once? These kinds of questions can be answered by applying some basic probability rules that you will

Transcription

1 TOPIC 30 Probability Rules What is the probability that a randomly selected movie will be one that neither you nor your friend has seen yet? What is the probability that you have not seen the film, given that your friend has not seen it? If you apply to two different graduate programs, how likely are you to be accepted by at least one of them? For how many days must you play a lottery game in order to have a 50/50 chance of winning at least once? These kinds of questions can be answered by applying some basic probability rules that you will learn in this topic. Overview Topic 11 provided you with an introduction to probability, which is the mathematical study of randomness and uncertainty. Probability is important for studying statistics because, as you have seen, well-designed studies make deliberate use of randomness, either by selecting random samples from a population or by randomly assigning subjects to experimental groups. In Topic 11, you began to understand that probability refers to the long-run fraction of times that an event occurs when a random process is repeated over and over. You also learned how to use simulation to approximate probabilities, and you saw how to calculate exact probabilities in situations where outcomes are equally likely. This topic introduces you to a somewhat more formal and mathematical examination of probability. You will learn mathematical rules that enable you to determine probabilities, whether or not the random process involves equally likely outcomes. You will encounter and learn how to apply rules for calculating probabilities of one event or another event, one event and another event, and conditional probabilities of one event given another event. You will also examine the concept of independence as it applies to random events. Rossman/Chance, Workshop Statistics, 4/e 1

2 Preliminaries 1. If you have a.4 probability of being accepted by one graduate school and a.3 probability of being accepted by another graduate school, is the probability that you are accepted by at least one of them (one or the other or both) equal to.7? 2. If you have a.7 probability of being accepted by one graduate school and a.6 probability of being accepted by another graduate school, is the probability that you are accepted by at least one of them equal to 1.3? Does this imply you are guaranteed to be accepted by at least one school? 3. If you learn that you have been accepted by one of the graduate schools, would you suspect that your probability of being accepted by the other would increase, decrease, or remain the same? In-Class Activities In Topic 11, you learned how to calculate probabilities for outcomes of a random process (like giving four babies to their mothers at random or picking a random number in a lottery game) when the outcomes were equally likely. This probability told you the long-run proportion of times that the outcome would occur if the random process were repeated a large number of times. In this topic, you will learn more formal notation and terminology, as well as some useful rules for calculating and combining probabilities. A collection of outcomes is called an event. Examples of events from the Random Babies activity in Topic 11 include the outcome for which all four mothers get the correct baby and the set of outcomes for which at least two mothers get the correct baby. In the notation of Activity 11-1, this latter event Rossman/Chance, Workshop Statistics, 4/e 2

3 consists of the outcomes {1234, 1243, 1324, 1432, 2134, 3214, 4231}. We will often use the following short-hand notation: capital letters to denote an event and Pr(E) to denote the probability of event E. Recall from Topic 11 that all probabilities are between 0 and 1, inclusive, so 0 Pr(E) 1. Activity 30-1: Watching Films 6-17, 6-18, 30-1, 30-3, In 1998, the American Film Institute created a list of the top 100 American films ever made ( Two people, Allan and Beth, classified each film according to whether they have seen it. The results are summarized in the following 2 2 table: Beth Yes Beth No Total Allan Yes 42 6 Allan No Total 100 Suppose that one of these 100 films is chosen at random, which means each of the 100 films is equally likely to be chosen. a. Determine the probability that Allan has seen the film. (Hint: Because the 100 films are equally likely to be chosen, you can calculate this probability by counting how many films he has seen and then dividing that value by 100.) b. Similarly, determine the probability that Beth has seen the film. c. Determine the probability that Allan has not seen the film. Also describe two different ways to find this probability. The complement rule asserts that the probability an event does not occur equals one minus the probability the event does occur: Pr(not E) = 1 Pr(E). Rossman/Chance, Workshop Statistics, 4/e 3

4 d. Use the complement rule to determine the probability that Beth has not seen the film. e. From the table, determine the probability that both Allan and Beth have seen the film, Pr(A and B). f. From the table, determine the probability that neither Allan nor Beth has seen the film, Pr(not A and not B). g. Add the values in the appropriate cells of the table at the beginning of this activity to determine the probability that either Allan or Beth (or both) have seen the film, Pr(A or B). h. You might think that Pr(A or B) = Pr(A) + Pr(B). Calculate this sum, and indicate whether it is greater or less than Pr(A or B) and by how much. (Hint: Remember you calculated Pr(A or B) in part g.) Explain why this result makes sense, and indicate how to adjust the right side of this expression to make the equality valid. Rossman/Chance, Workshop Statistics, 4/e 4

