Section 1.1 - Introduction to Sets Definition: A set is a well-defined collection of objects usually denoted by uppercase letters. Definition: The elements, or members, of a set are denoted by lowercase letters. Set Notations: 1. Roster Notation: Lists each element between braces Example 1: 2. Set-builder Notation: A rule is given that describes the property an object x must satisfy to qualify for membership in the set. Example 2: Notation: If a is an element of a set A, we write a A. If a doesn t belong to A we write a / A. Example 3: Let A = {1,2,3}. Definition: Two sets A and B are equal, written A = B, if and only if they have exactly the same elements. (Note: The elements do NOT have to be in the same order.) Example 4: Let A = {1,2,3}, B = {2,1,3}, C = {1,2,3,4} Definition: If every element of a set A is also an element of a set B, then we say that A is a subset of B and write A B. Definition: A is a proper subset of B if A is a subset of B but A does not equal B. (i.e. We write A B if A B and there exists at least one element in B that is not in A) Example 5: Let A = {1,2,3}, B = {2,1,3}, C = {1,2,3,4} Note:, represent containment between sets. To show an element is part of a set, we use. 1
Definition: The set that contains no elements is called the empty set and is denoted by /0 (not {/0}). The empty set is a subset of every set. Example 6: List all subsets of A = {a,b,c} Definition: A universal set, U, is the set of all elements of interest in a particular matter. We have different universal sets for different problems. We use Venn Diagrams to visually represent sets. The Universal Set U is denoted by a rectangle. Subsets of U are represented by circles inside the rectangle. Set Operations: 1. If U is a universal set and A is a subset of U, then the set of all elements in U that are NOT in A is called the complement of A, denoted A c. Example 7: A c = {x U x / A} 2. The intersection of sets A and B is the set of all elements that belong to both A and B. Example 8: A B = {x U x A and x B} 3. The union of sets A and B is the set of all elements that belong to A or B. Example 9: A B = {x U x A or x B } 2
Set Complementation: U c = /0 /0 c = U (A c ) c = A A A c = U A A c = /0 Example 10: Let s check some of the above properties using Venn Diagrams. Definition: Two sets A and B are disjoint if A B = /0. Example 11: Let A = {1,3,5,7}, B = {2,4,6,8}. Are A and B disjoint? Properties of Set Operations: Commutative Law: Associative Law: Distributive Law: A B = B A A B = B A A (B C) = (A B) C A (B C) = (A B) C A (B C) = (A B) (A C) A (B C) = (A B) (A C) 3
DeMorgan s Laws (A B) c = A c B c (A B) c = A c B c Example 12: Let s prove DeMorgan s Law using Venn Diagrams Example 13: Let A, B, and C be subsets of a universal set U. Shade the following regions on a Venn Diagram: a) A B C c b) (A B) c C 4
Example 14: Let U = {1,2,3,4,5}, A = {1,2,3}, B = {1,3,5}. Find a) A c b) A B c) A A c d) A A c e) (A B) c Example 15: Let U denote the set of all cars in a dealer s lot and A = {x U x is equipped with automatic transmission} B = {x U x is equipped with air conditioning} C = {x U x is equipped with side air bags} Find an expression in terms of A, B, and C for each of the following sets: a) The set of cars with at least one of the given options b) The set of cars with automatic transmission and side air bags but no air conditioning c) The set of cars with exactly one of the given options Section 1.1 Homework Problems: 1-7 (odd), 11-51 (odd) 5
Section 1.2 - The Number of Elements in a Set Definition: For a set A, we denote the number of elements in A as n(a). Example 1: If A = {x x is a letter in the English Alphabet}, B = {a,b,c}, and C = /0, find: a) n(a) b) n(b) c) n(c) You will need to remember the following formula: Example 2: Does the formula make sense? n(a B) = n(a) + n(b) n(a B) Example 3: In a recent survey of 200 members of a local sports club, 100 members indicated that they plan to attend the next Summer Olympic Games, 60 indicated that they plan to attend the next Winter Olympic Games, and 40 indicated that they plan to attend both games. How many members of the club plan to attend a) At least one of the two games? b) The Summer Olympic Games only? c) Exactly one of the games? d) None of the games? 6
Example 4: If n(a) = 12,n(B) = 12,n(A B) = 5,n(A C) = 5,n(B C) = 4,n(A B C) = 2, and n(a B C) = 25, find n(c). Example 5: To help plan the number of meals to be prepared in a college cafeteria, a survey was conducted, and the following data were obtained: 8 students ate only breakfast. 80 students ate only lunch. 96 students ate exactly 2 meals. 68 students ate breakfast and lunch. 58 students ate all three meals. 100 students did not eat dinner. 112 students ate breakfast and dinner. 99 students ate exactly 1 meal. How many students were surveyed? 7
