MTH281 Probability and Statistics Chapter 3 PDF

MTH281: Probability and Statistics Chapter 3: Probability Zayed University College of Natural and Health Sciences Spring 2025 Departm...

MTH281: Probability and Statistics Chapter 3: Probability Zayed University College of Natural and Health Sciences Spring 2025 Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 1 / 61  Outline 1 Basic Probability Concepts Probability Experiments Fundamental Counting Principle Types of Probability 2 Elementary Probability Rules Complement Rule Addition Rule 3 Conditional Probability and Independence Conditional Probability Independence Multiplication Rule Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 2 / 61  Outline Objectives In this chapter, you learn: Describe sample spaces and events for random experiments with graphs, tables, lists, or tree diagrams. Interpret probabilities and use the probabilities of outcomes to calculate probabilities of events in discrete sample spaces. Calculate the probabilities of joint events such as unions and in- tersections from the probabilities of individual events. Interpret and calculate conditional probabilities of events. Determine the independence of events and use independence to calculate probabilities. Use permutations and combinations to count the number of out- comes in both an event and the sample space. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 3 / 61  Basic Probability Concepts Table of Contents 1 Basic Probability Concepts Probability Experiments Fundamental Counting Principle Types of Probability 2 Elementary Probability Rules Complement Rule Addition Rule 3 Conditional Probability and Independence Conditional Probability Independence Multiplication Rule Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 4 / 61  Basic Probability Concepts Probability Experiments Probability Experiments Definition (Probability Experiment) A Probability experiment is any process of observation that has an uncertain outcome. The process must be defined so that on any single repetition of the experiment, one and only one of the possible outcomes will occur. Examples: Obtaining blood types from a group of individuals. Measuring the pH reading of a water sample. Determining the brand of the laptop owned by a student. Recording the time of a chemical reaction. Counting the number of errors in a code. Determining whether a network is configured correctly. Identifying Vitamin D status. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 5 / 61  Basic Probability Concepts Probability Experiments Basic Principles: Sample Space Definition (Sample Space) The sample space, S, is the set of all possible outcomes of a proba- bility experiment. Examples: 1 Examine a fuse for a defect and note N or D. S = {N , D} 2 Examine two fuses in sequence and note the outcome. S = {N N , N D, DN , DD} 3 Genders of three children from oldest to youngest. S = {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG} 4 Run a computer program. S = {compiles, does not compile} Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 6 / 61  Basic Probability Concepts Probability Experiments Sample spaces can also be described graphically with tree diagrams. When a sample space can be constructed in several steps or stages, we can represent each of the n1 ways of completing the Tree Diagram first step as a branch of a tree. Each of the ways of completing the second step can be represented as n2 branches starting from the ends of the original branches, and so forth. A tree diagram is a graphical representation that helps in visu- xample 2-3 alizing a multiple-step Message experiment. Delays Each message in a digital communication system is classified as to whether i Example: Each message ed is received within the time specifi inbyathedigital communication system design. system If three messages are classified,isuse a ree diagram to represent the sample space of possible outcomes. classified Each message as toeither can be received whether on time oritlate. is received The possible within results for the time specified three messages by by can be displayed ight branches thein the tree diagram system shown in If design. Fig.three 2-5. messages are classified, use a tree Practical Interpretation: A tree diagram can effectively represent a sample space. Even if a tree becomes too large to diagram onstruct, it can to represent still conceptually the sample clarify the sample space. space of possible outcomes. Message 1 On time Late Message 2 On time Late On time Late Message 3 On time Late On time Late On time Late On time Late IGURE 2-5 S Tree= {OOO, diagram for three messages. OOL, OLO, OLL, LOO, LOL, LLO, LLL} Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 7 / 61  Basic Probability Concepts Probability Experiments Basic Principles: Events Definition (Event) An event is a subset of the sample space of a random experiment. Example: Genders of three children. S = {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG} A= Two girls. A = {BGG, GBG, GGB} B= All three children are of the same gender. B = {BBB, GGG} C= At most one boy. C = {GGG, BGG, GBG, GGB} Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 8 / 61  Basic Probability Concepts Probability Experiments Basic Principles: Simple Events Definition (Simple Event) An event is said to be simple if it consists of exactly one outcome. An event that consists of more than one outcome is not a simple event. Example: Genders of three children. S = {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG} A= Three boys. C = {BBB} ⇒ Simple event B= Two girls. A = {BGG, GBG, GGB} ⇒ Not a simple event Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 9 / 61  Basic Probability Concepts Probability Experiments Exercises 1 Describe the sample space for the following experiments. a Vitamin D status of a selected student. b Inspect chips from a manufacturing process until the first defec- tive is discovered. c A light bulb is observed so that the length of its useful life might be recorded. d The number of hits is recorded at ZU Web site in a day. 2 Determining a person’s blood type (A, B, AB, O) and Rh-factor (positive, negative). a Develop a tree diagram for this experiment. b Describe the sample space of the experiment. c Define and classify (simple or not) the following events: I A= A person has both type O blood and the Rh– factor. II B= A person has Rh+ factor. III C= A person has type O blood. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 10 / 61  Basic Probability Concepts Fundamental Counting Principle Fundamental Counting Principle In certain situations, determining the outcomes in the sample space (or an event) becomes more difficult. Instead, counts of the numbers of outcomes in the sample space and various events are used to analyze the random experiments using counting techniques. Fundamental Counting Principle If one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence is m × n. For a sequence of k events in which the first event can occur n1 ways, the second event can occur n2 ways, the third event can occur n3 ways, and so on, the total number of ways of completing the operation is n1 × n2 × · · · × nk Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 11 / 61  Basic Probability Concepts Fundamental Counting Principle Examples 1 The design for a Website is to consist of four colors, five fonts, and three positions for an image. From the multiplication rule, Number of different designs = 4 × 5 × 3 = 60 2 In one year, three awards (research, teaching, and service) will be given for a class of 25 graduate students in a computer science department. If each student can receive at most one award, how many possible selections are there? Number of ways = 25 × 24 × 23 = 13, 800 3 How many passwords can be generated if it must consist of 4 different lower-case letters? Number of different passwords = 26 × 25 × 24 × 23 = 358, 800 Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 12 / 61  Basic Probability Concepts Fundamental Counting Principle Exercises 1 New designs for a wastewater treatment tank have proposed three possible shapes, four possible sizes, three locations for input valves, and four locations for output valves. How many different product designs are possible? 2 Passwords consist of eight characters from the 10 digits and the 26 letters of the alphabet, and they are case sensitive. a How many different passwords can be formed? b How many of these consist of different characters? c At a speed of 10 million passwords per second, how long will take a spy program to try all of them? 3 A software company is hiring for two positions: a software de- velopment engineer and a sales operations manager. How many ways can these positions be filled if there are 15 people applying for the engineering position and 18 people applying for the man- agerial position? Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 13 / 61  Basic Probability Concepts Fundamental Counting Principle Exercises 4 An access code consists of a letter followed by four digits. Any letter can be used, the first digit cannot be 0, and the last digit must be even. a How many different access codes could be generated? b How many of these access codes contain only even numbers? c How many of these access codes contain no vowels? 5 An experimenter is studying the effects of temperature, pressure, and type of catalyst on yield from a chemical reaction. Three different temperatures, four different pressures, and five different catalysts are under consideration. a If any particular experimental run involves the use of a single temperature, pressure, and catalyst, how many experimental runs are possible? b How many experimental runs involve use of the lowest tempera- ture and two lowest pressures? Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 14 / 61  a. Identify the Basicevent. Probability Concepts Types of Probability b. Decide whether the probability is determined by knowing all possible Probability outcomes, whether the probability is estimated from the results of an experiment, or whether the probability is an educated guess. c. Make a conclusion. Probability of an Event Answer: Page A35 Probability is a measure of the chance that an experimental outcome A probability cannot be negative or greater than 1, as stated in the rule will occurbelow. when an experiment is carried out. Probability R AAxioms NGE OF PROBABILITIES RULE If E is anThe experimental probability of anoutcome, then 0Pand(E) event E is between denotes 1, inclusive. Thatthe is, probability that E will occur and: 0 … P1E2 … 1. 1 The probability of any outcome is between 0 and 1, i.e. When the probability of an event is 1, the event is certain to occur. When the ≤ Pis(E) probability of an event is 0, the0event ≤ 1 A probability of 0.5 indicates impossible. that an event has an even chance of occurring or not occurring. 2 The figure The possible below shows range the possible range of probabilities of probabilities and and their meanings. their meanings: Impossible Unlikely Even chance Likely Certain 0 0.25 0.5 0.75 1 3 The An sum of that event theoccurs probabilities of allofthe with a probability 0.05 outcomes equals or less is typically 1. considered unusual. Unusual events are highly unlikely to occur. Later in this course you will identifyand Department of Mathematics unusual eventsMTH281: Statistics when studying Probabilityinferential statistics. and Statistics Spring 2025 15 / 61  Basic Probability Concepts Types of Probability Types of Probability 1 The classical (or theoretical) probability: ▶ The classical method makes certain assumptions (such as equally likely, independence) about situation. ▶ Equally likely events are events that have the same probability of occurring. ▶ If an experiment has n possible equally likely outcomes, this method would assign a probability of n1 to each outcome. ▶ If the sample space outcomes are equally likely to occur, then: Number of outcomes in A P (A) = Number of outcomes in S ▶ Examples: 1 The probability of guessing an ATM PIN is 10,000 = 0.0001. 