# Probability. Vocabulary

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 is a measurement of how likely it is for something to happen. An experiment is a process that yields an observation. Example 1. Experiment 1. The experiment is to toss a coin and observe whether it lands heads or tails. Experiment 2. The experiment is to toss a die and observe the number of spots. An outcome is the result of an experiment. Example 2. Experiment 1. One possible outcome is heads, which we will designate H. Experiment 2. One possible outcome is 5. Sample space is the set of all possible outcomes, we will usually represent this set with S. Example 3. Experiment 1. S = {H, T } Experiment 2. S = {1, 2, 3, 4, 5, 6}. An event is any subset of the sample space. Events are frequently designated as E or E 1, E 2, etc. if there are more than one. Example 4. Experiment 1. One possible event is landing heads up. E = {H}. Experiment 2. One event might be getting an even number, a second event might be getting a number greater than or equal to 5. E 1 = {2, 4, 6}, E 2 = {5, 6}. A certain event is guaranteed to happen, a sure thing. An impossible event is one that cannot happen. Example 5. From our Experiment 1. If the event E 1 is landing heads up or landing tails up, then E 1 = {H, T } and we can see that E 1 is a certain event. If the event E 2 is getting a 6, then E 2 = {} and we can see than E 2 is impossible. Outcomes in a sample space are said to be equally likely if every outcome in the sample space has the same chance of happening, i.e. one outcome is no more likely than any other.

2 2 Geometry Basic Computation of a Probability If E is an event in an equally likely sample space, S, then the probability of E, denoted p(e) is computed p(e) = n(e) n(s) Example 6. Experiment 1. E 1 = {H, T }, so p(e 1 ) = n(e 1) n(s) = 2 2 = 1. Experiment 2. E 2 = {5, 6}, so p(e 2 ) = n(e 2) n(s) = 2 6 = 1 3. You should note that since n( ) = 0, p( ) = 0 and since n(s) = 1, p(s) = 1. n(s) Relative frequency is the number of times a particular outcome occurs dvided by the number of times the experiment is performed. Example 7. Suppose you toss a fair coin 10 times and observe the results: heads occurred 6 times and tails occurred 4 times. In this case the relative frequency of heads would be 6 10 = 0.6. Now suppose you toss a fair coin 1000 times and observe the results: heads occurred 528 times and tails occured 472 times. Now the relative frequency of heads is = Something which can be observed from the previous two experiments is called the Law of Large Numbers. The Law of Large Numbers: If an experiment is repeated a large number of times, the relative frequency tends to get closer to the probability. Example 8. In the previous example, we observed the outcome heads. This was a fair coin so we know p(h) = 0.5. Notice that when we performed the experiment 1000 times, the relative frequency was closer to 0.5 than when we performed the experiment 10 times. Odds The odds for an event, E, with equally likely outcomes are: and the odds against the event E are:. Note, these are read as a to b. o(e) = n(e) : n(e) o(e) = n(e) : n(e) Example 9. Toss a fair coing twice and observe the outcome. The sample space is If E is getting two tails, then the odds for E are 1 to 3. Also S = {HH, HT, T H, T T }. o(e) = 1 : 3, o(e) = 3 : 1,

3 MAT Module 3 the odds against E are 3 to 1. Some Applications Probability is used in genetics. When Mendel experimented with pea plants he discovered that some genes were dominant and others were recessive. This means that given one gene from each parent, the dominant trait will show up unless a recessive gene is received from each parent. This is often demonstrated by using a Punnett square. Here is a typical Punnett square: R R w wr wr w wr wr The letters along the top of the table represent the gene contribution from one parent and the letters down the left-hand side of the table represent the gene contribution from the other parent. Each cell of the table contains a genetic combination for a possible offspring. This particular Punnett square represent the crossing of a pure red flower pea with a pure white flower pea. The offspring will each inherit one of each gene; since red is dominant here, all offspring will be red. Example 10. Suppose we cross a pure red flower pea plant with one of the offspring that has one of each gene. The resulting Punnett square would be R R w wr wr R RR RR From this we can see that the probability of producing a pure red flower pea is 2 4 = 1 2, and we can see that each offspring would produce red flowers. Example 11. This time we will cross two of the offspring which have one of each gene. The Punnett square is w R w ww wr R Rw RR Here we can find that the probability of an offspring producing white flowers is 1 4. Example 12. Sickle cell anemia is inherited. This is a co-dominant disease. A person with two sickle cell genes will have the disease while a person with one sickle cell gene will have a mild anemia called sickle cell trait. Suppose a healthy parent (no sickle cell gene) and a parent with sickle cell trait have children. Use a Punnett square to determine the following probabilities. (1) the child has sickle cell (2) the child has sickle cell trait (3) the child is healthy Solution: We will use S for the sickle cell gene and N for no sickle cell gene. Our Punnett square is