5 The addition rule says to determine the probability of one event or the other (or both) occurring, you add the probabilities of the individual events but then subtract the probability of the intersection of the two events those outcomes that are part of both events. For any two events E and F, Pr(E or F) = Pr(E) + Pr(F) Pr(E and F). The probability of the intersection (E and F) needs to be subtracted because otherwise it is counted twice, once in Pr(E) and once in Pr(F). Here, the 42 films that both Allan and Beth have seen were included in both the 48 films that Allan had seen and also among the 59 films that Beth had seen. i. Use this addition rule as a second way to calculate the probability that Allan or Beth has seen the movie, verifying your answer to part g. j. As a third way to calculate this probability, first describe which movies belong to the complement (opposite) of the event Allan or Beth has seen the film. Then find the probability of this complement from the table at the beginning of the activity. Then use the complement rule to determine Pr( Allan has seen the film or Beth has seen the film ) using the complement rule. Have you confirmed your answers to part g and part i? k. In what circumstance is it valid simply to add the probabilities of two events to calculate the probability that one or the other event occurs? In other words, what would have to be true about events E and F in order for Pr(E or F) = Pr(E) + P(F)? Rossman/Chance, Workshop Statistics, 4/e 5

6 Two events E and F are mutually exclusive when it is impossible for both of them to occur simultaneously. In this special case, Pr(E or F) = Pr(E) + Pr(F). Note that the addition rule still applies here because Pr(E and F) = 0 when E and F are mutually exclusive. An example from Topic 11 is that Pr( 0 or 1 mother gets the correct baby ) = Pr( 0 mothers get the correct baby ) + Pr( 1 mother gets the correct baby ). The intersection, both 0 and 1 mother getting the correct babies at the same time, can t happen. Watch Out When you speak of the probability of E or F, you include three possibilities: E and not F, F and not E, both E and F. Do not assume that or means that only one of the events occurs. Always be on the lookout for whether calculating the probability of the complement of the event you re looking for might be easier than calculating the probability of that event itself. In such cases, you can first find the probability of the complement and then subtract from one to calculate the probability of interest. Remember to subtract the probability of the intersection (E and F) when applying the addition rule. You do not necessarily need to use one of these rules when calculating probabilities. If you can perform the calculation using other methods, that s fine. Better yet, calculate the probability in a couple of different ways to verify your answer. Activity 30-2: Graduate School Applications 30-2, 30-4, 30-6 Suppose you have applied to two graduate schools and believe you have a.30 probability of being accepted by school C, a.60 probability of being accepted by school D, and a.25 probability of being accepted by both schools. In symbols, Pr(C) =.30, Pr(D) =.60 and Pr(C and D) =.25. Rossman/Chance, Workshop Statistics, 4/e 6

7 a. Organize this information into the following probability table. (Hint: Determine the appropriate places to enter the values.30,.60, and.25. Do not fill in the other missing entries yet.) Accepted by C Not Accepted by C Total Accepted by D Not Accepted by D Total 1.0 b. Now go ahead and fill in the remaining entries in the table. (Hint: All of the values in the table should be probabilities and should, therefore, be between 0 and 1.) c. Determine the probability of being accepted by at least one of these two schools, first by simply adding the appropriate entries in the table. d. Confirm your answer to part c by using the general addition rule to calculate Pr(C or D). e. Determine the probability of being rejected by both schools (i.e., being accepted by neither school). f. Determine the probability of being accepted by one school but not both schools. Activity 30-3: Watching Films 6-17, 6-18, 30-1, 30-3, Reconsider the 2 2 table pertaining to the AFI top 100 films and whether Allan or Beth has seen the film: Rossman/Chance, Workshop Statistics, 4/e 7

