events by collecting data empirically or by establishing a theoretical probability model.
Here are two situations involving chance:
Obstacle Courses 1. Introduction. Here are two situations involving chance: (i) Someone rolls a die three times. (People usually roll dice in pairs, so dice is more common than die, the singular form.)
Whenever possible, ask your child to tell you the time to the nearest 5 minutes. Use a clock with hands as well as a digital watch or clock.
Can you tell the time? Whenever possible, ask your child to tell you the time to the nearest 5 minutes. Use a clock with hands as well as a digital watch or clock. Also ask: What time will it be one hour
1. The chance of getting a flush in a 5-card 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
CSC/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
The topic for the third and final major portion of the course is Probability. We will aim to make sense of statements such as the following:
CS 70 Discrete Mathematics for CS Spring 2006 Vazirani Lecture 17 Introduction to Probability The topic for the third and final major portion of the course is Probability. We will aim to make sense of
STAT Statistics I Midterm Exam One. Good Luck!
STAT 515 - Statistics I Midterm Exam One Name: Instruction: You can use a calculator that has no connection to the Internet. Books, notes, cellphones, and computers are NOT allowed in the test. There are
Compound Probability. Set Theory. Basic Definitions
Compound Probability Set Theory A probability measure P is a function that maps subsets of the state space Ω to numbers in the interval [0, 1]. In order to study these functions, we need to know some basic
(a) Suppose you flip a coin and roll a die. Are the events obtain a head and roll a 5 dependent or independent events?
Unit 6 Probability Name: Date: Hour: Multiplication Rule of Probability By the end of this lesson, you will be able to Understand Independence Use the Multiplication Rule for independent events Independent
Discrete 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
Junior 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
Math 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,
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
Week 1: Probability models and counting
Week 1: Probability models and counting Part 1: Probability model Probability theory is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model
Teaching the TERNARY BASE
Features Teaching the TERNARY BASE Using a Card Trick SUHAS SAHA Any sufficiently advanced technology is indistinguishable from magic. Arthur C. Clarke, Profiles of the Future: An Inquiry Into the Limits
Week 1. 1 What Is Combinatorics?
1 What Is Combinatorics? Week 1 The question that what is combinatorics is similar to the question that what is mathematics. If we say that mathematics is about the study of numbers and figures, then combinatorics
Lesson 8: The Difference Between Theoretical Probabilities and Estimated Probabilities
Lesson 8: The Difference Between Theoretical and Estimated Student Outcomes Given theoretical probabilities based on a chance experiment, students describe what they expect to see when they observe many
Sec 5.1 The Basics of Counting
1 Sec 5.1 The Basics of Counting Combinatorics, the study of arrangements of objects, is an important part of discrete mathematics. In this chapter, we will learn basic techniques of counting which has
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Statistics Homework Ch 5 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A coin is tossed. Find the probability
Section 6.1 #16. Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?
Section 6.1 #16 What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? page 1 Section 6.1 #38 Two events E 1 and E 2 are called independent if p(e 1
Contemporary Mathematics Math 1030 Sample Exam I Chapters Time Limit: 90 Minutes No Scratch Paper Calculator Allowed: Scientific
Contemporary Mathematics Math 1030 Sample Exam I Chapters 13-15 Time Limit: 90 Minutes No Scratch Paper Calculator Allowed: Scientific Name: The point value of each problem is in the left-hand margin.
