Chapter 4. Probability

Transcription

1 Chapter 4. Probability Chapter Problem: Are polygraph instruments effective as lie detector? Table 4-1 Results from Experiments with Polygraph Instruments Did the Subject Actually Lie? No (Did Not Lie) Yes (Lied) Positive Test Result (Polygraph test indicated (False positive) (True positive) that the subject lied.) Negative Test Result 32 9 (Polygraph test indicated (True negative) (False negative) that the subject did not lie) 1

2 How statistician think: 4.1 Review and Preview Rare Event Rule for Inferential Statistics If, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct. Re-phrase: 1. Assumption 2. Find the probability of an event, under the assumption, is very small 3. Conclude that the assumption is probably not correct 2

3 4.2 Basic Concepts of Probability Part 1 Definition An event is any collection of outcomes of a procedure. A simple event is an outcome or an event that cannot be further broken down into simpler components. The sample space for a procedure consists of all possible simple events. e.g.1 Procedure Example of event Complete sample space Single birth female (simple event) {f, m} 3 births 2 female and a male {fff, ffm, fmf, fmm, mff, mfm, mmf, mmm} 3

4 4.2 Basic Concepts of Probability Part 1 Notion for Probabilities P denotes a probability A, B, and C denote specific events P(A) denotes the probability of event A occurring. Three different approaches to find the probability of an event: 1. Relative frequency approximation of probability 2. Classical approach to probability (requires equally likely outcome) 3. Subjective probabilities 4

5 4.2 Basic Concepts of Probability Part 1 1: Relative Frequency Approximation of Probability Conduct (or observe) a procedure, and count the number times that event A actually occurs. Based on these actual result, P(A) is estimated as follows number of times A occurred P( A) number of times the trial was repeated When trying to determine the probability that an individual car Crashes in a year, we must examine past results to determine the number of cars in use in a year and the number of them that crashed, then we find the ratio of the number of cars that crashed to the total number of cars. For a recent year, the result is a probability of (see example 2) 5

6 4.2 Basic Concepts of Probability Part 1 2: Classical Approach to Probability Assume that a given procedure has n different simple event and that each of those simple events has an equal chance of occurring. If event A can occur is s of these n ways, then number of ways A can occurr s P( A) number of different simpleevents n When trying to determine the probability of winning the grand prize in a lottery by selecting 6 numbers between 1 and 60, each combination has an equal chance of occurring. The probability of winning is , which can be found by using methods presented later in this chapter. 6

7 4.2 Basic Concepts of Probability Part 1 3: Subjective Probabilities P(A), the probability of event A, is estimated by using knowledge of the relevant circumstances. When trying to estimate the probability of an astronaut surviving a mission in space shuttle, experts consider past event along with changes in technologies and conditions to develop an estimate of the probability. As of this writing, that probability has be estimated by NASA scientists as

8 4.2 Basic Concepts of Probability Part 1 Law of Large Numbers. As a procedure is repeated again and again, the relative frequency probability (from Rule 1) of an event tends to approach the actual probability. Probability and Outcomes that are not always equally likely. One common mistake is to incorrectly assume that outcomes are equally likely just because we know nothing about the likelihood of different outcomes. 8

9 4.2 Basic Concepts of Probability Part 1 e.g.2 Probability of a car crash. Find the probability that a randomly selected car in the U.S. will be in crash this year. Sol. For a recent year, there were 6,511,100 cars that crashed among the 135,670,000 cars registered in the U.S. (Statistical Abstract of the U.S.). We use the relative frequency approach: number of cars that crashed 6,511,100 P( Crash) total number of cars 135,670,

10 4.2 Basic Concepts of Probability Part 1 e.g.3 Probability of a Positive Test Result. Refer to table 4-1 included in the chapter problem. Assuming that one of the 98 test results summarized in table 4-1 is randomly selected, find the probability that it is a positive test result. Sol. The sample space consists of the 98 test results in table 4-1. Among the 98 results, 57 of them are positive results ( ). Since each test result is equally likely to be selected, we can apply the classical approach: P(positive test result from table 4-1) number of positive test results totalnumber of results

11 4.2 Basic Concepts of Probability Part 1 e.g.4 Genotype When studying the affect of heredity on height, we can express each individual genotype, AA, Aa, aa, and aa, on an index card and shuffle the four cards randomly and select one of them. What is the probability that we select a genotype in which the two components are different. Ans. Classical approach, 0.5 e.g.5 Probability of a President from Alaska. Find the probability that the next President of the U.S. is from Alaska. Ans. Subjective estimate, e.g.6 Stuck in an Elevator. What is the probability that you will get stuck in the next elevator that you ride? Ans. Subjective estimate,