4 4 Geometry N N S SN SN N NN NN (1) No offspring have two sickle cell genes so this probability is 0. (2) We see that two of the offspring will have one of each gene, so this probability is 2 4 = 1 2. (3) Two of the children will have no sickle cell genes, this probability is 2 4 = 1 2. Basic Properties of Probability We will need to start with one more new term. Two events are mutually exclusive if they cannot both happen at the same time, i.e. E F =. So, if p(a B) = 0 then A and B are mutually exclusive, We have the following rules for probability: Basic Probability Rules p( ) = 0, p(s) = 1, 0 p(e) 1 Example 13. Roll a die and observe the number of dots; S = {1, 2, 3, 4, 5, 6}. Given E is the event that you roll a 15, F is the event that you roll a number between 1 and 6 inclusive, and G is the event that you roll a 3. Find p(e), p(f ), and p(g). Solution: Since 15 is not in the sample space E = and p(e) = 0. F = S, so p(f ) = 1. Lastly, G = {3}, so p(g) = n(g) 1. n(s) 6 Example 14. Roll a pair of dice, one red and one white. Find the probabilities: (1) the sum of the pair is 7 (2) the sum is greater than 9 (3) the sum is not greater than 9 (4) the sum is greater than 9 and even (5) the sum is greater than 9 or even (6) the difference is 3 Solution: This sample space is too large to list as there are 6 6 = 36 outcomes. Hence, we will just count up the number of ways to achieve each event and them divide by 36. (1) The pairs that sum to 7 are {(6, 1), (1, 6), (5, 2), (2, 5), (4, 3), (3, 4)}, hence this probability is 6 = (2) Sums greater than 9 are 10, 11 and 12, the pairs meeting this condition are {(6, 4), (4, 6), (5, 5), (6, 5), (5, 6), (6, 6)}, this gives the probability as 6 36 = 1 6. (3) The sum is not greater than 9 would be all of those not included in the previous part, so there must be 36 6 = 30 outcomes in this set. Hence, this probability is = 5 6. (4) Those with a sum greater than 9 and even would be those that sum to 10 or 12, {(6, 4), (4, 6), (5, 5), (6, 6)}. Since this set contains 4 outcomes, this probability is 4 36 = 1 9.

5 MAT Module 5 (5) Half of the pairs would sum to an even number and part (3) gives us that there are 6 with sums greater than 9. Note that (4) gives us the number in the intersection. Using the counting formula n(a B) = n(a) + n(b) n(a B), this set contains = 20 and the probability is 20 = (6) Those pairs whose difference is 3 are {(6, 3), (3, 6), (5, 2), (2, 5), (4, 1), (1, 4)}, hence the probability is 6 =

6 6 Geometry We have two more formulae for probability that will be useful. Basic Probability Rules p(e) + p(e) = 1, p(e F ) = p(e) + p(f ) p(e F ) Use of Venn Diagrams for Probability. It is often helpful to put the information given in a Venn Diagram to organize the information and answer questions. The following is an example of such a case. Example 15. Zaptronics makes CDs and their cases for several music labels. A recent sampling indicated that there is a 0.05 probability that a case is defective, a 0.97 probability that a CD is not defective, and a 0.07 probability that at lease one of them is defective. (1) What is the probability that both are defective? (2) What is the probability that neither is defective? Solution: We start by writing down the information given. Let C represent a defective case and let D represent a defective CD. Then we are given: p(c) = 0.05, p(d) = 0.97, p(c D) = 0.07 Using the property p(e) + p(e) = 1, we get p(d) = 1 p(d) = = Using the property p(e F ) = p(e) + p(f ) p(e F ), we get p(c D) = p(c) + p(d) p(c D) = = Put this together in a Venn diagram and recall that p(u) = 1. From this we can see the answers we need. picture is missing here (1) This is the middle football shape so, 0.01 (2) This is everything outside of the circles, so 0.93 Expected Value Once again, we will need some new vocabulary. The payoff is the value of an outcome. Example 16. You toss a coin, if it comes up heads you win \$1, if it comes up tails, you pay me \$0.50. The outcome H, has the payoff 1. The outcome T, has the payoff The expected value of an event is a long term average of payoffs of an experiment. If the experiment were repeated often enough, the actual profit/loss will get close to the expected value. To compute the expected value, you first multiply each payoff by the probability of getting that payoff. Once you have done all of the multiplications, you add your results together. This sum is the expected value of an experiment. Let m 1, m 2, m 3,..., m k be the payoffs associated with the k outcomes of an experiment. Let p 1, p 2, p 3,..., p k, repectively, be the probabilities of those outcomes. Then the expected value is found as follows. Expected Value: E = m 1 p 1 + m 2 p 2 + m 3 p m k p k