4 When rolling two dice, the probability of getting a sum of 9 is 36. The probability of getting three children all of the same gender when three children are born (assuming that boys and girls are equally likely) is 82 = 0.25. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 16 / 61  Basic Probability Concepts Types of Probability Types of Probability 2 Empirical (or statistical) probability: ▶ Empirical probability (Also called Relative frequency method) as- signs probabilities based on experimentation or historical data. ▶ Let E be an outcome of an experiment. Then, if the experiment is performed N times, P (E) is the relative frequency of E, i.e., Number of times E occurs P (E) = N ▶ Examples: A random sample of 500 college students were surveyed and it was determined that 425 have Facebook account. The probability that a college student selected at random has Facebook account 425 is = 0.85(85%). 500 Based on Worldmeter (August 13, 2023), the probability of re- 1, 054, 525 covery from Corona virus in UAE is = 0.9883. 1, 067, 030 AVG Free detects 99.7% of real-world malware samples. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 17 / 61  Basic Probability Concepts Types of Probability Types of Probability 3 The subjective probability: ▶ Subjective methods assign probabilities based on a combination of an individual’s past experience, educated guess, personal opinion, and analysis of a particular situation. ▶ The subjective probability, differs from the other two approaches because subjective probability differs from person to person. ▶ Subjective probability is especially useful in making decisions in situations in which you cannot use a classical probability or em- pirical probability. ▶ Examples: The probability that you will get an A in this course. A technology analyst said that there is a 90% chance that the launch event of the iPhone 17 will be in early September 2025. The Oxford Vaccine Group announced in April 2020 that there is an 80% chance of developing an effective COVID-19 vaccine by September. However, two months later, they announced that the vaccine has only 50% chance of working. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 18 / 61  Basic Probability Concepts Types of Probability Helpful Hints on Probability Calculations If the sample space outcomes are equally likely to occur, then: Number of outcomes in A P (A) = Number of outcomes in S n(A) = n(S) Probability of “at least one”: ▶ “At least one” is equivalent to “one or more.” Hence, the com- plement of getting at least one item of a particular type is that you get no items of that type. ▶ To find the probability of “at least one of something”, calcu- late the probability of none and then subtract that result from 1. That is, P (at least one) = 1–P (none) Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 19 / 61  Basic Probability Concepts Types of Probability Examples 1 Classify each statement as an example of classical probability, empirical probability, or subjective probability. Explain your rea- soning. a A healthcare researcher says that the probability that UAE will have a 7th wave of COVID-19 in January 2023 is 0.25. This probability is most likely based on an educated guess. It is an example of subjective probability. b The probability that a randomly selected Emirati adult is obese is 0.32. This statement is most likely based on a survey of a sample of Emirati adults, so it is an example of empirical probability. c The probability of winning a 1000-ticket raffle with one ticket is 1 1000. Because you know the number of outcomes and each is equally likely, this is an example of classical probability. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 20 / 61  Basic Probability Concepts Types of Probability Examples 2 A computer system uses passwords that consist of four letters followed by a single digit. a Find the probability that a randomly chosen password will not have any repeated letters. n(S) = 26 × 26 × 26 × 26 × 10 = 10 × 264 = 4, 569, 760 n(A) = 26 × 25 × 24 × 23 × 10 = 3, 588, 000 3, 588, 000 P (A) = = 0.7852 4, 569, 760 b Find the probability that a randomly chosen password will have at least one vowel. P (At least one vowel) = 1 − P (No vowels) 21 × 21 × 21 × 21 × 10 = 1− 4, 569, 760 = 1 − 0.4256 = 0.5744 Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 21 / 61  Basic Probability Concepts Types of Probability Exercises 1 Classify the statement as classical probability, empirical probabil- ity, or subjective probability. Explain your reasoning. a A doctor may feel a patient has a 90% chance of a full recovery. b A customer will randomly select an iPhone from an inventory of four white and one black iPhones. c According to company records, the probability that a laptop will need repairs during a six-year period is 0.10. d More than 30% of the results from major search engines for the keyword phrase “ring tone” are fake pages. e The meteorologist reports that there is a 70% chance that the rainfall in UAE during October 2020 will be above normal. f A survey showed that 44% of online Internet shoppers experience some kind of technical failure at checkout. g The probability that a randomly selected ATM PIN number start- ing with 6 is 0.10. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 22 / 61  Basic Probability Concepts Types of Probability Exercises 2 Three cybersecurity experts were asked to assess the probability that a major data breach will occur in a multinational corporation within the next year. The experts stated their probabilities as 0.4, 0.6 and 0.9. a What method of probability assessment are the three experts using? b Why did the experts provide different probability assessments? c Which expert is expressing the least uncertainty in the probability assessment? 