12 4.2 Basic Concepts of Probability Part 1 Finding the total number of outcomes. e.g.7 Gender of children. Find the probability that when a couple has 3 children, they will have exactly 2 boys. Assume that the boys and girls are equally likely and that the gender of any child is not influenced by the gender of any other child. 3 P( 2 boysin 3 birth)

13 4.2 Basic Concepts of Probability Part 1 Finding the total number of outcomes. e.g.8 American Online Survey. AOL asked users this question about KFC, will KFC gain or lose business after eliminating trans fats? Among the responses received, 1941 said that KFC would gain business, 1260 said that KFC business would remain the same, and 204 said that KFC would lose business. Find the probability that a randomly selected response states that KFC would gain business P( responseof a gain in business) Interpretation. There is a probability that if a response is randomly selected, it was a response of a gain in business. Important: Note that the survey involves a voluntary response sample because the AOL users themselves decided whether to respond. Consequently, when interpreting the results of this survey, keep in mind that they do not necessary reflect the opinions of the general population. The response reflect only the opinions of those who chose to respond. 13

14 4.2 Basic Concepts of Probability Part 1 1 Certain Likely Chance Unlikely 0 Impossible Figure 4 2 Possible Values for Probabilities 14

15 4.2 Basic Concepts of Probability Part 1 The probability of an impossible event is 0 The probability of an event that is certain to occur is 1 For any event A, the probability of A is between 0 and 1 inclusive. That is 0 P(A) 1 e.g.9 Find the probability that Thanksgiving Day will be on (a) Wednesday, (b) Thursday. P(Thanksgiving on Wednesday) = 0 P(Thanksgiving on Thursday) = 1 15

16 4.2 Basic Concepts of Probability Part 1 Complementary Events The complement of event A, denoted by A, consists of all outcomes in which event A does not occur. e.g.10 Guessing on an SAT Test. A typical question on an SAT test requires the test taker to select one of five possible choices: A, B, C, D, or E. Because only one answer is correct, if you make a random guess, your probability of being correct is 1/5 or 0.2. Find the probability of making a random guess and not being correct. Ans

17 4.2 Basic Concepts of Probability Part 1 Rounding off Probabilities. Either simple fractions or round the decimal to three significant digits. e.g (e.g.2) 1/ ½ = 0.5 (no need to write as 0.500) 1941/ (e.g.8) 17

18 4.2 Basic Concepts of Probability Part 1 e.g.12 Unusual Events? In a clinical experiment of the Salk vaccine for polio, 200,745 children were given a placebo 201,229 other children were treated with the Salk Vaccine 115 cases of polio among those in the placebo group 33 cases in the treatment group If we assume that the vaccine has no effect, the probability of getting such test results is found to be less than Is an event with a probability less than an unusual event? Ans. Yes, the event is unusual What does that probability imply about the effectiveness of the vaccine? Ans. we conclude that the vaccine appears to be effective Rare event rule for inferential statistics! (beginning of the chapter) 18

19 4.2 Basic Concepts of Probability Part 2 Part 2. Beyond the Basics of Probability: Odds expressions of likelihood are often given as odds, such as 50:1 Definition The actual odds against event A occurring are the ratio P(A)/P(A). It is usually expressed in the form a : b The actual odds in favor of event A occurring are P(A)/P(A). If odds against event A is a : b, then odds in favor of A is b : a. The payoff odds against event A represents the ratio of net profit (if you win) to the amount bet. payoff odds against event A = (net profit):(amount bet) 19

20 4.2 Basic Concepts of Probability Part 2 Odds e.g.13 If you bet $5 on the number 13 in roulette, your probability of winning is 1/38 and the payoff odds are given by the casino as 35:1. a. Find the actual odds against the out come of P( A), P( A), soodds against winning is P( A) : P( A) 37 : b. How much net profit would you make if you win by betting 13? By definition 35:1 = (net profit):5, therefore, net profit is 35(5) = $175 20