7 MAT Module 7 In mathematics, the capital greek letter sigma, Σ, is used to tell us to add up the things that follow. Using this idea, an informal but perhaps more palatable form of the formula for expected value is Some example should be done here. E = Σ(probability) (payoff). Conditional Probability We will introduce the idea of conditional probability with an example. Example 17. Two coins are tossed. Event E is getting exactly one tail. Event F is getting at least one tail. The sample space for the experiment is S = {HH, HT, T H, T T }. The event has probability p(e) = 2/4. E {HT, T H} If I tell you that at least one of the coins is showing a tail, then you know that event F has occurred and that the outcome HH is no longer possible. Hence, we effectively reduce the size of the sample space to F = {HT, T H, T T }. With F as the sample space, the probability of event E = 2/3. Thus, the probability of one event occurring is changed by knowing that another event has already occured. This is conditional probability. Conditional probability is a probability that is based on knowing that some event within a sample space has already occurred. The notation for the probability of the event E occurring when it is known that the event F has occurred is p(e F ) and is read the probability of E given F. The formula for computing the conditional probability is p(e F ) = p(e F ). p(f ) Example 18. In our previous example we found p(e F ) = 2/3 by reducing the size of our sample space and using the basic probability formula. We could have found it using the formula. E F = {HT, T H} so p(e F ) = 1/2 Using the formula, we get p(e F ) = 1/2 3/4 = = 2 3, which is exactly the same value we got by counting.

8 8 Geometry Some examples should be inserted here. Associated with conditional probability is the idea of independence of two events. Two events are independent events if knowing that one of them has occurred does not change the probability that the other occurs; p(e F ) = p(e). Some examples need to be inserted here. to be enhanced

### + 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

### Math/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability

Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock

### 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

### Section 6.2 Definition of Probability

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 that it will

### GENETIC CROSSES. Monohybrid Crosses

GENETIC CROSSES Monohybrid Crosses Objectives Explain the difference between genotype and phenotype Explain the difference between homozygous and heterozygous Explain how probability is used to predict

### Basic concepts in probability. Sue Gordon

Mathematics Learning Centre Basic concepts in probability Sue Gordon c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Set Notation You may omit this section if you are

### 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)

### 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

### 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

### 11.1 The Work of Gregor Mendel

11.1 The Work of Gregor Mendel Lesson Objectives Describe Mendel s studies and conclusions about inheritance. Describe what happens during segregation. Lesson Summary The Experiments of Gregor Mendel The

### V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay \$ to play. A penny and a nickel are flipped. You win \$ if either

### Chapter 13 & 14 - Probability PART

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

### Mendelian Genetics. I. Background

Mendelian Genetics Objectives 1. To understand the Principles of Segregation and Independent Assortment. 2. To understand how Mendel s principles can explain transmission of characters from one generation

### 2 GENETIC DATA ANALYSIS

2.1 Strategies for learning genetics 2 GENETIC DATA ANALYSIS We will begin this lecture by discussing some strategies for learning genetics. Genetics is different from most other biology courses you have

### Probability Review Solutions

Probability Review Solutions. A family has three children. Using b to stand for and g to stand for, and using ordered triples such as bbg, find the following. a. draw a tree diagram to determine the sample

### AP: LAB 8: THE CHI-SQUARE TEST. Probability, Random Chance, and Genetics

Ms. Foglia Date AP: LAB 8: THE CHI-SQUARE TEST Probability, Random Chance, and Genetics Why do we study random chance and probability at the beginning of a unit on genetics? Genetics is the study of inheritance,

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

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

### Chapter 4 Probability

The Big Picture of Statistics Chapter 4 Probability Section 4-2: Fundamentals Section 4-3: Addition Rule Sections 4-4, 4-5: Multiplication Rule Section 4-7: Counting (next time) 2 What is probability?

