Chapter 4 – Probability (Page 1 of 24) 4.1 What is Probability? Probability is a numerical measure between 0 and 1 that describes the likelihood that an event will occur. Probabilities closer to 1 indicate that the event is more likely to occur. Probabilities near 0 indicate that the event is less likely to occur. Probability Notation P( A) , read “P of A,” denotes the probability of event A. P( A) = 1 means event A is certain to occur P( A) = 0 means event A is impossible. Probability based on Relative Frequency The probability of an event E occurring can be estimated by the relative frequency of the occurrence of the event. P( E) = frequency of E f = sample size n Example A From a random sample of 100 lab reports 40 had erroneous results. What is the probability that a lab report selected at random has an erroneous result? If event E is “has an erroneous result,” then f 40 P(E) = = = 0.40 n 100 So, approximately 40% of lab reports are erroneous. The estimate gets better as n gets larger and larger. Chapter 4 – Probability (Page 2 of 24) Law of Large Numbers In the long run, as the sample size increases, the relative frequency of outcomes gets closer and closer to the actual probability. Example B 1. Estimate the probability of “heads” occurring when tossing a coin. Simulate the experiment by using the TI-83 random number generator. STAT / 2: SortA( MATH / PRB / 5: randInt(min.range, max.range, # of numbers) a. Use 5 tosses b. Use 25 tosses c. Use 200 tosses Probability when the Outcomes are Equally Likely Probability of event E = Number of outcomes favorable to E Total number of outcomes Example C a. What is the probability of rolling a 3 on a die? b. What is the probability of rolling a 2 or 4 on a die? Chapter 4 – Probability (Page 3 of 24) Guided Exercise 1 Assign a probability for the following events and state if the technique was intuition, relative frequency, or the formula for equally likely outcomes. a. The health center director at a college found that 375 students out of 500 tested needed corrective lenses. What is the probability that a randomly selected student from the college needs corrective lenses? b. Four members on a cleanup committee draw lots to see who will clean the barbecue grills. What is the probability that George will clean the grill? c. Brad Gilbert, a tennis commentator, said that Roger Federer had an advantage over Marcos Baghdatis in winning the 2006 Australian Open. What do you suppose is a reasonable number for the probability that Federer wins (before the outcome is known). Chapter 4 – Probability (Page 4 of 24) Statistical Experiment Terminology a. A statistical experiment or statistical observation is any random activity that results in a definite outcome. b. An outcome (simple event) is the observable or measurable result of an experiment. c. The sample space, S, is the set of all possible outcomes of an experiment. d. An event E is any set of outcomes. Number of Outcomes Favorable to E n(E) e. P(E) = n(S) Total Number of Outcomes Example D Identify the experiment, sample space and event. Then answer the question. 1. A coin is tossed, what is the probability of it landing heads up? 2. A die is tossed, what is the probability of a 2 or 4? 3. A card is drawn from a standard deck of 52 cards, what is the probability that a diamond is drawn. 4. Two dice are tossed, what is the probability that the sum is 6? Chapter 4 – Probability (Page 5 of 24) Example 1 Eye color is controlled by a pair of genes (one from the mother and one from the father) called a genotype. Brown eye color, B, is dominant over blue eye color, b. Therefore, in the genotype Bb the brown-eye gene, B, dominates and a person with a Bb genotype has brown eyes. If both parents have brown eyes and genotype Bb, what is the probability that the child will have blue eyes? What is the probability that the child will have brown eyes? Guided Exercise 2 Professor Gill has 3 true-false questions on an exam. In order to assure the pattern of t-f answers is random he lists all combinations of three t-f answers on sheets of paper and randomly selects one from a hat. a. List the sample space in a tree diagram. b. What is P(all three answers will be false)? c. What is P(exactly two will be true)? Chapter 4 – Probability (Page 6 of 24) Complement of an Event If P( A) is the probability that event A will occur, then the complement of that, P( Ac ) , is the probability that event A will not occur. So, A and Ac together make up the sample space, and 1. P( Ac ) = 1! P( A) 2. P( A) + P( Ac ) = 1 Guided Exercise 3 If you breed two cream-colored guinea pigs, the probability that the offspring will be pure white is 0.25. What is the probability that the offspring will not be pure white? Chapter 4 – Probability (Page 7 of 24) Probability Summary 1. The probability of event A is denoted P( A) . 