8 Beth Yes Beth No Total Allan Yes Allan No Total Suppose again that one of these 100 films is chosen at random. But then suppose you are provided with the information that Allan has seen the film. a. Given the information that Allan has seen the film, what is the updated (conditional) probability that Beth has seen it? (Hint: Restrict your consideration to films that Allan has seen, and ask yourself, what fraction of them has Beth seen?) b. How does this conditional probability of Beth having seen the film given that Allan has seen it compare with the original (unconditional) probability of Beth having seen the film in the first place? Does the knowledge that Allan has seen the film make it more or less likely (or neither) that Beth has seen it? c. Suggest how this conditional probability could have been calculated from Pr(A and B), Pr(A), and Pr(B). Also answer this question: Which of these three values is not needed? Denote the conditional probability of an event B given that event A has occurred by Pr(B A). This conditional probability can be calculated as Pr(B A) = Pr(A and B) / P(A). Rossman/Chance, Workshop Statistics, 4/e 8

9 Note: When using the complement rule with conditional probabilities, be sure to condition on the same event. In other words, Pr(not E F) = 1 Pr(E F). d. Use this definition of conditional probability to calculate Pr(not A not B) in this context. Also explain using your own words what the resulting probability means. Now consider these hypothetical data concerning the number of these films seen by Chuck and by Donna: Donna Yes Donna No Total Chuck Yes Chuck No Total e. Compare Donna s (unconditional) probability of having seen the film with the conditional probability that she has seen it given that Chuck has. Does the knowledge that Chuck has seen the film change the probability that Donna has seen it? Pr(D) = Pr(D C) = Two events E and F are said to be independent if Pr(E F) = Pr(E). This says that knowing that event F has occurred does not change the probability that event E will occur as well. An equivalent condition for E and F to be independent is that Pr(E and F) = Pr(E) Pr(F). If two events are not independent, they are said to be dependent. Note that if events E and F are independent, then events not E and not F are also independent, as are E and not F, and so on. Rossman/Chance, Workshop Statistics, 4/e 9

10 f. Are the events Allan has seen the film and Beth has seen the film independent? What about the events Chuck has seen the film and Donna has seen the film? Explain how you are determining your answers, based on your earlier calculations. Also discuss why your answer makes sense in this context (friends seeing films) and how the nature of the dependence differs between the two cases. Now suppose you are told that Ellen has seen 80% of the films that Donna has seen. g. Express the value of.80 as a conditional probability involving the events E = Ellen has seen the film and D = Donna has seen the film. Pr( ) =.80 h. Can you use the information given about Donna and Ellen to determine the proportion of films that both Donna and Ellen have seen? If so, please do. (Hint: Solve for Pr(D and E) from the expression for Pr(E D).) The multiplication rule asserts that Pr(E and F) = Pr(E) Pr(F E). This can equivalently be written as Pr(E and F) = Pr(F) P(E F). When the two events are independent, this becomes Pr(E and F) = Pr(E) Pr(F). In applying the multiplication rule, think about which event probabilities have been given to you in order to decide which version of the multiplication rule to use. In checking the reasonableness of your final calculated values, one thing to notice is that when you determine the probability of at least one event occurring, you tend to add the values, resulting in a higher probability than the individual events. To Rossman/Chance, Workshop Statistics, 4/e 10

11 determine the probability of two events occurring simultaneously, you multiply, resulting in a lower probability. Watch Out Don t confuse the property of independence with the property of being mutually exclusive. These properties are very different. In fact, if two events (with nonzero probabilities) are mutually exclusive events, then they cannot be independent because Pr(E F) = 0 when E and F are mutually exclusive. Many students are tempted to always multiply (unconditional) probabilities to calculate the probability that both events occur. Be careful to do this only when the events are independent or when one of the probabilities is a conditional one. Do not be too quick to assume independence. Think about the context of the random process to judge whether assuming the events are independent is reasonable (e.g., complete strangers picking a movie vs. two friends). With real data, you will almost never find perfect (exact) independence, but you can determine whether the events are close to independent. Activity 30-4: Graduate School Applications 30-2, 30-4, 30-6 Recall the scenario from Activity 30-2, in which you have applied to two graduate schools and believe that you have a.3 probability of being accepted by school C, a.6 probability of being accepted by school D, and a.25 probability of being accepted by both. a. Are the events acceptance by C and acceptance by D independent? Answer without calculating any new probabilities, and explain your answer. Rossman/Chance, Workshop Statistics, 4/e 11

