# Elementary probability

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Elementary probability Many of the principal applications of calculus are to questions of probability and statistics. We shall include here an introduction to elementary probability, and eventually some discussion of what calculus has to do with it. he three basic rules Probability is assigned to certain repeatable events. he probability P (E) ofanevente is a measure of the relative frequency with which those events occur. It is therefore a number lying between 0and 1. If it is equal to 0, that means the event cannot occur at all. If it is 1, that means it is certain. For example, if you toss a coin in the air, it will presumably land heads exactly as often as it lands tails, so the probability of either event is 1/2. If you throw a single die, the probability of any one of the 6 faces coming out on top is 1/6. Even elementary events like this can be combined to make up more complicated ones, as we shall see in a moment. For example, if you throw a single die, what is the probability that the score is 3 or less? here are three basic rules which tell us how to compute the probability of more complicated events without a great deal of trouble. Complementarity. If E is an event with probability P (E), then the probability that E doesn t occur is 1 P (E). Sum. If E and F are two events which cannot occur simultaneously, then the probability that either E or F occurs is P (E)+P (F ). Product. If E and F are two independent events, then the probability that both occur is P (E)P (F ). he meaning of these will become clear soon. Even now we can make a few remarks, however. If we combine the first and second rule, we have this consequence: If an event has possible distinct outcomes E 1, E 2,..., E n then the sum of the probabilities P (E i )is1. Informally, you can just say that, after all, something has to happen! And if you have listed all the possibilities, then it has to be one of them. As for the product rule, independent events are ones which do not affect each other at all, and the product rule should be clear. For example, if I throw two coins simultaneously a zillion times, how one of them lands will not affect how the other does. he first coin will come up heads about half a zillion times, and among the half a zillion times it comes up heads the second coin will also come up heads. his means that about one-quarter of the time they will both come up heads, and since 1/4 =(1/2)(1/2) this illustrates the product rule. hese rules are pretty simple, but in order to apply them correctly you will have to be careful. ossing coins Let s look in detail at what happens when we toss two coins simultaneously. Being careful here means we should keep in mind that although the two coins look alike, they are really different. We could, in principle, mark them in order to make them distinguishable. Suppose we mark them as A and B. If we toss them both, there are altogether four possible outcomes: (1) they both come up H; (2) A comes up H and B comes up ; (3) A comes up and B comes up H; (4) they both come up. On grounds of symmetry, we can see that each of these occurs with the same probability, and since the sum of the probabilities of all possible outcomes must equal 1, they must each have probability 1/4.

2 ossing coins 2 A B Probability H H 1/4 H 1/4 H 1/4 1/4 We could also have applied the product rule to see that the probability is 1/4, since the outcomes for the two coins are independent, and 1/4 =(1/2)(1/2). Exercise 1. Suppose one of the coins is biased, so that the probability of coming up heads is p 1/2. What is the probability of getting tails? Make a table for this situation analogous to the one above. Exercise 2. How many distinct outcomes if we throw two dice? Make a list of them all. In practice, when we examine the coins to see what the outcome of a toss is, we won t be able to tell the difference betwen them. he only thing that is easily observable is the total number which come up heads. Based on this observation, there are three possible observable outcomes: 0, 1, or 2 heads in total. What are the probabilities for these? We have to express these compound events in terms of the elementary ones. o get 0heads, both coins must come up tails, so the probability is 1/4. o get 2 heads, both must be heads; probability 1/4. But to get 1 head, we can have either H or H. Each of these has probability 1/4, and they cannot happen at the same time, so we apply the sum rule. he probability is 1/4+1/4 =1/2. We get this table: n = 2 Number of heads Probability 0 1/4 1 1/2 2 1/ For comparison, we can make up a similar table for the simpler case of a single coin: n = 1 Number of heads Probability 0 1/2 1 1/2 Exercise 3. Make up a similar table and graph for a pair of coins for which the probability of heads is 0.4. What happens for three coins A, B, C? We can make up a list of all possible outcomes in the following way. What happens to the third coin C is independent of what happens to the first two. So to make a list of all possible outcomes for three coins we write down the list of outcomes for two, and then for each of these we put in a different outcome for the third. Since there were four cases for two coins, there are 8 for 3 coins. his is the list we get: 0 1

