7.1: Sets. What is a set? What is the empty set? When are two sets equal? What is set builder notation? What is the universal set?

Transcription

1 7.1: Sets What is a set? What is the empty set? When are two sets equal? What is set builder notation? What is the universal set? Example 1: Write the elements belonging to each set. a. {x x is a natural number less than 5} b. {x x is a state that borders Florida} What is a subset?

2 Example 2: Decide whether the following statements are true or false. Explain. a. {3,4,5,6} = {4,6,3,5} b. {5,6,9,12} {5,6,7,8,9,10,11} A property of Subsets: Example 3: List all possible subsets for each set. a. {7,8} b. {a,b,c} A set with k elements has subsets. Example 4: Find the number of subsets for each set. a. {3,4,5,6,7} b. What is a Venn diagram? How can we use Venn diagrams to represent sets? What is the complement of a set?

3 Example 5: Let U = {1,2,3,4,5,6,7,8,9,10,11} and A = { 1,2,4,5,7} and B={2,4,5,7,9,11}. Find each set. a. A b. B c. d. (A ) What is the intersection of two sets? What is the union of two sets? What does it mean for sets to be disjoint? Example 6: Let A = {1,3,5,7,9,11}, B= {3,6,9,12}, C = {1,2,3,4,5} and the universal set U={1,2,3,4,5,6,7,8,9,10,11,12}. Find each set. a. A B = b. A C = c. (AB) C = Example 7: A department store classifies credit card applicants by gender, marital status, and employment status. Let the universal set be all credit card applicants, M be the set of male applicants, S be the set of single applicants and E be the set of employed applicants. Describe each set in words. a. M E = b. M S = c. M S =

4 7.2: Applications of Venn Diagrams Venn diagrams can be used to represent the intersection and union of sets. Example 1: Draw Venn diagrams and shade regions to represent each set. a. A B b. A B c. A (B C ) Example 2: A researcher is collecting data on 100 households and finds that 76 have a DVD player 21 have a Blu-ray player 12 have both Answer the following questions. a. How many do not have a DVD player? b. How many have neither a DVD player nor a Blu-ray player? c. How many have a Blu-ray player but not a DVD player?

5 Example 3: A survey of 77 freshman business students at a large university produced the following results 25 of the students read Bloomberg Businessweek; 27 read The Wall Street Journal; 27 do not read Fortune; 11 read Bloomberg Businessweek but not The Wall Street Journal; 11 read The Wall Street Journal and Fortune; 13 read Bloomberg Businessweek and Fortune; 9 read all three. a. How many students read none of the publications? b. How many read only Fortune? c. How many read Bloomberg Businessweek and The Wall Street Journal, but not Fortune? RULE: Union Rule for Sets: Example 4: A group of 10 students meet to plan a school function. All are majoring in accounting or economics or both. Five of the students are economics majors and 7are majors in accounting. How many major in both subjects?

6 Example 5: The following table gives the number of threatened and endangered animal species in the world as of May Source: U.S. Fish and Wildlife Service. Endangered (E) Threatened (T) Totals Amphibians and Reptiles (A) Arachnids and Insects (I) Birds (B) Clams, Crustaceans, corals, snails ( C) Fishes (F) Mammals Totals Using the letters given in the table to denote each set, find the number of species in each of the following sets. a. E B= b. E B = c. (F M) T

7 7.3: Introduction to Probability What is an experiment? What is a trial? What is an outcome? What is a sample space? Example 1: Give the sample space for each experiment: a. A spinner like the one shown is spun. b. For the purpose of a public opinion poll, respondents are classified as young, middle-aged, or senior, and as male or female. c. An experiment consists of studying the numbers of boys and girls in families with exactly 3 children. Let b represent boy and g represent girl. What is an event?

8 Example 2: For the sample space in Example 1(c), write the following events. a. Event H: the family has exactly 2 girls b. Event K: the three children are the same sex c. Event J: the family has 3 girls An event with only one possible outcome is called a. If an event, E, equals the sample space S, then E is called a. Example 3: Suppose a coin is flipped until both a head and a tail appear, or until the coin has been flipped four times, whichever comes first. Write each of the following events in set notation. a. The coin is flipped exactly three times. b. The coin is flipped at least three times. c. The coin is flipped at least two times. d. The coin is flipped fewer than two times. RULE: Set operations for Events: o Let E and F be events for a sample space S E F occurs when E F occurs when E occurs when

9 Example 4: A study of workers earning the minimum wage grouped such workers into various categories, which can be interpreted as events when a worker is selected at random. Consider the following events: E: worker is under 20; F: worker is white; G: worker is female. Describe the following events in words. Source: Economic Policy Institute. a. E b. F G c. E G DEFINITION: Mutually Exclusive Events Example of mutually exclusive event: DEFINITION: Basic Probability Principle: Example 5: Suppose a single fair die is rolled. Use the sample space S = {1,2,3,4,5,6} and give the probability of each event. a. E: the die shows an even number b. F: the die shows a number less than 10 c. G: the die shows an 8.

10 For any event E, It is not always possible to establish exact probabilities for events. Instead, useful approximations are often found by drawing past experience. This is known as. Example 6: The following table lists the estimated number of injuries in the US associated with recreation equipment. Source: National Safety Council Equipment Number of Injuries Bicycles 515,871 Skateboards 143,682 Trampolines 107,345 Playground Climbing Equipment 77,845 Swings or swing sets 59,144 Find the probability that a randomly selected person whose injury is associated with recreation equipment was hurt on a trampoline.