### Chapter 5. Discrete Probability Distributions

Chapter 5. Discrete Probability Distributions Chapter Problem: Did Mendel s result from plant hybridization experiments contradicts his theory? 1. Mendel s theory says that when there are two inheritable

### I. WHAT IS PROBABILITY?

C HAPTER 3 PROBABILITY Random Experiments I. WHAT IS PROBABILITY? The weatherman on 0 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and

### 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

### Introductory Probability. MATH 107: Finite Mathematics University of Louisville. March 5, 2014

Introductory Probability MATH 07: Finite Mathematics University of Louisville March 5, 204 What is probability? Counting and probability 2 / 3 Probability in our daily lives We see chances, odds, and probabilities

### Math 3C Homework 3 Solutions

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

### Cell Division. Use Target Reading Skills. This section explains how cells grow and divide.

Cell Processes and Energy Name Date Class Cell Processes and Energy Guided Reading and Study Cell Division This section explains how cells grow and divide. Use Target Reading Skills As you read, make a

### 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

### PROBABILITY 14.3. section. The Probability of an Event

4.3 Probability (4-3) 727 4.3 PROBABILITY In this section In the two preceding sections we were concerned with counting the number of different outcomes to an experiment. We now use those counting techniques

### STAT 319 Probability and Statistics For Engineers PROBABILITY. Engineering College, Hail University, Saudi Arabia

STAT 319 robability and Statistics For Engineers LECTURE 03 ROAILITY Engineering College, Hail University, Saudi Arabia Overview robability is the study of random events. The probability, or chance, that

### Chapter 4: Probability and Counting Rules

Chapter 4: Probability and Counting Rules Learning Objectives Upon successful completion of Chapter 4, you will be able to: Determine sample spaces and find the probability of an event using classical

### Probability OPRE 6301

Probability OPRE 6301 Random Experiment... Recall that our eventual goal in this course is to go from the random sample to the population. The theory that allows for this transition is the theory of probability.

### Definition and Calculus of Probability

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

### LAB : THE CHI-SQUARE TEST. Probability, Random Chance, and Genetics

Period Date LAB : THE CHI-SQUARE TEST Probability, Random Chance, and Genetics Why do we study random chance and probability at the beginning of a unit on genetics? Genetics is the study of inheritance,

### 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

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

What does it mean for something to be random? An event is called random if the process which produces the outcome is sufficiently complicated that we are unable to predict the precise result and are instead

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

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

### Genetics 1. Defective enzyme that does not make melanin. Very pale skin and hair color (albino)

Genetics 1 We all know that children tend to resemble their parents. Parents and their children tend to have similar appearance because children inherit genes from their parents and these genes influence

### Chapter 3. Probability

Chapter 3 Probability Every Day, each us makes decisions based on uncertainty. Should you buy an extended warranty for your new DVD player? It depends on the likelihood that it will fail during the warranty.

### 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

### Homework 3 Solution, due July 16

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

### 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

### Probability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes

Probability Basic Concepts: Probability experiment: process that leads to welldefined results, called outcomes Outcome: result of a single trial of a probability experiment (a datum) Sample space: all

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

6.7. Probability. The study of probability has increased in popularity over the years because of its wide range of practical applications. In probability, each repetition of an experiment is called a trial,

### Variations on a Human Face Lab

Variations on a Human Face Lab Introduction: Have you ever wondered why everybody has a different appearance even if they are closely related? It is because of the large variety or characteristics that

### Mendelian Inheritance & Probability

Mendelian Inheritance & Probability (CHAPTER 2- Brooker Text) January 31 & Feb 2, 2006 BIO 184 Dr. Tom Peavy Problem Solving TtYy x ttyy What is the expected phenotypic ratio among offspring? Tt RR x Tt

### 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

### Probability: The Study of Randomness Randomness and Probability Models. IPS Chapters 4 Sections 4.1 4.2

Probability: The Study of Randomness Randomness and Probability Models IPS Chapters 4 Sections 4.1 4.2 Chapter 4 Overview Key Concepts Random Experiment/Process Sample Space Events Probability Models Probability

