Math 166 Ch 1 Sets and Probability Texas A&M Spring 2016

Transcription

1 1.1 Introduction to Sets Sets Set is a collection of items, referred to as its elements or members. A set is represented by a capital letter. For example, A = {1, 2, 3, 4, 5} is a set A containing elements 1, 2, 3, 4, and 5. A set can be written in two notations: Roster Notation: explicitly writing all the elements. e.g. A = {1, 2, 3, 4, 5} Set-builder Notation: define the set in terms of its properties. e.g. A = {x x is a set of first five natural numbers} READ AS : the set of all x such that x is one of the first five natural numbers. An element of or belongs to ( ) Given the set A = {1, 2, 3, 4, 5} We can write 2 A and it is read as 2 is an element of set A., i.e. 2 is contained in set A. Example 1: Write the set containing the letters in the word BADGE. Example 2: Write the set containing the letters in the word MATHEMATICS. Subset If every element of a set B is also in set A, then B is a subset of A, and is written as B Í A If every element in set B is also in set A, and B ¹ A, then B is a PROPER SUBSET of A and is written as B Ì A Every set is a subset of itself. Empty set is a subset of every set. 1

2 Empty and Universal Set Empty Set: A set with NO ELEMENTS. Represented as { } or Æ. Universal Set: Set of ALL elements being considered. It's denoted by U. Example 3 : Let A = {1, 2, 3, 4} B = {2, 4, 3, 1} C = {1, 3} D = {1, {2}, 3} State whether each of the following is True or False. a) B Í A b) C Í A c) C Ì A d) B Ì A e) A Í A f ) 2 B g) {4, 1} Ì C h) 6 Ï B i ) {2} B j) {2} Ë B k) {4, 1} Í C l) 4, 1 Í C m) {2} Ì {1,2,3} n) Æ Ì B o) Æ Í B p) {2} Ì D q) {2} D Example 4. Given the set M = {a, b, c}. a) List all the subsets of the set M. b) List all the proper subsets of the set M. c) How many subsets does M have? d) How many PROPER subsets does M have? 2

3 Set Operations Consider the sets: U = { 1, 2, 3, 4, 5, 6, 7, 8 } A = { 1, 2, 3 } ; C = { 5, 7 } B = { 2, 4, 6 } ; D = { 3, 2, 1 } ; E = { 2, 3, 4, 5 } 1. Set Union (Logic : OR) Union of two sets is a set of all elements that belong to A, or B, or both. In other words, Union of two sets is a set of all elements that belong to ATLEAST ONE of A or B. It's represented as A Ç B. 2. Set Intersection (Logic : AND) Intersection of two sets is a set of all elements that belong to BOTH, A and B. It's represented as Two sets A and B are disjoint if they have no elements in common, i.e. if A È B = Æ. 3. Complement of a Set (Logic : NOT) Given a set A. The complement of the set A is the set of all the elements that are in set U, but not in A. It's represented as A C 3

4 Example 5. State whether the following are True or False, given T = { 1, 2, 3, 5, 7} a) T = {x x is a set of prime numbers between 0 and 10} b) If S = {x x is a set of odd numbers less than 10}, then S È T has 15 proper subsets. c) {5} T d) {7, 5, 2} Í T Example 6. Given the sets U, A, B, and C, find the indicated sets: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9} A = {1, 2, 3, 4, 5} B = {2, 4, 6, 8} C = {1, 3, 5, 9} (a) (A Ç B ) C (b) ( B Ç C ) È A C (c) A È (B C Ç C ) (d) A C È B È C C (e) B Ç B C f) C È C C 4

5 Venn Diagrams: Universal set is represented by a rectangle. All other sets are usually represented by circles. Example 7. Draw the Venn diagrams for each of the following and clearly indicate the final answer: a) A Ç B b) A È B c) A È (B C Ç C ) d) A C È B È C C 5

6 Laws of Set Operation (Can be verified using Venn Diagrams) Commutative Laws : A Ç B = B Ç A A È B = B È A Associative Laws : A Ç (B Ç C ) = ( A Ç B ) Ç C A È (B È C ) = ( A È B ) È C Distributive Laws : A Ç (B È C ) = (A Ç B ) È ( A Ç C ) A È (B Ç C ) = (A È B ) Ç ( A È C ) De Morgan's Laws : ( A Ç B ) C = A C È B C ( A È B ) C = A C Ç B C Proof of De-Morgan's Laws a) ( A Ç B ) C = A C È B C b) ( A È B ) C = A C Ç B C Example 8. Let U be the set of major league baseball players and let N = {x x plays for the New York Yankees} S = {x x plays for the San Francisco Giants } F = {x x is an outfielder} H = {x x has hit 20 homers in one season } Write the set that represent the following descriptions a) Outfielders for the New York Yankees. b) San Francisco Giants who have never hit 20 homers in a season. c) Major league outfielders who have hit 20 homers in a season and do not play for the NY Yankees or the SF Giants. d) NY Yankees or SF Giants who have hit 20 homers in a season. 6