2. 0 ! P( A) ! 1 3. The sum of the probabilities of all outcomes in a sample space is 1. 4. P( A) + P( Ac ) = 1 5. If P( A) = 0 , then event A is impossible. 6. If P( A) = 1 , then event A is certain. Probability versus Statistics Probability is the field through which statistical work is done. * Probability is the field of study that makes statements about what will occur when samples are drawn from a known population. * Statistics is the field of study that describes how samples are to be obtained and how inferences are to be made about unknown populations. Example of Probability Questions Suppose a box contains 3 green balls, 5 red balls, and 4 white balls. a. If one ball is drawn, what is P(green)? b. If 3 balls are drawn, what is P(one is white and two are red)? c. If 4 balls are drawn, what is P(none are red)? Example of a Statistics Exploration Suppose a box contains a collection of colored balls. The central question is to discover the (relative) number and color of each ball in the box by random sampling. Chapter 4 – Probability Rules (Page 8 of 24) 4.2 Compound Events (2 or more events) Independent Events Two events are independent if the outcome of one event does not affect the outcome of the other event (and Visa Versa). Mathematically, two events A and B are independent if and only if P(A and B)= P(A)! P(B) Example E Identify if the events described are independent or not. a. Roll a fair die twice. b. Draw two cards from a standard deck of 52 cards, without replacing the first card before drawing the second. Multiplication Rules: Probability of Event A and Event B 1. Two Events A and B are independent if and only if P(A and B)= P(A)! P(B) ) 2. For Any Events A and B [dependent or independent] P(A and B)= P(A)! P(B, given A has ocurred)= P(A)! P(B | A) Example 3 Suppose two fair dice are thrown. What is the probability of getting a five on each die? a. Solve using the multiplication rule. b. Solve by drawing the sample space. Chapter 4 – Probability Rules (Page 9 of 24) Example 4 Find the probability of drawing two Aces from a standard deck of 52 cards if a. the first card is replaced into the deck and the deck is shuffled before drawing the second card (called “with replacement”). b. the cards are drawn in sequence without replacement of the first card (called “without replacement”). c. In part b, explain why (mathematically) the two events are not independent. Guided Exercise 4 Andrew is 55, and the probability that he will be alive in 10 years is 0.72. Ellen is 35, and the probability that she will be alive in 10 years is 0.92. What is the probability that both will be alive in 10 years? Chapter 4 – Probability Rules (Page 10 of 24) Guided Exercise 5 A quality control procedure for testing Ready-Flash disposable cameras is done by randomly drawing 2 cameras from each lot of 100 (w/o replacement). If both are defective, then the entire lot is rejected. Find the probability that the lot will be rejected (i.e. both cameras will be defective) if the lot contains 10 defective cameras. a. What is the probability of getting a defective camera on the first draw? b. What is the probability of getting a defective camera on the second draw? c. Find the probability that the lot will be rejected (i.e. both cameras will be defective) if the lot contains 10 defective cameras. Example D Suppose a fair coin is tossed, then a fair die is rolled, and finally a card is drawn from a standard deck of 52 cards. What is the probability of the outcome of “heads” on the coins and 5 on the die and an ace for the card? Chapter 4 – Probability Rules (Page 11 of 24) Mutually