### Genetics Copyright, 2009, by Dr. Scott Poethig, Dr. Ingrid Waldron, and Jennifer Doherty Department of Biology, University of Pennsylvania 1

Genetics Copyright, 2009, by Dr. Scott Poethig, Dr. Ingrid Waldron, and Jennifer Doherty Department of Biology, University of Pennsylvania 1 We all know that children tend to resemble their parents in

### 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

### Ch. 13.3: More about Probability

Ch. 13.3: More about Probability Complementary Probabilities Given any event, E, of some sample space, U, of a random experiment, we can always talk about the complement, E, of that event: this is the

### 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

### Chapter 3: The basic concepts of probability

Chapter 3: The basic concepts of probability Experiment: a measurement process that produces quantifiable results (e.g. throwing two dice, dealing cards, at poker, measuring heights of people, recording

### Probability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection.

1 3.1 Sample Spaces and Tree Diagrams Probability This section introduces terminology and some techniques which will eventually lead us to the basic concept of the probability of an event. The Rare Event

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

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

### Worked examples Basic Concepts of Probability Theory

Worked examples Basic Concepts of Probability Theory Example 1 A regular tetrahedron is a body that has four faces and, if is tossed, the probability that it lands on any face is 1/4. Suppose that one

### Counting principle, permutations, combinations, probabilities

Counting Methods Counting principle, permutations, combinations, probabilities Part 1: The Fundamental Counting Principle The Fundamental Counting Principle is the idea that if we have a ways of doing

### Unit 21: Binomial Distributions

Unit 21: Binomial Distributions Summary of Video In Unit 20, we learned that in the world of random phenomena, probability models provide us with a list of all possible outcomes and probabilities for how

### Statistical Inference. Prof. Kate Calder. If the coin is fair (chance of heads = chance of tails) then

Probability Statistical Inference Question: How often would this method give the correct answer if I used it many times? Answer: Use laws of probability. 1 Example: Tossing a coin If the coin is fair (chance

### 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

### 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

### 4.4 Conditional Probability

4.4 Conditional Probability It is often necessary to know the probability of an event under restricted conditions. Recall the results of a survey of 100 Grade 12 mathematics students in a local high school.

### number of favorable outcomes total number of outcomes number of times event E occurred number of times the experiment was performed.

12 Probability 12.1 Basic Concepts Start with some Definitions: Experiment: Any observation of measurement of a random phenomenon is an experiment. Outcomes: Any result of an experiment is called an outcome.

### Heredity. Sarah crosses a homozygous white flower and a homozygous purple flower. The cross results in all purple flowers.

Heredity 1. Sarah is doing an experiment on pea plants. She is studying the color of the pea plants. Sarah has noticed that many pea plants have purple flowers and many have white flowers. Sarah crosses

### INTRODUCTION TO GENETICS USING TOBACCO (Nicotiana tabacum) SEEDLINGS

INTRODUCTION TO GENETICS USING TOBACCO (Nicotiana tabacum) SEEDLINGS By Dr. Susan Petro Based on a lab by Dr. Elaine Winshell Nicotiana tabacum Objectives To apply Mendel s Law of Segregation To use Punnett

### Solutions to Self-Help Exercises 1.3. b. E F = /0. d. G c = {(3,4),(4,3),(4,4)}

1.4 Basics of Probability 37 Solutions to Self-Help Exercises 1.3 1. Consider the outcomes as ordered pairs, with the number on the bottom of the red one the first number and the number on the bottom of

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

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

### Unit 19: Probability Models

Unit 19: Probability Models Summary of Video Probability is the language of uncertainty. Using statistics, we can better predict the outcomes of random phenomena over the long term from the very complex,

### Probability Using Dice

Using Dice One Page Overview By Robert B. Brown, The Ohio State University Topics: Levels:, Statistics Grades 5 8 Problem: What are the probabilities of rolling various sums with two dice? How can you

### A trait is a variation of a particular character (e.g. color, height). Traits are passed from parents to offspring through genes.