3 Can you write with your left hand? Use the students in your statistics class to estimate the percentage of people who are left- handed. How can your result be thought of as an estimate for the probability that a person chosen at random is left-handed? What type of probability is being used in this estimation? Explain why it is considered that type. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 23 / 61  Basic Probability Concepts Types of Probability Exercises 4 Each of two parents has the genotype brown/blue, which consists of the pair of alleles that determine eye color, and each parent contributes one of those alleles to a child. Assume that if the child has at least one brown allele, that color will dominate and the eyes will be brown. (The actual determination of eye color is more complicated than that.) a List the different possible outcomes (Assume equally likely). b What is the probability that a child of these parents will have the blue/blue genotype? c What is the probability that the child will have brown eyes? d How would your answers to the above questions change if the outcomes were not equally likely? Explain how you would ap- proach calculating the probabilities in this case. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 24 / 61  Basic Probability Concepts Types of Probability Exercises 5 During a health awareness campaign at a shopping mall, 250 peo- ple walked by a free blood pressure screening station. Of these, 90 stopped to have their blood pressure checked, and 35 were advised to follow up with a doctor due to high readings. a What is the probability that a person who walks by the blood pressure screening station will stop to have their blood pressure checked? b What is the probability that a person who stops to have their blood pressure checked will be advised to follow up with a doctor? c What is the probability that a person who walks by the blood pressure screening station will stop and be advised to follow up with a doctor? Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 25 / 61  Basic Probability Concepts Types of Probability Exercises 6 A security code consists of a person’s first and last initials followed by four digits. a Find the probability of guessing a person’s code on the first try. b You know a person’s first name and that the last digit is odd. Find the probability of guessing his/her code on the first try. c The statements in parts (a) and (b) are examples of which prob- ability? Explain your reasoning. 7 An 8-character database password is formed from 10 digits, 26 lower-case and 26 upper-case letters. a How many different passwords can be formed? b What is the probability that a randomly selected password does not have repeated characters? c What is the probability that a randomly selected password con- tains at least one digit? Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 26 / 61  Basic Probability Concepts Types of Probability Exercises 8 The password of an electronic equipment consists of a string of seven characters such that the first three are lower case letters and the last four are numbers. a How many possible passwords can be formed? b What is the probability that the string of three letters begins with a “w” and the string of four numbers begins with a “4”? c What is the probability that a randomly selected password con- tains no vowels? d What is the probability that a randomly selected password con- tains at least one odd number? 9 What is the probability that in a class of 10 students no two students will have the same birth date? (Hint: This is called the Birthday Problem). Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 27 / 61  Elementary Probability Rules Table of Contents 1 Basic Probability Concepts Probability Experiments Fundamental Counting Principle Types of Probability 2 Elementary Probability Rules Complement Rule Addition Rule 3 Conditional Probability and Independence Conditional Probability Independence Multiplication Rule Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 28 / 61  1. P1A2 1 P1A 2 5 1 Elementary Probability 2. P1event Rules Complement A does not occur2 Rule 5 P1Ac 2 5 1  P1A2 (3) Complementary 3 Events Complement of an EventExample The probability that a college student who has not received a flu shot will get the flu is 0.45. What is the probability that a college student will not get the flu if the student Definition (Complement) has not had the flu shot? Solution: In this case, we have The complement of an event denoted by A′ (sometimes Ac or Ā), A,5 0.45 P1will get flu2 P1will not get flu2 5 1  P1will get flu2 5 1  0.45 5 0.55 is the set of all outcomes in S that are not contained in A. Figure 4-1 A' The Event A and Its Complement Ac A Sample space Complementary Events For any event A, Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 19917_ch04_ptg01_hr_142-195.indd 148 30/09/16 5:53 PM P (A′ ) = 1 − P (A) Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 29 / 61  Elementary Probability Rules Complement Rule Example Among 32 dieters following a similar routine, 18 lost weight, 5 gained weight, and 9 remained the same weight. If one of these dieters is randomly chosen, find the probability that he or she: a Gained weight (G). 5 P (G) = = 0.1563 32 b Lost weight (L). 18 P (L) = = 0.5625 32 c Did not lose weight (L′ ). P (L′ ) = 1 − P (L) = 1 − 0.5625 = 0.4375 Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 30 / 61  Elementary Probability Rules Addition Rule are mutually exclusive when A and B cannot occur at Event Operations: Intersection Definition (Intersection) s show the relationship The intersection between of two events events A that and B, are mutually denoted by “A and B”, is at are the event consisting of all outcomes that are in bothAAand not mutually exclusive. Note that when events and B (Also e, they havebynoAoutcomes denoted ∩ B). in common, so P1A and B2 = 0.        