Example 6: A survey was conducted of College Station residents to determine what activities they participated in during the 4th of July weekend. It was found that 955 residents watched fireworks. 50 residents only went swimming. 528 residents participated in exactly two of these activities. 1168 residents watched fireworks or ate BBQ. 250 residents only watched fireworks and ate BBQ. 60 residents did not watch fireworks and did not eat BBQ. 425 residents watched fireworks, ate BBQ, and went swimming. 523 residents went swimming and ate BBQ. How many residents did not eat BBQ? Section 1.2 Homework Problems: 1-29 (odd) 8
Section 1.3 - Sample Spaces and Events Definition: An experiment is an activity with an observable result. Definition: The outcome is the result of an experiment. Definition: The sample space is the set of all possible outcomes of an experiment. Definition: An event is a subset of the sample space. (Note: An event E is said to occur whenever E contains the observed outcome.) Definition: E and F are mutually exclusive if E F = /0. Note: All of the set operations (union, intersection, complement) work the same with events. Example 1: Let s consider the experiment of rolling a fair six-sided die and observing the number that lands uppermost. a) Determine the sample space. b) Find the event E where E = {x x is an even number}. c) Find the event F where F = {x x is a number greater than 2} d) Find the event (E F) e) Find the event (E F) f) Are the events E and F mutually exclusive? g) Are the events E and F complementary? Example 2: Let s consider the experiment of flipping a fair coin two times and observing the resulting sequence of heads and tails. a) Determine the sample space. b) Find the event E where E = {x x has one or more heads} c) Find the event F where F = {x x has more than 2 heads} d) List all events of this experiment. 9
Note: We could have used a tree diagram in the previous example to determine all of the possible outcomes. Example 3: An experiment consists of casting a pair of fair six sided dice and observing the number that falls uppermost on each die. a) Find the sample space for this experiment. b) Determine the event that the sum of the numbers falling uppermost is less than or equal to 6. c) Determine the event that the number falling uppermost on one die is a 4 and the number falling uppermost on the other die is greater than 4. 10
Example 4: A die is rolled. If the die shows a 1 or a 6, a coin is tossed. What is the sample space for this experiment? Example 5: The manager of a local bank observes how long it takes a customer to complete his transactions at the ATM. a) Describe an appropriate sample space for this experiment. b) Describe the event that it takes a customer between 2 and 3 minutes, inclusive, to complete his transactions at the ATM. Section 1.3 Homework Problems: 1-27 (odd) 11
Section 1.4 - Basics of Probability Definition: The probability of an event is a number between 0 and 1 that represents the likelihood of the event occuring. The larger the probability, the more likely the event is to occur. Definition: A sample space in which all of the outcomes are equally likely is known as a uniform sample space. Definition: If S is a finite uniform sample space and E is any event, then the probability of E, P(E), is given by P(E) = n(e) n(s) where n(e) is the number of elements in E and n(s) is the number of elements in S. Example 1: Suppose a fair six-sided die is rolled and the number that lands uppermost is observed. a) Find the sample space for this experiment. b) Find the probability that an odd number is rolled. c) Find the probability that a 9 is rolled. d) Find the probability that a number less than 8 is rolled. Example 2: Suppose a single card is randomly drawn from a standard 52 card deck. Determine the probability of each of the following events: a) A king is drawn. b) A heart is drawn. 12
Definition: If an experiment is performed and the frequencies of each outcome are recorded, the probabilities resulting from this experiment are referred to as empirical probabilities. Example 3: In a survey of 1000 randomly selected consumers, 50 said they bought brand A cereal, 60 said they bought brand B, and 80 said they bought brand C. What is the empirical probability that a randomly selected consumer purchased brand C? Definition: The table that lists the probability of each outcome in an experiment is known as the probability distribution. Example 4: A pair of fair six-sided dice is cast and the sum of the numbers landing uppermost is observed. Find the probability distribution for this experiment. Example 5: Let S = {a,b,c,d,e} be the sample space associated with an experiment having the following frequency table: If F = {a,b} and G = {b,c,e}, what is P(F c G)? Outcome a b c d e Frequency 1 3 6 2 2 Section 1.4 Homework Problems: 1-13 (odd), 17, 21, 23, 25, 31, 33 13
Section 1.5 - Rules for Probability Properties: 1. P(E) 0 for any event E 2. P(S) = 1 3. If E and F are mutually exclusive (that is, E F = /0), then P(E F) = P(E) + P(F) 4. P(E F) = P(E) + P(F) P(E F) 5. P(E c ) = 1 P(E) Example 1: Let E and F be two events of an experiment with sample space S. Suppose P(E) = 0.7, P(F) = 0.5, and P(E F) = 0.3. Compute: a) P(E F) b) P(E c ) c) P(F c ) d) P(E c F c ) e) P(E c F) Example 2: An experiment consists of selecting a card at random from a 52-card deck. What is the probability that a diamond or a king is drawn? 14