5 ossing n coins 5 Exercise 5. Write down the next two rows. What we would like, however, is a way to generate any row without necessarilty knowing all the rows above it. Here is that rule: o write down the n-th row, (a) Start with a 1. Call this the 0-number in that row. (b) o get from the k-th number to the k + 1-st, multiply by n k and divide by k + 1. Here, for example, is the fourth row with the calculation sdispalyed: n = (4/1) 1 6 = (3/2) 4 4 = (2/3) 6 1 = (1/4) 4 We can detect the pattern of calculation. In the n-throwwestartwith1. henextis(n/1) 1 = n; then (n 1)/2 timesn, orn(n 1)/2. In the k-th column we shall get n(n 1)(n 2)...(n k +1)/1(2)(3)...(k). Exercise 6. Apply this rule to get the 5-th row. Exercise 7. Apply this rule to get the 10-th row. Exercise 8. What is the probability of getting 5 heads in a toss of 10 coins? 5 heads in a toss of 11 coins?

### Events. Independence. Coin Tossing. Random Phenomena

Random Phenomena Events A random phenomenon is a situation in which we know what outcomes could happen, but we don t know which particular outcome did or will happen For any random phenomenon, each attempt,

### 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

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

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

### Discrete probability and the laws of chance

Chapter 8 Discrete probability and the laws of chance 8.1 Introduction In this chapter we lay the groundwork for calculations and rules governing simple discrete probabilities. These steps will be essential

### Examples of infinite sample spaces. Math 425 Introduction to Probability Lecture 12. Example of coin tosses. Axiom 3 Strong form

Infinite Discrete Sample Spaces s of infinite sample spaces Math 425 Introduction to Probability Lecture 2 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 4,

### An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event

An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event is the sum of the probabilities of the outcomes in the

### MATHEMATICAL THEORY FOR SOCIAL SCIENTISTS THE BINOMIAL THEOREM. (p + q) 0 =1,

THE BINOMIAL THEOREM Pascal s Triangle and the Binomial Expansion Consider the following binomial expansions: (p + q) 0 1, (p+q) 1 p+q, (p + q) p +pq + q, (p + q) 3 p 3 +3p q+3pq + q 3, (p + q) 4 p 4 +4p

### Combinations and Permutations

Combinations and Permutations What's the Difference? In English we use the word "combination" loosely, without thinking if the order of things is important. In other words: "My fruit salad is a combination

### A1. Basic Reviews PERMUTATIONS and COMBINATIONS... or HOW TO COUNT

A1. Basic Reviews Appendix / A1. Basic Reviews / Perms & Combos-1 PERMUTATIONS and COMBINATIONS... or HOW TO COUNT Question 1: Suppose we wish to arrange n 5 people {a, b, c, d, e}, standing side by side,

### 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

### 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

### MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution

### Chapter 15. Definitions: experiment: is the act of making an observation or taking a measurement.

MATH 11008: Probability Chapter 15 Definitions: experiment: is the act of making an observation or taking a measurement. outcome: one of the possible things that can occur as a result of an experiment.

### Probability experiment. Toss two coins Toss three coins Roll two dice

Probability experiment Toss two coins Toss three coins Roll two dice Activity 1: An Experiment with 2 coins WE TOSS TWO COINS When we toss two coins at the same time, the possible outcomes are: (two Heads)

### 4. Binomial Expansions

4. Binomial Expansions 4.. Pascal's Triangle The expansion of (a + x) 2 is (a + x) 2 = a 2 + 2ax + x 2 Hence, (a + x) 3 = (a + x)(a + x) 2 = (a + x)(a 2 + 2ax + x 2 ) = a 3 + ( + 2)a 2 x + (2 + )ax 2 +

### 34 Probability and Counting Techniques

34 Probability and Counting Techniques If you recall that the classical probability of an event E S is given by P (E) = n(e) n(s) where n(e) and n(s) denote the number of elements of E and S respectively.

### Toss a coin twice. Let Y denote the number of heads.

! Let S be a discrete sample space with the set of elementary events denoted by E = {e i, i = 1, 2, 3 }. A random variable is a function Y(e i ) that assigns a real value to each elementary event, e i.