A and B A B B utually Departmentexclusive. A and of Mathematics and Statistics B are MTH281: not mutually Probability and Statistics exclusive. Spring 2025 31 / 61  course, if you bought a convertible with Probability Elementary leather upholstery, Rules that would Addition Rulebe fine, too. Pictorially, the shaded portion of Figure 4-4(b) represents the outcomes satisfying the Event Operations: Union or condition. Notice that the condition A or B is satisfied by any one of the following conditions: 1. Any outcome in A occurs. 2. Any outcome in B occurs. Definition (Union) 3. Any outcome in both A and B occurs. TheIt is important uniontoofdistinguish between the two events combinations Aorand and the and by B, denoted combina- “A or B” is the event tions because we apply different rules to compute their probabilities. consisting of all outcomes that are either in A or in B or in both events (Also denoted by A ∪ B). (b) The Event A or B A or B B A B Sample space Sample space Combining Events Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 32 / 61  Elementary Probability Rules Addition Rule DEFINITION Mutually Exclusive Events Two events A and B are mutually exclusive when A and B cannot occur at the same time. Definition (Mutually Exclusive Events) TwoVenn The events A andshow diagrams B arethemutually relationshipexclusive whenthat between events A and B cannot are mutually xclusive and events that are not mutually exclusive. occur at the same time (i.e., A and B = ϕ). Note that when events A and are mutually exclusive, they have no outcomes in common, so P1A and B2 = 0.        A and B A A B B A and B are mutually exclusive. A and B are not mutually exclusive. EDepartment X A MofPMathematics L E and 1 Statistics MTH281: Probability and Statistics Spring 2025 33 / 61  Elementary Probability Rules Addition Rule The prob Addition Rule P1A If events Addition Rule P1A or B For any two events A and B, number P (A or B) = P (A) + P (B) − P (A and B) In wo the individ both occu B avoids dou A and B EXAM A Using 1.  You Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 34 / 61 YOU SHOULD LEARN Elementary Probability Rules Mutually Exclusive Events The Addition Rule Addition Rule A Summary of Probab MUTUALLY EXCLUSIVE EVENTS o determine whether two Addition Rule for3.2,Mutually s are mutually exclusive In Section you learned how to Exclusive find the probability ofEvents two events, A an o use the Addition Rule occurring in sequence. Such probabilities are denoted by P1A and B2. In d the probability of section, you will learn how to find the probability that at least one of two ev vents Addition Rule forwillMutually Exclusive occur. Probabilities Events such as these are denoted by P1A or B2 and depen whether the events are mutually exclusive. For any two mutually exclusive events A and B, DEFINITION P (A Two or B) events =BPare(A) A and + P (B) mutually exclusive when A and B cannot occur the same time. In general, if the events A1 , A2 ,... , AN are mutually exclusive then, p The Venn diagrams show the relationship between events that are mut P (A or A2 or exclusive ility and statistics, the s usually used as1an... or and ANevents B are mutually = that )exclusive, P (A are not mutually exclusive. Note that when events A 1 )have they + Pno(A 2 ) +.in.common, outcomes. + P (A so N ) P1A and B2 or” rather than an or.” For instance,        three ways for A and B or B” to occur. A urs and B does cur. A B urs and A does B cur. B both occur. A and B are mutually exclusive. A and B are not mutually exclusi Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 35 / 61  Elementary Probability Rules Addition Rule Example (1) A survey of homeowners showed that during the past year, 45% pur- chased a smart device to monitor energy consumption, 54% purchased a smart device to monitor water usage, and 30% purchased smart de- vices to monitor both energy consumption and water usage. Let A = purchased a smart device to monitor energy consumption B = purchased a smart device to monitor water usage Then, we have P (A) = 0.45, P (B) = 0.54 and P (A and B) = 0.30. a What is the probability that a homeowner did not purchase a smart device to monitor energy consumption? P (A′ ) = 1 − P (A) = 1 − 0.45 = 0.55 Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 36 / 61  Elementary Probability Rules Addition Rule Example (1) b What is the probability that a homeowner did purchase a smart device to monitor energy consumption or water usage? P (A or B) = P (A) + P (B) − P (A and B) = 0.45 + 0.54 − 0.30 = 0.69 c What is the probability that a homeowner purchased neither smart device? P (Neither) = 1 − P (A or B) = 1 − 0.69 = 0.31 d Are “purchase a smart device to monitor energy consumption” and “purchase a smart device to monitor water usage” mutually exclusive? Explain. No, since P (A and B) ̸= 0. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 37 / 61  Elementary Probability Rules Addition Rule Example (2) A blood bank catalogs the types of blood, including positive or negative Rh-factor, given by donors during the last five days. The number of donors who gave each blood type is shown in the table. A donor is selected at random. Find the probability that the donor Blood type Rh-factor O A B AB Total Positive (Rh+) 156 139 37 12 344 Negative (Rh-) 28 25 8 4 65 Total 184 164 45 16 409 a has type O or type A blood. The events are mutually exclusive (a donor cannot have type O blood and type A blood) P (O or A) = P (O) + P (A) 184 164 348 = + = = 0.8509 409 409 409 Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 38 / 61  Elementary Probability Rules Addition Rule Example (2) b has type B blood or is Rh-negative. The events are not mutually exclusive (a donor can have type B blood and be Rh-negative) P (B or Rh-) = P (B) + P (Rh-) − P (B and Rh-) 45 65 8 102 = + − = = 0.2494 409 409 409 409 c has type O blood or is Rh-positive. The events are not mutually exclusive (a donor can have type O blood and be Rh-positive) P (O or Rh+) = P (O) + P (Rh+) − P (O and Rh+) 184 344 156 372 = + − = = 0.9095 409 409 409 409 Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 39 / 61  Elementary Probability Rules Addition Rule Exercises 1 Out of 800 purchases made at a computer retailer, 336 were PCs, 398 were laptops, and 66 were printers. As part of an audit, one purchase record is sampled at random. a What is the probability that it is a PC? b What is the probability that it is not a printer? c What is the probability that it is a PC or a laptop? 