Problem Set 2. Counting
Problem Set 2. Counting 1. (Blitzstein: 1, Q3 Fred is planning to go out to dinner each night of a certain week, Monday through Friday, with each dinner being at one of his favorite ten restaurants. i
The study of probability is concerned with the likelihood of events occurring. Many situations can be analyzed using a simplified model of probability
The study of probability is concerned with the likelihood of events occurring Like combinatorics, the origins of probability theory can be traced back to the study of gambling games Still a popular branch
Background. Game Theory and Nim. The Game of Nim. Game is Finite 1/27/2011
Background Game Theory and Nim Dr. Michael Canjar Department of Mathematics, Computer Science and Software Engineering University of Detroit Mercy 26 January 2010 Nimis a simple game, easy to play. It
Chapter 2. Permutations and Combinations
2. Permutations and Combinations Chapter 2. Permutations and Combinations In this chapter, we define sets and count the objects in them. Example Let S be the set of students in this classroom today. Find
Math 1313 Section 6.2 Definition of Probability
Math 1313 Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability
Grade 6 Math Circles Combinatorial Games - Solutions November 3/4, 2015
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles Combinatorial Games - Solutions November 3/4, 2015 Chomp Chomp is a simple 2-player
Probability: Terminology and Examples Spring January 1, / 22
Probability: Terminology and Examples 18.05 Spring 2014 January 1, 2017 1 / 22 Board Question Deck of 52 cards 13 ranks: 2, 3,..., 9, 10, J, Q, K, A 4 suits:,,,, Poker hands Consists of 5 cards A one-pair
Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results:
Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results: Outcome Frequency 1 8 2 8 3 12 4 7 5 15 8 7 8 8 13 9 9 10 12 (a) What is the experimental probability
Page 1 of 22. Website: Mobile:
Exercise 15.1 Question 1: Complete the following statements: (i) Probability of an event E + Probability of the event not E =. (ii) The probability of an event that cannot happen is. Such as event is called.
2 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
Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance Free-Response 1. A spinner with regions numbered 1 to 4 is spun and a coin is tossed. Both the number spun and whether the coin lands heads or tails is
1. 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
Games for Drill and Practice
Frequent practice is necessary to attain strong mental arithmetic skills and reflexes. Although drill focused narrowly on rote practice with operations has its place, Everyday Mathematics also encourages
SMT 2014 Advanced Topics Test Solutions February 15, 2014
1. David flips a fair coin five times. Compute the probability that the fourth coin flip is the first coin flip that lands heads. 1 Answer: 16 ( ) 1 4 Solution: David must flip three tails, then heads.
Statistics 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
Table 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 800-522-4700. Blackjack Hands down, Blackjack is the most
Math Steven Noble. November 24th. Steven Noble Math 3790
Math 3790 Steven Noble November 24th 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
Subtraction Step Down
Face Off - or Subtraction War Materials Needed: 1 set of Dominoes per player (remove any domino with a blank end), 1 pair of dice per player, one game board Skills: subtracting, outcomes chart and probability
Dealing with some maths
Dealing with some maths Hayden Tronnolone School of Mathematical Sciences University of Adelaide August 20th, 2012 To call a spade a spade First, some dealing... Hayden Tronnolone (University of Adelaide)
10-1. Combinations. Vocabulary. Lesson. Mental Math. able to compute the number of subsets of size r.
Chapter 10 Lesson 10-1 Combinations BIG IDEA With a set of n elements, it is often useful to be able to compute the number of subsets of size r Vocabulary combination number of combinations of n things
3 The multiplication rule/miscellaneous counting problems
Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1 Suppose P (A 0, P (B 05 (a If A and B are independent, what is P (A B? What is P (A B? (b If A and B are disjoint, what is
BLACKJACK Perhaps the most popular casino table game is Blackjack.
BLACKJACK Perhaps the most popular casino table game is Blackjack. The object is to draw cards closer in value to 21 than the dealer s cards without exceeding 21. To play, you place a bet on the table
3 The multiplication rule/miscellaneous counting problems
Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1. Suppose P (A) = 0.4, P (B) = 0.5. (a) If A and B are independent, what is P (A B)? What is P (A B)? (b) If A and B are disjoint,
Probability. 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
To Your Hearts Content
To Your Hearts Content Hang Chen University of Central Missouri Warrensburg, MO 64093 hchen@ucmo.edu Curtis Cooper University of Central Missouri Warrensburg, MO 64093 cooper@ucmo.edu Arthur Benjamin [1]
Math Games Ideas. For School or Home Education. by Teresa Evans. Copyright 2005 Teresa Evans. All rights reserved.