Exclusive Events Two events are mutually exclusive if they cannot occur at the same time. Mutually Exclusive Events A B Sample Space, S Events that can occur together A B Sample Space, S Example E Suppose a single card is drawn from a well-shuffled bridge deck. a. Are the events of drawing a Jack or King mutually exclusive? b. Are the events of drawing a Jack or Diamond mutually exclusive? Addition Rules: Probability of Event A or Event B 1. For mutually exclusive events A and B P(A or B)= P(A)+ P(B) 2. For any events A and B P(A or B)= P(A)+ P(B) - P(A and B) Chapter 4 – Probability Rules (Page 12 of 24) Addition Rules: Probability of Event A or Event B 1. For mutually exclusive events A and B P(A or B)= P(A)+ P(B) 2. For any events A and B P(A or B)= P(A)+ P(B) - P(A and B) Example F a. What is the probability of drawing a Jack or a King from a standard deck in one draw? b. What is the probability of drawing a Jack or a diamond from a standard deck in one draw? Guided Exercise 7 If you purchase a pair of slacks in your waist size from a second hand clothing store the probability that they will be too tight is 0.30 and the probability that they will be too loose is 0.10. a. Are the events mutually exclusive? b. If you choose a pair of slacks in your waist size, what is the probability that they will be too tight or too loose? Chapter 4 – Probability Rules (Page 13 of 24) Guided Exercise 8 In a program to prepare for a high school equivalency exam it is found that 80% of the students need work in math, 70% in English, and 55% in both areas. Draw a Venn diagram and find the probability that a randomly selected student will need work in a. Math and English b. Math or English c. Math, but not English d. English, but not Math e. Neither Math nor English Exercise 14 About 14% of senior citizens (65 years or older) get the flu each year, and about 24% of the people under 65 years old get the flu each year. In the general population, there are 12.5% senior citizens. What is the probability that a person selected at random a. is a senior citizen who will get the flu? b. is a person under 65 who will get the flu? c. Draw a tree diagram for this problem. Chapter 4 – Probability Rules (Page 14 of 24) Exercise 20 Diagnostic tests of medical conditions have several results. The test result can be positive of negative, whether or not the patient has the condition (+ indicates the patient tested positive for the condition). Consider a random sample of 200 patients, some of whom have a medical condition and some of whom do not. Results of a new Condition Condition diagnostic test for Present Absent the condition are 110 20 shown. For a person Test + Test 20 50 selected at random Column Total 130 70 compute the following. a. P(+, given condition present) b. P(-, given condition present) c. P(-, given condition absent) d. P(+, given condition absent) e. P(condition present and +) f. P(condition present and -) g. Are the events “Test +“ and “Condition Absent” independent? Explain mathematically. Row Total 130 70 200 Chapter 4 – Probability Rules (Page 15 of 24) Exercise 26 An alcoholic treatment program has two phases: Phase 1 lasts 10 weeks; phase 2 lasts 1 year. The probability that a client will relapse in phase 1 is 0.27. However, if the client did NOT have a relapse in phase 1, then the probability that a client will have a relapse in phase 2 is 0.10. If the client did have a relapse in phase 1, then the probability that a client will have a relapse in phase 2 is 0.70. Let R1 be the event that the client had a relapse in phase 1, and R2 be the event that the client had a relapse in phase 2. Draw a tree diagram showing all possible outcomes, and find the probability of each of the following. a. P(R1 ), P(not R 1 ) b. P(R 2 , given not R1 ), P(not R 2 , given not R1 ) c. P(R 2 , given not R1 ), P(not R 2 , given not R1 ) e. P(R 2 ) Chapter 4 – Trees & Counting Techniques (Page 16 of 24) 4.3 Tree Diagrams and Counting Techniques Example 7 Jackie needs to take psychology, anatomy, and Spanish. There are 4 sections of psychology, 2 of anatomy, and 3 of Spanish offered that do not conflict with each other. Use a tree diagram to list all the possible schedules Jackie can take. How many possible