### 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

### 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,

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

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

### MATH 10: Elementary Statistics and Probability Chapter 3: Probability Topics

MATH 10: Elementary Statistics and Probability Chapter 3: Probability Topics Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you should

### AQA Statistics 1. Probability. Section 2: Tree diagrams

Notes and Examples AQA Statistics Probability Section 2: Tree diagrams These notes include sub-sections on; Reminder of the addition and multiplication rules Probability tree diagrams Problems involving

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

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

### Performance Assessment Task Fair Game? Grade 7. Common Core State Standards Math - Content Standards

Performance Assessment Task Fair Game? Grade 7 This task challenges a student to use understanding of probabilities to represent the sample space for simple and compound events. A student must use information

### + Section 6.2 and 6.3

Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities

### 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

### 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

### Massachusetts Institute of Technology

n (i) m m (ii) n m ( (iii) n n n n (iv) m m Massachusetts Institute of Technology 6.0/6.: Probabilistic Systems Analysis (Quiz Solutions Spring 009) Question Multiple Choice Questions: CLEARLY circle the

### JANUARY TERM 2012 FUN and GAMES WITH DISCRETE MATHEMATICS Module #9 (Geometric Series and Probability)

JANUARY TERM 2012 FUN and GAMES WITH DISCRETE MATHEMATICS Module #9 (Geometric Series and Probability) Author: Daniel Cooney Reviewer: Rachel Zax Last modified: January 4, 2012 Reading from Meyer, Mathematics

### 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

### Feb 7 Homework Solutions Math 151, Winter 2012. Chapter 4 Problems (pages 172-179)

Feb 7 Homework Solutions Math 151, Winter 2012 Chapter Problems (pages 172-179) Problem 3 Three dice are rolled. By assuming that each of the 6 3 216 possible outcomes is equally likely, find the probabilities

### Regions in a circle. 7 points 57 regions

Regions in a circle 1 point 1 region points regions 3 points 4 regions 4 points 8 regions 5 points 16 regions The question is, what is the next picture? How many regions will 6 points give? There's an

### PROBABILITY. Thabisa Tikolo STATISTICS SOUTH AFRICA

PROBABILITY Thabisa Tikolo STATISTICS SOUTH AFRICA Probability is a topic that some educators tend to struggle with and thus avoid teaching it to learners. This is an indication that teachers are not yet

### 1. The sample space S is the set of all possible outcomes. 2. An event is a set of one or more outcomes for an experiment. It is a sub set of S.

1 Probability Theory 1.1 Experiment, Outcomes, Sample Space Example 1 n psychologist examined the response of people standing in line at a copying machines. Student volunteers approached the person first

### Real Numbers and Monotone Sequences

Real Numbers and Monotone Sequences. Introduction. Real numbers. Mathematical analysis depends on the properties of the set R of real numbers, so we should begin by saying something about it. There are

### GENERATING FUNCTION. and f( 1) = a m ( 1) m = a 2k a 2k+1 So a2k+1 = 1 (f(1) f( 1)).

GENERATING FUNCTION. Examples Question.. Toss a coin n times and find the probabilit of getting exactl k heads. Represent H b x and T b x 0 and a sequence, sa, HTHHT b (x )(x 0 )(x )(x )(x 0 ). We see

### Binomial Probability Distribution

Binomial Probability Distribution In a binomial setting, we can compute probabilities of certain outcomes. This used to be done with tables, but with graphing calculator technology, these problems are

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### Probability. Vocabulary

MAT 142 College Mathematics Probability Module #PM Terri L. Miller & Elizabeth E. K. Jones revised January 5, 2011 Vocabulary In order to discuss probability we will need a fair bit of vocabulary. Probability

### Partial Fractions. (x 1)(x 2 + 1)

Partial Fractions Adding rational functions involves finding a common denominator, rewriting each fraction so that it has that denominator, then adding. For example, 3x x 1 3x(x 1) (x + 1)(x 1) + 1(x +

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

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

### H + T = 1. p(h + T) = p(h) x p(t)

Probability and Statistics Random Chance A tossed penny can land either heads up or tails up. These are mutually exclusive events, i.e. if the coin lands heads up, it cannot also land tails up on the same

### 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

### Combinatorial Proofs

Combinatorial Proofs Two Counting Principles Some proofs concerning finite sets involve counting the number of elements of the sets, so we will look at the basics of counting. Addition Principle: If A

### Gap Closing. Decimal Computation. Junior / Intermediate Facilitator's Guide

Gap Closing Decimal Computation Junior / Intermediate Facilitator's Guide Module 8 Decimal Computation Diagnostic...5 Administer the diagnostic...5 Using diagnostic results to personalize interventions...5

### USING PLAYING CARDS IN THE CLASSROOM. What s the Pattern Goal: To create and describe a pattern. Closest to 24!

USING PLAYING CARDS IN THE CLASSROOM You can purchase a deck of oversized playing cards at: www.oame.on.ca PRODUCT DESCRIPTION: Cards are 4 ½ by 7. Casino quality paperboard stock, plastic coated. The