7 1.2 The Number of Elements in a Set Union Rule or Addition Rule of Sets If A and B are two sets such that A Ì U and B Ì U, and A and B are not disjoint then n(a Ç B) = Proof : If A and B are disjoint sets such that A Ì U and B Ì U, then n(a Ç B) = Example If n(u) = 200, n(b) = 100, n(a Ç B) = 175, n(a È B) = 40, what is n(a), n(a È B) C, n(b C ), n( A C Ç B C ). 7

8 Example 9. If n(u) = 200, n(a Ç B) = 150, what is n( A C È B C )? Example 10. If n(b) = 200, n(a È B È C ) = 40, n(a È B È C C ) = 20, n(a C È B È C ) = 50, what is n(a C È B È C C ). Example 11. In a survey of 500 people, a pet food manufacturer found that 200 owned a dog but not a cat, 150 owned a cat but not a dog, and 100 owned neither a dog or cat. a) How many owned both a cat and a dog? b) How many owned a dog? 8

9 Example 12. A survey of 600 people over the age of 50 found that 200 owned some stocks or real estate but no bonds, 220 owned some real estate or bonds but no stock, 60 owned real estate but no stocks or bonds, and 130 owned both stocks and bonds. How many owned none of the three? 9

10 Example 13. A survey of 500 people found that 190 played golf, 200 skied, 95 played tennis, 100 played golf but did not ski or play tennis, 120 skied but did not play golf or tennis, 30 played golf and skied but did not play tennis, and 40 did all three. a) How many played golf and tennis but did not ski? b) How many played tennis but did not play golf or ski? c) How many participated in at least one of the three sports? d) How many did not play any or the sport? e) How many participated in exactly one sport? f) How many play exactly two of these sport? 10

11 1.3 Sample Space and Events Experiment : is an activity that has observable results. The result of an experiment is known as its outcome. Standard deck of 52 playing cards: A standard deck of 52 playing cards has four 13-card suits: clubs, diamonds, hearts, and spades. The diamonds and hearts are red, the clubs and spades are black. Each 13-card suit contains cards numbered from 2 to 10, a jack(11), a queen(12), a king(13), and an ace(14). Sample Space ( S ): of an experiment is the set of all possible outcomes of the experiment (similar to the Universal Set: set of all possible elements). Each repetition of an experiment is known as a Trial. Examples: Experiment Sample Space a) Flip a coin b) Roll a 6-side fair die h) Flip a coin three times in a row i) Roll a die, and if the number on the die is even, then flip a coin. 11

12 Event: Given a sample space S for an experiment, an event is any subset of S. An elementary (or simple) event is an event with a single outcome. Since Æ is a subset of every set, Æ Í S. Æ is the impossible event since it means no outcome has occurred. e.g.: Flip a coin, the event that both heads and tails show is an impossible event. H È T = Æ. Every set is a subset of itself. Hence, S Í S. S is the certainty event, since any outcome of an experiment must be in S. Event Flip two coins and find the event that at least one tail comes up. Roll two dice and find the event that the numbers of top faces of both dice are even. Roll two sided dice and find the event that the sum of numbers on faces of both dice is less than 5. Pick a card from a stack of 52-cards and observe the suit. Example 14. Two fair 6-faced dice are rolled and the numbers on each are observed. a) How many elements (outcomes) are there in the sample space? b) How many events are possible for this sample space? c) Find the event the sum of the numbers on the top faces of both the dice is observed. 12

13 Uniform Sample Space : is one where each event is equally likely. Example 15. An experiment consists of selecting a digit from the number and observing it. a) What is the uniform sample space for this experiment? b) Indicate the outcomes in the event that an even digit is selected. c) Indicate the outcomes in the event that an odd digit is selected Union, Intersection, and Complement of Events (Recall : Set Operations) If E and F are two events of an experiment, with sample space S, where S = { a, b, c, d, e, f, g, h } E = { a, b, c, d } F = { c, a, f } a) E Ç F: is the union of E and F, and consists of the set of outcomes that are in E or in F (or both). If an element exists in both the sets, it will be written only once in set E Ç F. b) E È F : is the intersection of E and F, and consists of the set of outcomes that are in both E and F. If E È F =Æ, E and F are said to be mutually exclusive events (Recall: Disjoint sets). That is, the sets E and are F disjoint. c) E C : is the complement of, E and consists of the set of outcomes not in E. In other words, E' is the event that E does not occur. Example 16. Let a card be chosen from a standard deck of 52 cards. Let E be the event of drawing a 5. Let F be the event of drawing a heart. Let G be the event of drawing a black. a) List the elements of E, F. b) What is E È F, G È F, E È G. c) Are any of these events mutually exclusive? d) How many elements are there in E È G C? E Ç G C? 13