Math Games Ideas For School or Home Education by Teresa Evans Copyright 2005 Teresa Evans. All rights reserved. Permission is given for the making of copies for use in the home or classroom of the purchaser
Take one! Rules: Two players take turns taking away 1 chip at a time from a pile of chips. The player who takes the last chip wins.
Take-Away Games Introduction Today we will play and study games. Every game will be played by two players: Player I and Player II. A game starts with a certain position and follows some rules. Players
Game, Set, and Match Carl W. Lee September 2016
Game, Set, and Match Carl W. Lee September 2016 Note: Some of the text below comes from Martin Gardner s articles in Scientific American and some from Mathematical Circles by Fomin, Genkin, and Itenberg.
Finite Mathematics MAT 141: Chapter 8 Notes
Finite Mathematics MAT 4: Chapter 8 Notes Counting Principles; More David J. Gisch The Multiplication Principle; Permutations Multiplication Principle Multiplication Principle You can think of the multiplication
Classic Dominoes. Number of Players: 2-4
Classic Dominoes Number of Players: 2-4 First, all dominoes must be turned face down and mixed. Each player then draws five dominoes and stands them up on end in front of them so the backs of the dominoes
Detailed Solutions of Problems 18 and 21 on the 2017 AMC 10 A (also known as Problems 15 and 19 on the 2017 AMC 12 A)
Detailed Solutions of Problems 18 and 21 on the 2017 AMC 10 A (also known as Problems 15 and 19 on the 2017 AMC 12 A) Henry Wan, Ph.D. We have developed a Solutions Manual that contains detailed solutions
UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis
![UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis](/thumbs/82/84793934.jpg "UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis")
UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis Lecture 7 Class URL: http://vlsicad.ucsd.edu/courses/cse21-s14/ Lecture 7 Notes Goals for this week: Unit FN Functions
Unit 6: What Do You Expect? Investigation 2: Experimental and Theoretical Probability
Unit 6: What Do You Expect? Investigation 2: Experimental and Theoretical Probability Lesson Practice Problems Lesson 1: Predicting to Win (Finding Theoretical Probabilities) 1-3 Lesson 2: Choosing Marbles
Ex 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.
This artwork is for presentation purposes only and does not depict the actual table.
Patent Pending This artwork is for presentation purposes only and does not depict the actual table. Unpause Games, LLC 2016 Game Description Game Layout Rules of Play Triple Threat is played on a Roulette
3. If you can t make the sum with your cards, you must draw one card. 4. Players take turns rolling and discarding cards.
1 to 10 Purpose: The object of the game is to get rid of all your cards. One player gets all the red cards, the other gets all the black cards. Players: 2-4 players Materials: 2 dice, a deck of cards,
Roll & Make. Represent It a Different Way. Show Your Number as a Number Bond. Show Your Number on a Number Line. Show Your Number as a Strip Diagram
Roll & Make My In Picture Form In Word Form In Expanded Form With Money Represent It a Different Way Make a Comparison Statement with a Greater than Your Make a Comparison Statement with a Less than Your
Problem Set 10 2 E = 3 F
Problem Set 10 1. A and B start with p = 1. Then they alternately multiply p by one of the numbers 2 to 9. The winner is the one who first reaches (a) p 1000, (b) p 10 6. Who wins, A or B? (Derek) 2. (Putnam
100 square muddle. A game for two or three players
Cambridge University Press 978-1-107-62349-1 Cambridge Primary Mathematics Stage 2 Cherri Moseley and Janet Rees Excerpt More information 100 square muddle Maths focus: becoming familiar with the layout
Targets - Year 3. By the end of this year most children should be able to
Targets - Year 3 By the end of this year most children should be able to Read and write numbers up to 1000 and put them in order. Know what each digit is worth. Count on or back in tens or hundreds from
Probability Problems for Group 1 (Due by Oct. 26)
Probability Problems for Group (Due by Oct. 26) Don t Lose Your Marbles!. An urn contains 5 red, 6 blue, and 8 green marbles. If a set of 3 marbles is randomly selected, without replacement, a) what is