### SECTION 10-5 Multiplication Principle, Permutations, and Combinations

10-5 Multiplication Principle, Permutations, and Combinations 761 54. Can you guess what the next two rows in Pascal s triangle, shown at right, are? Compare the numbers in the triangle with the binomial

### 4.5 Finding Probability Using Tree Diagrams and Outcome Tables

4.5 Finding Probability Using ree Diagrams and Outcome ables Games of chance often involve combinations of random events. hese might involve drawing one or more cards from a deck, rolling two dice, or

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

### 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............................

### 94 Counting Solutions for Chapter 3. Section 3.2

94 Counting 3.11 Solutions for Chapter 3 Section 3.2 1. Consider lists made from the letters T, H, E, O, R, Y, with repetition allowed. (a How many length-4 lists are there? Answer: 6 6 6 6 = 1296. (b

### 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

### It is remarkable that a science, which began with the consideration of games of chance, should be elevated to the rank of the most important

PROBABILLITY 271 PROBABILITY CHAPTER 15 It is remarkable that a science, which began with the consideration of games of chance, should be elevated to the rank of the most important subject of human knowledge.

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### Games ~ 1. Triangle Sum. Poly Pick. In & Out. Skittles. Cover Up. Home Run. Cross Nim. To and Fro. Thirty one. Put Down. One or Two.

Games ~ 1 List of Contents Triangle Sum Poly Pick In & Out Skittles Cover Up Home Run Cross Nim To and Fro Thirty one Put Down One or Two Triangle Sum Players take turns moving a single marker or counter

### Pigeonhole Principle Solutions

Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such

### A (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes.

Chapter 7 Probability 7.1 Experiments, Sample Spaces, and Events A (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes. Each outcome

### Sampling Distributions and the Central Limit Theorem

135 Part 2 / Basic Tools of Research: Sampling, Measurement, Distributions, and Descriptive Statistics Chapter 10 Sampling Distributions and the Central Limit Theorem In the previous chapter we explained

### 4/1/2017. PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY

PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY 1 Oh the things you should learn How to recognize and write arithmetic sequences

### diffy boxes (iterations of the ducci four number game) 1

diffy boxes (iterations of the ducci four number game) 1 Peter Trapa September 27, 2006 Begin at the beginning and go on till you come to the end: then stop. Lewis Carroll Consider the following game.

### 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)

### For 2 coins, it is 2 possible outcomes for the first coin AND 2 possible outcomes for the second coin

Problem Set 1. 1. If you have 10 coins, how many possible combinations of heads and tails are there for all 10 coins? Hint: how many combinations for one coin; two coins; three coins? Here there are 2

### Solving Systems of Linear Equations. Substitution

Solving Systems of Linear Equations There are two basic methods we will use to solve systems of linear equations: Substitution Elimination We will describe each for a system of two equations in two unknowns,

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

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

### Patterns & Math in Pascal s Triangle!

Uses for Pascal s Triangle An entry in Pascal's Triangle is usually given a row number and a place in that row, beginning with row zero and place, or element, zero. For instance, at the top of Pascal's

### Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.

Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required

### 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.

### Introduction to Probability

Chapter 4 Introduction to Probability 4.1 Introduction Probability is the language we use to model uncertainty. We all intuitively understand that few things in life are certain. There is usually an element

### Area in Polar Coordinates

Area in Polar Coordinates If we have a circle of radius r, and select a sector of angle θ, then the area of that sector can be shown to be 1. r θ Area = (1/)r θ As a check, we see that if θ =, then the

### 3.2 Matrix Multiplication

3.2 Matrix Multiplication Question : How do you multiply two matrices? Question 2: How do you interpret the entries in a product of two matrices? When you add or subtract two matrices, you add or subtract

### Basic Probability Theory I

A Probability puzzler!! Basic Probability Theory I Dr. Tom Ilvento FREC 408 Our Strategy with Probability Generally, we want to get to an inference from a sample to a population. In this case the population

### Chapter 6: Random Variables

Chapter : Random Variables Section.3 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter Random Variables.1 Discrete and Continuous Random Variables.2 Transforming and Combining

### 4.1 4.2 Probability Distribution for Discrete Random Variables

4.1 4.2 Probability Distribution for Discrete Random Variables Key concepts: discrete random variable, probability distribution, expected value, variance, and standard deviation of a discrete random variable.