21 4.2 Basic Concepts of Probability Part 2 Odds e.g.9 If you bet $5 on the number 13 in roulette, your probability of winning is 1/38 and the payoff odds are given by the casino as 35:1 c. If the casino were operating just for the fun of it, and the payoff odds were changed to match the actual odds against 13, how much you win if the outcome is 13? Now 37:1 = (net profit):5 Net profit is 37(5) = $185 21

22 Notation for Addition Rule 4.3 Addition Rule Definition. A compound event is any event combining two or more simple events. Notation for Addition Rule P(A or B) = P(in a single trial, event A occurs or event B occurs or they both occur) Be careful about the notation P(A and B) 22

23 Notation for addition rule 4.3 Addition Rule Table 4-1 Results from Experiments with Polygraph Instruments Did the Subject Actually Lie? No (Did Not Lie) Yes (Lied) Positive Test Result (Polygraph test indicated (False positive) (True positive) that the subject lied.) 57 Negative Test Result 32 9 (Polygraph test indicated (True negative) (False negative) that the subject did not lie)

24 Notation for addition rule 4.3 Addition Rule e.g.1 Polygraph Test refer to table 4-1. if a subject is randomly selected from 98 subjects, find the probability of selecting a subject who had a positive test result or lied. Count: how many tested positive, or lied, or both. Three methods for counting method = 66 method = 66 method = 66 Ans. P(positive test result or lied) = 66/

25 Notation for addition rule 4.3 Addition Rule Formal Addition Rule: P(A or B) = P(A) + P(B) P(A and B) where P(A and B) denotes the probability that A and B both occur at the same time as an outcome in a trial of a procedure. Caution: avoid counting outcomes more than once. 25

26 Notation for addition rule 4.3 Addition Rule Intuitive Addition Rule To find P(A or B) find the sum of the number of ways event A can occur find the sum of the number of ways event B can occur adding in such a way that every outcome is counted only once P(A or B) is equal to that sum, divided by the total number of outcomes in the sample space. 26

27 4.3 Addition Rule Notation for addition rule Definition Event A and B are disjoint (or mutually exclusive) if they cannot occur at the same time. (That is, disjoint events do not overlap.) e.g.2 Polygraph Test (Table 4 1). Selecting one subject from 98. a. Determine whether the following events are disjoint: Event A: getting a subject with negative test result Event B: getting a subject who did not lie Not disjoint b. Find the probability of selecting a subject who had negative test result or did not lie. 56/

28 Notation for addition rule A B 4.3 Addition Rule A B Figure 4 3 Venn Diagram For Event That are Not Disjoint Figure 4 3 Venn Diagram For Event That are Disjoint This figure explains why P(A or B) = P(A) + P(B) P(A and B) 28

29 Complementary Events 4.3 Addition Rule Recall, The complement of event A, denoted by A, consists of all outcomes in which event A does not occur. A and A must be disjoint; it is impossible for an event and its complement to occur at the same time A either occur or does not occur, which means that either A or A must occur. Therefore P( Aor A) P( A) P( A) 1 29

30 Complementary Events 4.3 Addition Rule Rule of Complementary Events P( A) P( A) 1 P( A) P( A) 1 1 P( A) P( A) e.g.3 FBI data show that 62.4% of murders are cleared by arrests. We can express the probability of a murder being cleared by an arrest as P( cleared ) = Find P( cleared ) Ans. P( cleared ) = = A major advantage of the rule of complementary events is that its use can greatly simplify certain problems. 30

31 4.4 Multiplication Rule Notation P(A and B) = P(event A occurs in a first trial and event B occurs in a second trial) Consider following two question (1 st T or F, 2 nd multiple choice) 1. True or False: A pound of feathers is heavier than a pound of gold. 2. Who said that smoking is one of the leading causes of statistics? a. Philip Morris b. Smokey Robinson c. Fletcher Knebel d. R. J. Reynolds e. Virginia Slims Ten possible outcomes: 31

32 4.4 Multiplication Rule a Ta Figure 4 6 Tree Diagram of Test Answers T F b c d e a b c d e Tb Tc Td Te Fa Fb Fc Fd Fe 2 5 = 10 In this case P(T and c) = 1/10, P(T) = ½, P(c) = 1/5, Note that ½ (1/5) = 1/10, Is this suggesting P(T and c) = P(T)P(c)? 32

