Chapter 5: Probability in Our Daily Lives

1. Chapter 5: Probability in Our Daily Lives Section 5.2: How Can We Find Probabilities?

2. Learning Objectives • Sample Space • Event • Probabilities for a sample space • Probability of an event • Basic rules for finding probabilities about a pair of events • Probability of the union of two events • Probability of the intersection of two events

3. Learning Objective 1:Sample Space • For a random phenomenon, the sample space is the set of all possible outcomes.

4. Class Problem 1 • Election study: An election study records for each sampled subject either D = voted for Democratic candidate, R= voted for Republican candidate, I = voted for Independent candidate, or N = did not vote. • Show how to summarize possible outcomes for two elections with a tree diagram. For example, one possible outcome is DR, voting for the Democrat in the first election and for the Republican in the second election. • If the study considers three elections, how many possible outcomes are in the sample space?

5. Learning Objective 2:Event • An event is a subset of the sample space • An event corresponds to a particular outcome or a group of possible outcomes. • For example; • Event A = student answers all 3 questions correctly = (CCC) • Event B = student passes (at least 2 correct) = (CCI, CIC, ICC, CCC)

6. Learning Objective 3:Probabilities for a sample space Each outcome in a sample space has a probability • The probability of each individual outcome is between 0 and 1 • The total of all the individual probabilities equals 1.

7. Class Problem 2 • Random digit: A single random digit is selected using software or a random number table. • State the sample space for the possible outcomes. • State the probability for each possible outcome, based on what you know about the way random numbers are generated. • Each outcome in a sample space must have portability between 0 and 1, and the total of the probabilities must equal 1. Show that your assignment of probabilities in (b) satisfies this rule. (0,1,2,3,4,5,6,7,8,9) 1/10 = .1 = 10%

8. Learning Objective 4:Probability of an Event • The probability of an event A, denoted by P(A), is obtained by adding the probabilities of the individual outcomes in the event. • When all the possible outcomes are equally likely:

9. Class Problem 3 • Pop quiz: A teacher gives a four question unannounced true false pop quiz, with two possible answers to each question. • Use a tree diagram to show the possible response patterns in terms of whether any given response is correct or incorrect, How many outcomes are in the sample space? • An unprepared student guesses all the answers randomly. Find the probabilities of the possible outcomes on the tree diagram. • Refer to (b) Using the tree diagram, evaluate the probability of passing the quiz, which the teacher defines as answering at least three questions correctly. 16 All three correct: 1/16 = .0625 = 6.25% 5/16 = .3125 = 31.25%

10. Class Problem 4 • Rain tomorrow?: Tomorrow, it might rain sometime, or it might not rain at all. Since the sample space has two possible outcomes, each must have a probability ½. True or False? Explain. False-it is not certain that both possibilities are equally likely

11. Class Problem 5 • Insurance: Every year the insurance industry spends considerable resources assessing risk probabilities. To accumulate a risk of about one in a million of death, you can drive 100 miles, take a cross country plane flight, work as a police officer for 10 hours, work in a coal mine for 12 hours, smoke two cigarettes, be a nonsmoker but live with a smoker for two weeks, or drink 70 pints of beer in a year (Wilson and Crouch, 2001, pp. 208-209). Show that a risk of about one in a million of death is also approximately the probability of flipping 20 heads in a row with a balanced coin.

12. Class Problem 6 • All girls in a family: A family has four children, all girls. Is this unusual? • Construct a sample space for the possible genders of four children. (Hint: the outcome FMFF represents the cases where only the second child born is a male.) • If each child is equally likely to be a girl or a boy, find the probability that all four children are female. • What other assumption is necessary for the calculation in (b) to be valid?

13. Learning Objective 5:Basic rules for finding probabilities about a pair of events • Some events are expressed as the outcomes that • Do not include the specific event (complement of the event) • Are not connected with some other event (disjoint) • Are in one event and in another event (intersection of two events) • Are in one event or in another event (union of two events)

14. Learning Objective 5:Complement of an event • The complement of an event A consists of all outcomes in the sample space that are not in A. • The probabilities of A and of Ac add to 1 • P(Ac) = 1 – P(A)

15. Learning Objective 5:Disjoint events • Two events, A and B, are disjoint if they do not have any common outcomes

16. Learning Objective 5:Intersection of two events • The intersection of A and B consists of outcomes that are in both A and B

17. Learning Objective 5:Union of two events • The union of A and B consists of outcomes that are in A or B or in both A and B.

18. Learning Objective 6:Probability of the Union of Two Events • Addition Rule: • For the union of two events, • P(A or B) = P(A) + P(B) – P(A and B) • If the events are disjoint, P(A and B) = 0, so • P(A or B) = P(A) + P(B)

19. Learning Objective 6:Example • 80.2 million tax payers (80,200,000) • Event A = being audited • Event B = income greater than $100,000 • P(A and B) = 80/80200=.001 • You make $38,000 a year. What is the probability you are not audited?

20. Learning Objective 7:Probability of the Intersection of Two Events Multiplication Rule: For the intersection of two independent events, A and B, P(A and B) = P(A) x P(B)

21. What is the probability of getting 3 questionscorrect by guessing? If the probability of guessing correctly is .2 • What is the probability that a student answers at least 2 questions correctly? • P(CCC) + P(CCI) + P(CIC) + P(ICC) = • 0.008 + 3(0.032) = 0.104

22. Learning Objective 7:Assuming independence • Don’t assume that events are independent unless you have given this assumption careful thought and it seems plausible. • True-blooded Republicans: Of those who voted in the 2000 and 2004 U.S. Presidential elections, about 48% voted republican in 2000 and about 51% voted Republican in 2004. True or false: since 0.48 x 0.51 = 0.24, about 24% voted for the Republican (George W. Bush) in both elections. If true, explain why. If false, explain the flaw in the logic.

23. Events Often Are Not Independent • Example: A Pop Quiz with 2 Multiple Choice Questions • Data giving the proportions for the actual responses of students in a class in which the outcomes could be (I) for incorrect and (C) for correct Outcome: II IC CI CC Probability: 0.26 0.11 0.05 0.58 • Define the events A and B as follows: • A: {first question is answered correctly} • B: {second question is answered correctly} • P(A) = P{(CI), (CC)} = 0.05 + 0.58 = 0.63 • P(B) = P{(IC), (CC)} = 0.11 + 0.58 = 0.69 • P(A and B) = P{(CC)} = 0.58 • If A and B were independent, • P(A and B) = P(A) x P(B) = 0.63 x 0.69 = 0.43 • Thus, in this case, A and B are not independent!

24. Catalog sales: You are the marketing director for a museum that raises money by selling gift items from a mail order catalog. For each catalog sent to a potential customer the customer’s entry in the data file is Y if they ordered something and N if they did not. After you have mailed the fall and the winter catalogs, you estimate the probabilities of the buying patterns based on those who received the catalog as follows: Let F denote buying from the fall catalog and W denote buying from the winter catalog. Find P(F) and P(W). P(F) =.40 or 40% P(W) = .35 or 35% b) Explain what the event “F and W” means and find P(F and W). P(F and W) =.30 or 30% c) Are F and W independent events? Explain why you would not normally expect customer choices to be independent.