14 1.4, 1.5 Basics and Rules of Probability 1) If S is a uniform sample space, and n(s) be the number of elements in the sample space. If E is an event of the sample space, and n(e) be the number of elements in E, then the probability that E occurs is P(E) = 3. If E and F are two events of the sample space S, then a) 0 P(E) 1 b) P(S) = 1 c) P(Æ ) = 0 d) P(E) + P(E c ) = 1 Example 17. A fair coin is flipped three times. What is the probability that at least two heads occur? Example 18. An unfair 6-sided die is rolled 1000 times and the number on top face is observed. The table shows the results. Find the empirical probability that (a) a 2 will occur (b) an odd number will occur. Outcome Frequency

15 Example 19. An unfair 6-sided die is rolled and the probability distribution table is given: Outcome Frequency Probability of getting a 4 is the same as probability of getting a 5. Find the probability of getting a 4 or a 5. Example 20. Two 6-sided dice are rolled. Find the probability distribution table for the product of the numbers observed on the top faces. Example 21. Roll two 6-sided fair dice and observe the sum of the numbers on the top faces of the dice. What is the probability that the sum is a) between and including 3 and 6? b) less than 8? 15

16 Union Rule of Probability P(E Ç F) = P(E) + P(F) P(E È F) If E and F are mutually exclusive, then P(E Ç F) = P(E) + P(F) Example 21. A single card is randomly drawn from a standard deck of cards. What is the probability that the card drawn is a) red? b) a spade? c) either red or a king? Example 22.A basket contains 4 red and 5 green pens. Three pens are chosen at random. What is the probability that a) exactly two are green? b) at least one is red? c) exactly one is red and exactly two are green? d)exactly one is red or exactly two are green? 16

17 Example 23. A box contains 6 blue pens, 6 green pens, and 4 orange pens. A sample of 6 pens is to be picked from the box. What is the probability that a) exactly three pens are of the same color? b) at least one is blue? Example 24. A manufactured item is guaranteed for one year and has three critical parts. It has been decided that during the first year, the probability of failure of the first part is 0.03, of the second part is 0.02, of the third part is 0.01, of both the first and the second is 0.005, both the first and the third is 0.004, of both second and third is 0.003, and of all three parts is what is the probability that a) Exactly one of these parts will fail in the first year? b) At least one of these three parts fail in the first year? c) None of these parts fail in the first year? d) Exactly two of these parts fail in the first year? 17

18 Odds 1. Given: Probability, find Odds The odds in favor of an event E are : 2. Given: Odds, find Probability: If odds in favor of an event E are a/b or a:b, then P(E) = and P(E C ) = Example 25. Given P(E) = 0.26, P(F) = 0.28, P(E F) = 0.08, what are the odds that a) E occurs b) only E occurs c) E does not occur Example 26. If odds in favor of E are 13/37, then find a) P(E) b) P(E C ) 18

19 1.6, 1.7: Conditional Probability and Bayes' Theorem Conditional Probability: If E and F are two events, then we say that the probability that E will occur given that F occurs is P(E F ), and is the probability of E given F. P(E F ) = Example 12. P( E )= 0.3, P( F )= 0.7, P( E È F )= 0.2, find a) P(E C F ) b) P(E Ç F F ) c) P(E F C ) d) P(E È F C F ) Example 27. A card is randomly selected from a deck of 52 cards. What is the probability that the card selected is a King given that a card is higher than 10 and is black. 19

20 Probability Tree Example 28. We have two bags, A and B, each with 10 pens. Bag A has 2 red pens, 4 green pens, and the rest are pink, Bag B has 5 red and 2 pink pens. (R red pen, G green pen, P pink pen). Bag A is twice as likely to be picked as bag B. a) Draw the probability tree and find the missing probabilities. b) Find P(R B), P(G A). c) If a pen is picked, what is the probability that it is pink? d) If a green pen is picked, what is the probability that it came from bag B? e) If a pen was picked from bag A, what is the probability that is was red? 20

21 Example 29. Two bins contain transistors. The first bin has 10 defective and 20 non-defective transistors. The second bin has 5 defective and 25 non-defective transistors, If the probability of picking either bin is the same, what is the probability (of ) a) picking a defective transistor, given it's picked from the first bin? a) picking a good transistor and the second bin? b) picking a defective transistor and the first bin? c) that the second bin was chosen, given the transistor picked was defective? Example 30. Three machines turn out all the products in a factory, with the first machine producing 15% of the products, the second machine 30%, and the third machine 55%. The first machine produces defective products 11% of the time, the second machine 11% of the time and the third machine 8% of the time. If a defective product was chosen, what is the probability it came from the second machine? 21

22 Independent Events : Events E and F are said to be independent if P(E F ) = P(E ), P(F E ) = P( F ) For independent events, P(E ÈF ) = P(E ) P(F ) Example 31. A building has three elevators. The chance that elevator A is not working is 12%, the chance that elevator B is not working is 15%, the chance that elevator C is not working is 9%. If these three events are independent of each other, then what is the probability that a) no elevator is working? b) exactly one of the elevators is not working? c) exactly one of the elevators is working? d) all the elevators are working? 22