2 Suppose that after 10 years of service, 40% of computers have problems with motherboards (MB), 30% have problems with hard drives (HD), and 15% have problems with both MB and HD. Find the probability that a 10-year old computer a does not have problems with MB. b has problems with MB or HD. c still has fully functioning MB and HD. d Are the events “has a problem with MB” and “has a problem with HD” mutually exclusive? Explain. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 40 / 61  Elementary Probability Rules Addition Rule Exercises 3 Suppose a flu epidemic strikes a city. In 10% of families the mother has flu; in 15% of families the father has flu; and in 23% of families at least one of the parents have flu. a What is the probability that both mother and father have flu? b What is the probability that neither mother nor father have flu? c Are the events “mother has flu” and “father has flu” mutually exclusive? Explain. 4 A computer system uses passwords that are six characters, and each character is one of the 26 letters (a–z) or 10 integers (0–9). Uppercase letters are not used. Let A denote the event that a password begins with a vowel, and let B denote the event that a password ends with an even number. Suppose a hacker selects a password at random. Determine the following probabilities: a P (A) c P (A and B) b P (B) d P (A or B) Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 41 / 61  Elementary Probability Rules Addition Rule Exercises 5 In a study of the relationship between health risk and income, a large group of people were asked a series of questions. Some of the results are shown in the following table. Income Smoking Low Medium High Total Smoke 634 332 247 1,213 Don’t smoke 1,846 1,622 1,868 5,336 Total 2,480 1,954 2,115 6,549 Find the probability that a randomly selected participant is a a smoker. b not from the low income group. c from the low income group and smokes. d from the low income group or smokes. e from the low income group or from the medium income group. f neither from the low income group nor smokes. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 42 / 61  Conditional Probability and Independence Table of Contents 1 Basic Probability Concepts Probability Experiments Fundamental Counting Principle Types of Probability 2 Elementary Probability Rules Complement Rule Addition Rule 3 Conditional Probability and Independence Conditional Probability Independence Multiplication Rule Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 43 / 61  Conditional Probability and Independence Conditional Probability Conditional Probability According to the International Diabetes Federation (IDF) in 2017, 17.3% of the UAE population between the ages of 20 and 79 have type 2 diabetes. Therefore, if a 50-year old resident is selected at random from UAE population, the probability that he/she has type 2 diabetes is 0.173. Now assume that you learn that the selected resident is obese. With this extra information, the probability that he/she has type 2 diabetes becomes much greater than 0.173. A probability that is computed with the knowledge of additional information is called a conditional probability; a probability computed without such knowledge is called an unconditional probability. As this example shows, the conditional probability of an event can be much different than the unconditional probability. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 44 / 61  Conditional Probability and Independence Conditional Probability Conditional Probability Conditional Probability The probability of event A occurring, given that event B has occurred, is called the conditional probability of event A given event B, denoted P (A|B) is P (A and B) P (A|B) = P (B) provided P (B) > 0. Similarly, the conditional probability of event B given event A, denoted P (B|A) is P (A and B) P (B|A) = P (A) provided P (A) > 0. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 45 / 61  Conditional Probability and Independence Conditional Probability Example (1) The probability that a regularly scheduled flight departs on time is P (D) = 0.85; the probability that it arrives on time is P (A) = 0.92; and the probability that it departs and arrives on time is 0.78. a Find the probability that a plane arrives on time given that it departed on time. P (A and D) 0.78 P (A|D) = = = 0.9176 P (D) 0.85 b Find the probability that a plane departed on time given that it has arrived on time. P (A and D) 0.78 P (D|A) = = = 0.8478 P (A) 0.92 Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 46 / 61  Conditional Probability and Independence Conditional Probability Example (2) The following table summarizes blood groups and Rh types for ran- domly selected subjects. Blood type Rh-factor O A B AB Total Positive (Rh+) 156 139 37 12 344 Negative (Rh-) 28 25 8 4 65 Total 184 164 45 16 409 a Find the probability that a randomly selected person has Rh+ blood given that he/she is a type A blood type. P (Rh+ and A) 139 P (Rh+ |A) = = = 0.8476 P (A) 164 b Find the probability that a randomly selected person is a type O blood type given that he/she has Rh− blood. P (Rh− and O) 28 P (O|Rh− ) = − = = 0.4308 P (Rh ) 65 Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 47 / 61  Conditional Probability and Independence Conditional Probability Exercises 1 Consider the following events for a driver selected at random from the general population: A = driver is under 25 years old B = driver has received a speeding ticket Translate each of the following phrases into symbols. a The probability the driver has received a speeding ticket and is under 25 years old. b The probability a driver who is under 25 years old has received a speeding ticket. c The probability a driver who has received a speeding ticket is 25 years old or older. d The probability the driver is under 25 years old or has received a speeding ticket. e The probability the driver has not received a speeding ticket or is under 25 years old. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 48 / 61  Conditional Probability and Independence Conditional Probability Exercises 2 The probability that a data communications system will have high selectivity is 0.72, the probability that it will have high fidelity is 0.59, and the probability that it will have both is 0.33. a Find the probability that a system with high fidelity will also have high selectivity. b Find the probability that a system with high selectivity will also have high fidelity. 3 A recent survey shows that 85% of college students have a Face- book profile, 72% use YouTube regularly, and 65% do both. a Suppose that a randomly selected college student has a profile on Facebook, find the probability that he/she uses YouTube reg- ularly. b Given that a randomly selected college student uses YouTube regularly, what is the probability that he/she has a Facebook profile? Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 49 / 61  Conditional Probability and Independence Conditional Probability Exercises 4 A large shipment contains 600 wafers categorized by lot and by whether they conform to a thickness specification. A wafer is chosen at random from the shipment. Lot Conforming Nonconforming Total A 88 12 100 B 165 35 200 C 260 40 300 Total 513 87 600 a If the wafer is from Lot A, what is the probability that it is conforming? b If the wafer is conforming, what is the probability that it is from Lot A? c If the wafer is conforming, what is the probability that it is not from Lot C? d If the wafer is not from Lot C, what is the probability that it is conforming? Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 50 / 61  Conditional Probability and Independence Conditional Probability Exercises 5 Obesity is a growing public health concern worldwide. Adults with a high body mass index (BMI) of 25 or greater are considered overweight or obese. The following table shows the number of adults (in millions) who are overweight or obese in countries with different income levels, based on data from the WHO and the UN. Income level High BMI Low BMI Total High 549 414 963 Middle 900 2125 3025 Low 63 357 420 Total 1512 2896 4408 a Find the probability that a randomly selected adult has a high BMI? b Given that an adult comes from a middle-income country, what is the probability that he has a high BMI? c Given that a randomly selected adult has a high BMI, what is the probability that he comes from a high-income country? Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 51 / 61  Conditional Probability and Independence Independence Independence If the probability of event A is not changed by the occurrence of event B (i.e., conditional and unconditional probabilities are the same), we would say that events A and B are independent. Events that are not independent are often said to be dependent. Independence ▶ Two events A and B are said to be independent if and only if: P (A|B) = P (A) or P (B|A) = P (B) ▶ If A and B are independent, then A and B ′ , A′ and B, and A′ and B ′ are independent as well. Note: In many situations, we can determine whether events are independent just by understanding the circumstances surrounding the events. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 52 / 61  Conditional Probability and Independence Independence Example Determine whether the following pairs of events are independent: a A college student is chosen at random. The events are “being a freshman” and “being less than 20 years old.” These events are not independent. If the student is a freshman, the probability that the student is less than 20 years old is greater than for a student who is not a freshman. b A college student is chosen at random. The events are “having a smartphone” and “taking a statistics class.” These events are independent. If a student has a smartphone, this has no effect on the probability that the student takes a statistics class. c A person has diabetes and is obese. These events are not independent. If the person is obese, the probability that the he/she will develop diabetes is greater than for a person who is not a obese. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 53 / 61  Conditional Probability and Independence Independence Exercises 1 Determine whether the events are independent or dependent. a Smoking a pack of cigarettes per day and developing emphysema, a chronic lung disease. b A father having hazel eyes and a daughter having hazel eyes. c A father having hazel eyes and a daughter is obese. d Majoring in Computer Science and exercising daily. 2 Suppose that after 10 years of service, 40% of computers have problems with motherboards (MB), 30% have problems with hard drives (HD), and 15% have problems with both MB and HD. a Find the probability that a 10-year old computer has a problem with MB given that it has a problem with HD. b Find the probability that a 10-year old computer has a problem with HD given that it has a problem with MB. c Are the events “has a problem with MB” and “has a problem with HD” independent? Explain. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 54 / 61  Conditional Probability and Independence Independence Exercise 3 A researcher surveyed 800 college students to explore their percep- tions of global warming. The respondents were classified accord- ing to gender (female or male) and whether or not they believe in global warming. Results are summarized in the following table: Believe in Global Warming Gender Yes No Total Female 150 330 480 Male 50 270 320 Total 200 600 800 a Find the probability that a selected student believes in global warming. b If a female student is randomly selected, find the probability that she believes in global warming. c Does being female affect the probability of believing in global warming? Explain. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 55 / 61  Conditional Probability and Independence Independence Exercise 4 A recent research study revealed the following data on the magni- tude of change in smell sense attributed to COVID-19 by gender in patients with COVID-19 in the UAE population. Change in smell sense attributed to COVID-19 Gender Extreme reduction Moderate reduction No change Total Male 115 23 183 321 Female 63 19 94 176 Total 178 42 277 497 a Find the probability that a randomly selected patient had an ex- treme reduction in smell sense. b If a female patient is randomly selected, find the probability that she had an extreme reduction in smell sense. c Does being female affect the probability of having extreme re- duction in smell sense? Explain. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 56 / 61  Conditional Probability and Independence Multiplication Rule Multiplication Rule Multiplication Rule Given any two events A and B, P (A and B) = P (A|B)P (B) = P (B|A)P (A) If events A and B are independent, then the rule will be P (A and B) = P (A)P (B) In general, if E1 , E2 , · · · , Ek are independent events then P (E1 and E2 and · · · and Ek ) = P (E1 )P (E2 ) · · · P (Ek ) Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 57 / 61  Conditional Probability and Independence Multiplication Rule Examples 1 A survey of the apps in the App store, indicated that 30% were related to games. Forty percent of the game apps are free. What is the proportion of free games apps available in the App store? Let G be the event that the app is related to games and F be that the app is free. Then, we have P (G) = 0.30, P (F |G) = 0.40. Using the multiplication rule, we have P (G and F ) = P (F |G)P (G) = (0.40)(0.30) = 0.12 2 Mariam has applied for positions at Company A and Company B. The probability of getting an offer from Company A is 0.45, and the probability of getting an offer from Company B is 0.3. Assuming that the two job offers are independent of each other, what is the probability that she gets an offer from both companies? P (A and B) = P (A)P (B) = (0.45)(0.3) = 0.135 Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 58 / 61  Conditional Probability and Independence Multiplication Rule Exercises 1 Suppose that 40% of adults in UAE have membership in health clubs, and 65% of these health club members go to the club at least twice a week. What is the probability that a randomly selected adult is a health club member and goes to the club at least twice a week? 2 The probability that a doctor correctly diagnoses a particular ill- ness is 0.7. Given that the doctor makes an incorrect diagnosis, the probability that the patient files a lawsuit is 0.9. What is the probability that the doctor makes an incorrect diagnosis and the patient sues? 3 A computer program consists of two blocks written independently by two different programmers. The first block has an error with probability 0.25. The second block has an error with probability 0.35. If the program returns an error, what is the probability that there is an error in both blocks? Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 59 / 61  Conditional Probability and Independence Multiplication Rule Exercises 4 The probability that a particular rotator cuff surgery is successful is 0.9. Find the probability that a three rotator cuff surgeries are successful. b none of the three rotator cuff surgeries are successful. c at least one of the three rotator cuff surgeries is successful. 5 A system has three computers. Computer 1 works with a probabil- ity of 0.85; computer 2 works with a probability of 0.78; computer 3 works with a probability of 0.94. Suppose that the operations of the computers are independent of each other. a Suppose that the system works only when all three computers are working. What is the probability that the system works? b Suppose that the system works only if at least one computer is working. What is the probability that the system works? c Suppose that the system works only if at least two computers are working. What is the probability that the system works? Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 60 / 61  Conditional Probability and Independence Multiplication Rule Exercises 6 Suppose that we have a fuse box containing 20 fuses, of which 5 are defective. If 2 fuses are selected at random and removed from the box in succession without replacing the first, a what is the probability that both fuses are defective? b what is the probability that only one fuse is defective? c Repeat (a) and (b) assuming that the first fuse has been replaced before selecting the second one. 7 A lake contains 1,000 fish, 250 of which are tagged for monitoring migration patterns. A researcher captures two fish at random for examination. Let A represent the event that the first fish captured is tagged, and let B represent the event that the second fish captured is tagged. a Find P (A) and P (B|A). b Find P (A and B) c Are A and B independent? Is it reasonable to treat A and B as though they were independent? Explain. Department of Mathematics and Statistics MTH281: Probability and Statistics Spring 2025 61 / 61