12 b. Determine the conditional probability of acceptance by D given acceptance by C. How does it compare to the (unconditional) probability of acceptance by D? Now suppose you change your mind and apply to two different schools G and H. You regard the events acceptance by G and acceptance by H to be independent, with the probability of acceptance by G equal to.7 and the probability of acceptance by H equal to.8. c. Determine the probability of acceptance by both schools G and H. d. Determine the probability of acceptance by at least one school: G or H. (Hint: Use the addition rule and your answer to part c.) The general multiplication rule says that if E 1, E 2,, E k is a series of independent events, then Pr(E 1 and E 2 and E k ) = Pr(E 1 ) Pr(E 2 ) Pr(E k ). Suppose that in addition to applying to graduate schools G and H, you also apply to schools I and J, that you consider all acceptances to be independent of each other, and that you believe the probabilities of acceptance to be.5 and.9, respectively, at schools I and J. e. Determine the probability of acceptance by all four schools: G, H, I, and J. Rossman/Chance, Workshop Statistics, 4/e 12

13 f. Determine the probability of acceptance by at least one of these four schools. (Hint: First find the probability of the complement of this event.) g. Suppose you have a.7 probability of being accepted by a school and that given your acceptance, the conditional probability of receiving financial aid is.9. Determine the probability that you are both accepted and receive financial aid from this school. Self-Check Activity 30-5: Alarming Wake-Up 23-4, 30-5, 32-20, A study published in the journal Pediatrics (Smith et al., 2006) addressed the important issue of how to awaken children during a house fire so they can escape safely. Researchers worked with a volunteer sample of 24 healthy children ages 6 12 by training them to perform a simulated self-rescue escape procedure when they heard an alarm. Researchers then recorded the children s reactions to a conventional smoke alarm and to a personalized recording of the mother s voice saying the child s name and urging him or her to wake up. All 24 children were exposed to both kinds of alarms, with the order determined randomly. In the study, one child did not wake up to either alarm, 14 woke up to both alarms, and 9 woke up to the mother s voice but not the conventional alarm. Suppose one of these 24 children is chosen at random. a. Convert the given data into probabilities and display them in a 2 2 probability table. Also fill in the remaining cells of the table: Rossman/Chance, Workshop Statistics, 4/e 13

14 Awakened to Conventional Alarm Did Not Awaken to Conventional Alarm Awakened to Mother s Voice Did Not Awaken to Mother s Voice Total 1.0 b. Use the table to determine the probability that the randomly selected child awakened to at least one of the two kinds of alarms. Total c. Show how you could use the addition rule to calculate the probability in part b. Researchers also recorded whether the child successfully escaped the house within five minutes of the alarm sounding. They found that 20 children escaped with the mother s voice alarm, 9 children escaped with the conventional alarm, and 2 children did not escape with either alarm. d. Convert these counts into probabilities. Display them, and the remaining entries, in the following probability table: Escaped to Conventional Alarm Did Not Escape to Conventional Alarm Escaped to Mother s Voice Did Not Escape to Mother s Voice Total 1.0 e. Determine the probability that a randomly selected child escaped with one alarm but not the other. Total Rossman/Chance, Workshop Statistics, 4/e 14

15 f. Given the child escaped with one alarm but not the other, determine the conditional probability that the mother s voice alarm was the one he or she escaped with. g. Summarize what your analyses reveal about the relative effectiveness of these two kinds of alarms. Solution a. Awakened to Conventional Did Not Awaken to Conventional Total Alarm Alarm Awakened to Mother s Voice 14/ /24 = / Did Not Awaken to Mother s 0 1/ / Voice Total 14/ / b. The probability that the randomly selected child awakened to at least one of the two kinds of alarms is 9/ /24+0/24 = 23/ c. Let M denote the event that the child awakened to the mother s voice alarm and C be the event that the child awakened to the conventional alarm. Then the addition rule says that Pr(M or C) = Pr(M) + Pr(C) Pr(M and C) = 23/ /24 14/24 = 23/ d. Escaped to Conventional Did Not Escape to Conventional Total Alarm Alarm Escaped to Mother s Voice 7/ /24 = / Did Not Escape to Mother s 2/ / / Voice Total 9/24 = /24 = Rossman/Chance, Workshop Statistics, 4/e 15