33 4.4 Multiplication Rule e.g.1 If two of the subjects included in the table are randomly selected without replacement, find the probability that the 1 st selected person had a positive test result and the 2 nd selected person had a negative test result. Table 4-1 Results from Experiments with Polygraph Instruments Did the Subject Actually Lie? No (Did Not Lie) Yes (Lied) Positive Test Result (Polygraph test indicated (False positive) (True positive) that the subject lied.) Negative Test Result 32 9 (Polygraph test indicated (True negative) (False negative) that the subject did not lie) 33

34 Solution: 1 st selection 4.4 Multiplication Rule P(positive test result) = 57/98 2 nd selection P(negative test result) = 41/97 P(1 st subject has positive test result and 2 nd subject has negative result) = (57/98)(41/97) = The key point is that we must adjust the probability of the second event to reflect the outcome of the first event. 34

35 4.4 Multiplication Rule Notion for Conditional Probability P(B A) represents the probability of event B occurring after it is assumed that event A has already occurred. (We can read B A as B given A or as event B occurring after event A has already occurred. ) Definition Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. If A and B are not independent, they are said to be dependent. 35

36 4.4 Multiplication Rule Formal Multiplication Rule P(A and B) = P(A) P(B A) start P(A and B) Multiplication Rule P(A and B) = P(A) P(B) yes Are A and B Independent? no P(A and B) = P(A) P(B A) Figure 4 7 Applying Multiplication Rule Caution: when applying the multiplication rule, always consider whether the events are independent or dependent. 36

37 4.4 Multiplication Rule e.g.2 Quality Control in Manufacturing. Consider a small sample of 5 pacemakers, including three that are good (denoted by G) and two that are defective (denoted by D). A medical researcher randomly select two of the pacemakers for further experimentation. Find the probability that the first selected pacemaker is good (G) and the second pacemaker is also good (G). Use each of the following assumption: (a) with replacement; (b) without replacement. Ans. Independent, (3/5)(3/5) = 9/25 = 0.36 Ans. Dependent, (3/5)(2/4) = 0.3 Interpretation. Note that in part (b) we adjusted the second probability, to take into Account the selection in the first outcome. Also note that a medical researcher would not sample with replacement, as in part (a). However, in statistics we have a special interest in sampling with replacement. (see section 6-4) 37

38 4.4 Multiplication Rule Treating Dependent Events as Independent it is common practice to treat events as independent when small sample are draw from large populations. Guideline. If a sample size is no more than 5% of the population treats the selection as being independent. e.g.3 Quality Control in Manufacturing. Assume that we have a batch of 100,000 pacemakers, including 99,950 good ones and 50 defective ones. a. If two are randomly selected, find P(both are good). Ans. P(both good) = b. If 20 are randomly selected, find P(all 20 are good). Ans. P(all 20 are good) =

39 4.4 Multiplication Rule The following example illustrate the importance of carefully identifying the events being considered. e.g.4 Birthdays. Assume that two people are randomly selected and also assume that birthdays occur on the days of the week with equal frequencies. a. Find the probability that two people are born on the same day of the week. 1 P( Twoare on thesame day) 7 b. Find the probability that two people are born on Monday. 1 P( Twoare on Monday) 49 39

40 4.4 Multiplication Rule Important applications of the multiplication rule. E.g.5 Gives insight into hypothesis testing E.g.6 illustrates the principle of redundancy 40

41 4.4 Multiplication Rule e.g.5 Effectiveness of Gender Selection. A geneticist develops a procedure for increasing the likelihood of female babies. In an additional test, 20 couples uses the method and the results consist of 20 females among 20 babies. Assuming that gender selection procedure has no effect, find the probability of getting 20 females among 20 babies by chance. Based on the result, is there strong evidence to support the geneticist s claim that the procedure is effective in increasing the likelihood that babe s will female. (conclusion) Sol. If no effect, the P(all 20 offspring are female) = (1/2) 20 = Therefore gender-selection process appears to be effective in increasing the 41

42 4.4 Multiplication Rule e.g.6 Redundancy for Increased Reliability. Modern aircraft engines are now highly reliable. One design feature contributing to that reliability is the use of redundancy, whereby critical components are duplicated so that if one fails, the other will work. For example, single-engine aircraft now have two independent electrical systems so that if one electrical system fails, the other can continue to work so that the engine does not fail. For the purpose of this example, we assume that the probability of an electrical system failure is a. If the engine ins an aircraft has one electrical system, what is the probability that it will work? Ans. P(work) = b. If the engine in an aircraft has two independent electrical systems, what is the probability that the engine can function with a working electrical system? Ans. P(work) =