1 Biology Chapter 10 Study Guide Trait A trait is a variation of a particular character (e.g. color, height). Traits are passed from parents to offspring through genes. Genes Genes are located on chromosomes

### Probability & Probability Distributions

Probability & Probability Distributions Carolyn J. Anderson EdPsych 580 Fall 2005 Probability & Probability Distributions p. 1/61 Probability & Probability Distributions Elementary Probability Theory Definitions

### 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,

### AP Stats - Probability Review

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

### CCR Biology - Chapter 7 Practice Test - Summer 2012

Name: Class: Date: CCR Biology - Chapter 7 Practice Test - Summer 2012 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A person who has a disorder caused

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

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

### 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

### c. Law of Independent Assortment: Alleles separate and do not have an effect on another allele.

Level Genetics Review KEY Describe the 3 laws that Gregor Mendel established after working with pea plants. a. Law of Dominance: states that the effect of a recessive allele is not observed when a dominant

### Probabilities. Probability of a event. From Random Variables to Events. From Random Variables to Events. Probability Theory I

Victor Adamchi Danny Sleator Great Theoretical Ideas In Computer Science Probability Theory I CS 5-25 Spring 200 Lecture Feb. 6, 200 Carnegie Mellon University We will consider chance experiments with

### What about two traits? Dihybrid Crosses

What about two traits? Dihybrid Crosses! Consider two traits for pea: Color: Y (yellow) and y (green) Shape: R (round) and r (wrinkled)! Each dihybrid plant produces 4 gamete types of equal frequency.

### Decision Making Under Uncertainty. Professor Peter Cramton Economics 300

Decision Making Under Uncertainty Professor Peter Cramton Economics 300 Uncertainty Consumers and firms are usually uncertain about the payoffs from their choices Example 1: A farmer chooses to cultivate

### 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.

### 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

### AMS 5 CHANCE VARIABILITY

AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and

### Chapter 16: law of averages

Chapter 16: law of averages Context................................................................... 2 Law of averages 3 Coin tossing experiment......................................................

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

7 th Grade Probability Unit 7.S.8 Interpret data to provide the basis for predictions and to establish experimental probabilities. 7.S.10 Predict the outcome of experiment 7.S.11 Design and conduct an

### You flip a fair coin four times, what is the probability that you obtain three heads.

Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.

### Probability and Venn diagrams UNCORRECTED PAGE PROOFS

Probability and Venn diagrams 12 This chapter deals with further ideas in chance. At the end of this chapter you should be able to: identify complementary events and use the sum of probabilities to solve

### MANDELIAN GENETICS. Crosses that deviate from Mandelian inherintance

MANDELIAN GENETICS Crosses that deviate from Mandelian inherintance Explain codominant alleles. TO THE STUDENTS Calculate the genotypic and phenotypic ratio (1:2:1). Explain incomplete dominant alleles.

### Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution

Recall: Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.

### 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

### 7A The Origin of Modern Genetics

Life Science Chapter 7 Genetics of Organisms 7A The Origin of Modern Genetics Genetics the study of inheritance (the study of how traits are inherited through the interactions of alleles) Heredity: the

### Incomplete Dominance and Codominance

Name: Date: Period: Incomplete Dominance and Codominance 1. In Japanese four o'clock plants red (R) color is incompletely dominant over white (r) flowers, and the heterozygous condition (Rr) results in

### https://assessment.casa.uh.edu/assessment/printtest.htm PRINTABLE VERSION Quiz 10

1 of 8 4/9/2013 8:17 AM PRINTABLE VERSION Quiz 10 Question 1 Let A and B be events in a sample space S such that P(A) = 0.34, P(B) = 0.39 and P(A B) = 0.19. Find P(A B). a) 0.4872 b) 0.5588 c) 0.0256 d)

### PROBABILITY. SIMPLE PROBABILITY is the likelihood that a specific event will occur, represented by a number between 0 and 1.

PROBABILITY SIMPLE PROBABILITY SIMPLE PROBABILITY is the likelihood that a specific event will occur, represented by a number between 0 and. There are two categories of simple probabilities. THEORETICAL

### Basic Probability Concepts

page 1 Chapter 1 Basic Probability Concepts 1.1 Sample and Event Spaces 1.1.1 Sample Space A probabilistic (or statistical) experiment has the following characteristics: (a) the set of all possible outcomes

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

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

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

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

### HEREDITY (B) In domestic cats, the gene for Tabby stripes (T) is dominant over the gene for no stripes (t)

GENETIC CROSSES In minks, a single gene controls coat color. The allele for a brown (B) coat is dominant to the allele for silver-blue (b) coats. 1. A homozygous brown mink was crossed with a silverblue