schedules can she take? Guided Exercise 10 Louis plays three tennis matches. Use a tree diagram to list all the possible outcomes. How many possible outcomes can occur? Chapter 4 – Trees & Counting Techniques (Page 17 of 24) Example 8 a. Suppose there are 5 balls of identical size in an urn: 3 red and 2 blue. You are asked to draw out one ball, note its color and set it aside. Then draw another ball and note its color. List all possible outcomes and the probability for each outcome. b. Repeat, except replace the first ball before drawing the second ball. Chapter 4 – Trees & Counting Techniques (Page 18 of 24) Multiplication Rule of Counting If there are m possible outcomes for event E1 and n possible outcomes for event E2, then there are a total of m ! n outcomes for events E1 followed by E2. Example 9 An automobile comes in a choice of two body styles, 3 interior packages, 4 colors, and 2 types of transmission. How many distinct car orders are possible? Guided Exercise 12 A menu has a choice of 2 appetizers, 3 main courses, and 4 desserts. How many different full meals can be ordered? Chapter 4 – Trees & Counting Techniques (Page 19 of 24) Example G How many different ways can 4 people be seated at a dinner table with 4 chairs? Factorials For positive integer n, n! = 1! 2 !3!!(n " 1) ! n 0! = 1 e.g. 1! = 1 2! = 1! 2 = 2 3! = 1! 2 !3 = 6 4! = 1! 2 !3! 4 = 24 Example H Compute a. 6! b. 3!4! c. 8! (8 ! 5)! d. 10! 2!(10 ! 2)! Chapter 4 – Trees & Counting Techniques (Page 20 of 24) Permutations – Ordered Arrangements A permutation is an ordered arrangement. The number of ways to arrange n distinct objects, taken r at a time, is P = Pn,r = n r n! (n ! r)! (MATH>PRB>2: nPr) Example I How many different ways can 4 people be seated at a dinner table with 4 chairs? Example J How many different ways can 6 people be seated at a dinner table with 6 chairs? Example K How many different ways can 8 people be seated at a dinner table with 8 chairs? Chapter 4 – Trees & Counting Techniques (Page 21 of 24) Example L Alan, Bob, Cathy and Diane are friends. Find the number of ways to arrange a. all 4 in a line for a picture. List all the permutations. ABCD BACD CABD DABC ABDC BADC CADB DACB ACBD BCAD CBAD DBAC ACDB BCDA CBDA DBCA ADBC BDAC CDAB DCAB ADCB BDCA CDBA DCBA b. 3 of 4 in a line for a picture. List all the permutations. c. 2 of 4 in a line for a picture. List all the permutations. d. 1 of 4 in a line for a picture. List all the permutations. e. 0 of 4 in a line for a picture. List all the permutations. Chapter 4 – Trees & Counting Techniques (Page 22 of 24) Example 10 Find the number of ways to arrange 8 objects taken {8, 7, 6, 5, 4, 3, 2, 1, and 0} at a time. Guided Exercise 14 A board of directors has 12 members and must elect a president, vice president, and treasurer. In how many ways can this be done if no person can hold more than one office? Example M A committee has 5 members. In how many ways can a subcommittee of size 3 be selected? Size 2? Size 1? Size 4? Size 5? Size 0? Chapter 4 – Trees & Counting Techniques (Page 23 of 24) Combinations – Order Does Not Matter A combination is a subset, or sub-group, in which the order does not matter. The number of combinations of n objects taken r at a time is C = Cn,r = n r n! (MATH>PRB> 2:nCr) r!!(n " r)! Permutations vs. Combinations 1. Permutations consider groupings and order. 2. Combinations consider groupings only. 3. n Pr ! n Cr . That is, the number of permutations is greater than or equal to the number of combinations. Example I Suppose a committee has 5 members: Alice, Bob, Cathy, Dave, and Eve. How many subcommittees of size 0, 1, 2, 3, 4, and 5 can be formed. List all the possible subcommittees. Chapter 4 – Trees & Counting Techniques (Page 24 of 24) Example 11 A board of directors has 12 members and must send three to a convention. In how many ways can this be done? Guided Exercise 15 In how many different ways can you select 4 books from a list of 10 books? Open your books to exercises #22-30 in section 4.3. Show your work by showing the proper probability notation and the solution. Circle your answer. On exams I will not say whether to use a permutation or combination – identifying that is central to the problem.