Example 3: Among 500 freshman pursuing a business degree at a university, 320 are enrolled in an Economics course, 225 are enrolled in a Mathematics course, and 140 are enrolled in both an Economics and a Mathematics course. What is the probability that a freshman selected at random from this group is enrolled in: a) an Economics or Mathematics course? b) exactly one of these two courses? c) neither an Economics course nor a Mathematics course? Example 4: A salesman always makes a sale at at least one of the three stops in Atlanta. He makes a sale at only the first stop 30% of the time, 15% at only the second stop, and 20% at only the third stop. It was also found that he makes a sale at exactly two of the stops 25% of the time. Find the probability that the salesman makes a sale at all three stops in Atlanta. 15
Definition: If P(E) is the probability of an event E occuring, then a) The odds in favor of E occuring are P(E) 1 P(E) = P(E) P(E c ) [P(E) 1] b) The odds against E occuring are 1 P(E) P(E) = P(Ec ) P(E) [P(E) 0] Example 5: The probability of an event E occurring is.8. a) What are the odds in favor of E occurring? b) What are the odds against E occurring? Definition: If the odds in favor of an event E occuring are a to b, then the probability of E occurring is P(E) = a a + b Example 6: If a sports forecaster states that the odds of a certain boxer winning a match are 4 to 3, what is the probability that the boxer will win the match? Section 1.5 Homework Problems: 1, 5, 11, 17, 19, 23, 27, 31, 35 16
Section 1.6 - Conditional Probability Example 1: A survey is done of people making purchases at a gas station: a) What is the probability that a person buys a drink? buy drink (D) no drink (D c ) Total buy gas (G) 20 15 35 no gas (G c ) 10 5 15 Total 30 20 50 b) What is the probability that a person doesn t buy a drink? c) What is the probability that a person buys gas and a drink? d) What is the probability that a person buys gas but not a drink? e) What is the probability that a person who buys a drink also buys gas? f) What is the probability that a person who doesn t buy a drink buys gas? Definition: If E and F are events in an experiment and P(E) 0, then the conditional probability that the event F will occur given that the event E has already occurred is P(F E) = P(E F) P(E) Definition: The Product Rule is found by rearranging the above formula as follows: P(E F) = P(E) P(F E) 17
Example 2: Let s use a tree diagram to help us understand the product rule: Example 3: At a party, 1/3 of the guest are women. Seventy-five percent of the women wore sandals and 40% of the men wore sandals. a) What is the probability that a person chosen at random at the party is a man wearing sandals? b) What is the probability that a person chosen at random is wearing sandals? 18
Example 4: Consider drawing 3 cards from a standard deck of 52 cards without replacement. a) What is the probability that the 3 cards are hearts? b) What is the probability that the third card drawn is a heart given the first two cards are hearts? Definition: If A and B are independent events, then P(A B) = P(A) and P(B A) = P(B). Thus, two events A and B are independent if and only if P(A B) = P(A) P(B) Example 5: A medical experiment showed the probability that a new medicine was effective was.75, the probability of a certain side effect was.4, and the probability of both occuring was.3. Are the events independent? 19
Example 6: Meghan and Natalie go to Freebirds. After ordering their food they get to roll a pair of fair dice for a chance to get their meal for free. Each die has six sides with one of the sides having a backwards F on it. If both of the dice land on the backwards F, you win. What is the probability that at least one of the girls wins? Section 1.6 Homework Problems: 1, 5, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47 20
Section 1.7 - Bayes Theorem Example 1: If we are given information about P(F E), can we find P(E F)? Definition: The above formula is known as Bayes Theorem. Example 2: We are to choose a marble from a cup or a bowl. We flip a fair coin to decide whether to choose from the cup or the bowl. The bowl contains 1 red and 2 green marbles. The cup contains 3 red and 2 green marbles. What is the probability that a marble came from the bowl given that it is red? 21
Example 3: A crate contains 7 basketballs and 4 footballs. A bag contains 4 basketballs and 2 footballs. A ball is drawn at random from the crate and put in the bag. A ball is then drawn from the bag. Given that a basketball was chosen from the bag, what is the probability that a football was drawn from the crate? Example 4: Two cards are drawn in succession without replacement from a standard deck of 52 cards. What is the probability that the first card is a face card given that the second card is an ace? 22
Example 5: Complete the following tree diagram and use it to answer the following questions: a) Find P(E). b) Find P(A D). c) Find P(B E). d) Are E and B independent events? Section 1.7 Homework Problems: 1, 5, 9, 13, 17, 21, 25 23