### Basics of Probability

Basics of Probability August 27 and September 1, 2009 1 Introduction A phenomena is called random if the exact outcome is uncertain. The mathematical study of randomness is called the theory of probability.

### Section 6.2 ~ Basics of Probability. Introduction to Probability and Statistics SPRING 2016

Section 6.2 ~ Basics of Probability Introduction to Probability and Statistics SPRING 2016 Objective After this section you will know how to find probabilities using theoretical and relative frequency

### Most of us would probably believe they are the same, it would not make a difference. But, in fact, they are different. Let s see how.

PROBABILITY If someone told you the odds of an event A occurring are 3 to 5 and the probability of another event B occurring was 3/5, which do you think is a better bet? Most of us would probably believe

### Please circle A or B or I in the third column to indicate your chosen option in this table.

The choice of A means that the payoff to you will be 80 rupees if heads lands up from the coin toss and 130 rupees if tails lands up from the coin toss if this table is randomly chosen The choice of B

### I. WHAT IS PROBABILITY?

C HAPTER 3 PROAILITY Random Experiments I. WHAT IS PROAILITY? The weatherman on 10 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and

### ST 371 (IV): Discrete Random Variables

ST 371 (IV): Discrete Random Variables 1 Random Variables A random variable (rv) is a function that is defined on the sample space of the experiment and that assigns a numerical variable to each possible

### 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

### ALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite

ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,

### WORKED EXAMPLES 1 TOTAL PROBABILITY AND BAYES THEOREM

WORKED EXAMPLES 1 TOTAL PROBABILITY AND BAYES THEOREM EXAMPLE 1. A biased coin (with probability of obtaining a Head equal to p > 0) is tossed repeatedly and independently until the first head is observed.

### Grade 6 Math Circles. Binary and Beyond

Faculty of Mathematics Waterloo, Ontario N2L 3G1 The Decimal System Grade 6 Math Circles October 15/16, 2013 Binary and Beyond The cool reality is that we learn to count in only one of many possible number

### Session 29 Scientific Notation and Laws of Exponents. If you have ever taken a Chemistry class, you may have encountered the following numbers:

Session 9 Scientific Notation and Laws of Exponents If you have ever taken a Chemistry class, you may have encountered the following numbers: There are approximately 60,4,79,00,000,000,000,000 molecules

### 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

### Tossing a Biased Coin

Tossing a Biased Coin Michael Mitzenmacher When we talk about a coin toss, we think of it as unbiased: with probability one-half it comes up heads, and with probability one-half it comes up tails. An ideal

### 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

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### MATH 201. Final ANSWERS August 12, 2016

MATH 01 Final ANSWERS August 1, 016 Part A 1. 17 points) A bag contains three different types of dice: four 6-sided dice, five 8-sided dice, and six 0-sided dice. A die is drawn from the bag and then rolled.

### Teaching & Learning Plans. Plan 4: Outcomes of Coin Tosses. Junior Certificate Syllabus Leaving Certificate Syllabus

Teaching & Learning Plans Plan 4: Outcomes of Coin Tosses Junior Certificate Syllabus Leaving Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what the lesson,

### 4. Joint Distributions

Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 4. Joint Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an underlying sample space. Suppose

### Calculus for Middle School Teachers. Problems and Notes for MTHT 466

Calculus for Middle School Teachers Problems and Notes for MTHT 466 Bonnie Saunders Fall 2010 1 I Infinity Week 1 How big is Infinity? Problem of the Week: The Chess Board Problem There once was a humble

### Square roots by subtraction

Square roots by subtraction Frazer Jarvis When I was at school, my mathematics teacher showed me the following very strange method to work out square roots, using only subtraction, which is apparently

### Question of the Day. Key Concepts. Vocabulary. Mathematical Ideas. QuestionofDay

QuestionofDay Question of the Day What is the probability that in a family with two children, both are boys? What is the probability that in a family with two children, both are boys, if we already know

### Discrete mathematics is the study of techniques, ideas and modes

CHAPTER 1 Discrete Systems Discrete mathematics is the study of techniques, ideas and modes of reasoning that are indispensable in applied disciplines such as computer science or information technology.

### CS60003 Algorithm Design and Analysis, Autumn

CS60003 Algorithm Design and Analysis, Autumn 2010 11 Class test 1 Maximum marks: 40 Time: 07 08 09 10 Duration: 1 + ǫ hour Roll no: Name: [ Write your answers in the question paper itself. Be brief and