11 7.4: Basic Concepts of Probability RULE: Union Rule for Probability: Example 1: If a single card is drawn from an ordinary deck of cards, find the probability that it will be a red or a face card. Example 2: Suppose two fair dice are rolled. Find each probability. a. The first die shows a 2 or the sum of the two die is 6 or 7. b. The sum of the two die is 11 or the second die shows a 5. RULE: The Complement Rule: Example 3: If a fair die is rolled, what is the probability that any number but 5 will come up?

12 Example 4: If two fair die are rolled, find the probability that the sum of the numbers rolled is greater than 3. How do we give a probability statement in terms of odds? Example 5: Suppose the weather forecaster says the probability of rain tomorrow is 1/3. Find the odds in favor or rain tomorrow. Example 6: The odds that a particular package will be delivered on time are 25 to 2. a. Find the probability that the package will be delivered on time. b. Find the odds against the package being delivered on time. What is a probability distribution?

13 Example 8: The following table lists the major holders of US consumer credit (in billions of dollars) inn Source: The World Almanac and Book of Facts Holder Amount Depository institutions Finance companies Credit unions Federal government Nonfinancial business 48.5 Pools of securitized assets 49.9 a. Construct a probability distribution for the probability that a dollar of consumer credit is held by each type of holder. b. Find the probability that a dollar of consumer credit is held by a finance company or a credit union.

14 7.5: Conditional Probability; Independent Events DEFINITION: Conditional Probability: Example 1: The chart below shows results of a stock broker survey. Use the information in the chart to find the following probabilities. Picked stocks that Didn t pick stocks Totals went up (A) that went up (A ) Used research (B) Didn t use research (B ) Totals a. P(B A) b. P(A B) c. P(B A ) Example 2: Given P(E)= 0.4, P(F) = 0.5 and P(EF) = 0.7, find P(E F) Example 3: Two fair coins were tossed, and it is known that at least one was a head. Find the probability that both were heads.

15 Example 4: Two cards are drawn from a standard deck, one after another without replacement. Find the probability that the second is a red card, given that the first is a red card. RULE: Product Rule of Probability: Example 5: in a class with 2/5 women and 3/5 men, 25% of the women are business majors. Find the probability that a student chosen from the class at random is a female business major. Example 6: The Environmental Protection Agency is considering inspecting 6 plant for environmental compliance: 3 in Chicago, 2 in Los Angeles and 1 in New York. Due to a lack of inspectors, they decide to inspect 2 plants selected at random, one this month and one next month, with each plant equally likely to be selected, but no plant selected twice. What is the probability that one Chicago plant and one Los Angeles plant are selected?

16 Example 7: Two cards are drawn from a standard deck, one after another without replacement. a. Find the probability that the first card is a heart and the second card is a red. b. Find the probability that the second card is a red card. What is the difference between independent and dependent events? RULE: If events E and F are independent events, then RULE: Product Rule for Independent Events: Example 8: A calculator requires a keystroke assembly and a logic circuit. Assume that 99% of the keystroke assemblies are satisfactory and 97% of the logic circuits are satisfactory. Find the probability that a finished calculator will be satisfactory.

17 How can we show that two events are independent? Example 9: On a typical January day in Manhattan, the probability of snow is 0.10, the probability of a traffic jam is 0.80 and the probability of snow or a traffic jam (or both) is Are the event it snows and the event a traffic jam occurs independent?

18 7.6: Bayes Theorem Bayes Formula: Example 1: A manufacturer claims that its test will detect anabolic steroid use (that is, show positive for an athlete who uses steroids) 95% of the time. What the company does not tell you is that 6% of all anabolic steroid-free users also test positive (the false positive rate). At ESU, it is estimated that 10% of all the athletes use anabolic steroids. Let T denote the event that an athlete tests positive in the drug test and A denote the event that an athlete has used anabolic steroids. Create a tree diagram to represent the situation. Then find the following probabilities. P(A) = P(T A)= P(T A )= P(A )= P(T A)= P(T A )= Compute the probability that an athlete who tested positive is a steroid user.

19 Example 2: For a fixed length of time, the probability of a worker error on a certain production line is 0.1, the probability that an accident will occur when there is a worker error is 0.3, and the probability that an accident will occur when there is no worker error is 0.2. Find the probability of a worker error if there is an accident. Example 3: 1 % of women at age forty who participate in a routine screening have breast cancer. 80% of women with breast cancer will receive a positive mammography. 9.6% of women without breast cancer will also get a positive mammography. a. A woman in this age group had a positive mammography in a routine screening. What is the probability that she has breast cancer? b. Another woman in this age group has a negative mammography in her routine screening. What is the probability that she does not have breast cancer?

20 Example 4: Marie is getting married tomorrow, an at outdoor ceremony in the desert. In recent years, it has rained only 5 days each year. Unfortunately, the weatherman has predicted rain for tomorrow. When it actually rains, the weather man correctly forecasts rain 90% of the time. When it doesn t rain, he incorrectly forecasts rain 10% of the time. What is the probability that it will rain on the day of Marie s wedding? Example 5: An insurance company issues life insurance policies in three separate categories: standard, preferred, and ultra-preferred. Of the company s policy holders, 50% are standard, 40% are preferred, and 10% are ultra-preferred. Each standard policy holder has probability of dying in the next year, each preferred policyholder has probability of dying in the next year and each ultra-preferred policyholder as probability of dying in the next year. What is the probability that a deceased policy holder was ultra-preferred?