16 e. The probability that a randomly selected child escaped with one alarm but not the other is the sum of the probabilities in the top-right and bottom-left cells of the table: 13/24 + 2/24 = 15/24 =.625. f. Given the child escaped with one alarm but not the other, the conditional probability that the mother s voice alarm was the one he or she escaped with is (13/24)/(15/24) = 13/ g. Analyses of both waking up results and escaping results indicate these children were more likely to respond to the mother s voice alarm than to the conventional alarm. In terms of waking up, all but one child woke up to the mother s voice, and the exception did not wake up to the conventional alarm either. In terms of escaping the house within five minutes, these children were more likely to escape with the mother s voice alarm (probabilities of.833 vs..375). The probability is larger than.85 that a child who escaped with one alarm but not the other did so with the mother s voice alarm. Wrap-Up This topic continued your study of probability by introducing you to several rules that are useful for calculating new probabilities based on known ones. Two of these rules are the complement rule and the addition rule, which proved useful for calculations involving film watching and graduate school applications. The complement rule enables you to find the probability of the complement (opposite) of any event, so it s always wise to be on the lookout for whether the probability of an event s complement may be easier to calculate than the event s probability itself. For example, you can find the probability of being accepted by at least one graduate program by first calculating the probability of rejection by all programs because the complement of at least one is none. The addition rule enables you to determine the probability that one or the other of two events occurs, providing you know the probability of each event and the probability that both events occur. In fact, knowing any three of these probabilities allows you to determine the remaining one. Rossman/Chance, Workshop Statistics, 4/e 16

17 You also studied how to calculate conditional probabilities, which update uncertainty by taking partial information into account. A related idea that you studied is the concept of probabilistic independence. For example, you found that whether Allan has seen a film is not independent of whether Beth has seen it, because learning that Beth has seen a film increases the probability that Allan has seen it (indicating they have similar movie tastes). Another probability rule that you studied is the multiplication rule, which enables you to find the probability that two events both occur or even that multiple events all occur simultaneously. In Brief Some useful definitions to remember and habits to develop from this topic are All probabilities must be between 0 and 1, where 0 indicates the probability of an event that cannot possibly happen and 1 indicates the probability of an event that is guaranteed to happen. A probability table can be a useful device for organizing and calculating probabilities. The probability that an event does not happen can be calculated as one minus the probability that it does happen. This is known as the complement rule. The addition rule enables you to calculate the probability that one or the other (or both) of two events occurs: Pr(E or F) = Pr(E) + Pr(F) Pr(E and F). Two events are said to be mutually exclusive when it is impossible for both of them to occur simultaneously. Conditional probability provides an update in the uncertainty regarding one event given information that another event has occurred. A conditional probability is calculated as the probability that both events occur divided by the probability of the event that has already occurred: Pr(E F) = Pr(E and F)/Pr(F). Two events are said to be independent if the conditional probability of one given the other is the same as the event s unconditional probability in the first place, e.g., Pr(E F) = Pr(E). Rossman/Chance, Workshop Statistics, 4/e 17

18 The multiplication rule allows you to calculate the probability that two events both occur (intersection), based on one of their probabilities and the conditional probability of the other event given the first: Pr(E and F) = Pr(E F) Pr(F) = Pr(F E) Pr(E). In the case of independent events, the multiplication rule says the probability that both (or all) of the events occur is equal to the product of their individual probabilities: Pr(E and F) = Pr(E) Pr(F). You should be able to Use a probability table to organize and calculate probabilities. (Activities 30-1, 30-2, 30-3, 30-5) Apply the complement rule, addition rule, and multiplication rules to calculate probabilities. (Activities 30-1, 30-2, 30-3, 30-4, 30-5) Calculate conditional probabilities. (Activities 30-3, 30-5) Determine whether or not two events are independent. (Activity 30-4) The next topic continues to deepen your study of probability by introducing random variables and the concepts associated with random variables. Exercises Exercise 30-6: Graduate School Applications 30-2, 30-4, 30-6 Suppose you have applied to two graduate schools and believe you have a.30 probability of being accepted by school C and a.60 probability of being accepted by school D. a. Explain why it is impossible for the probability of being accepted by both schools to be.40. b. Explain why it is impossible for the probability of being accepted by neither school to be.05. c. Determine the largest possible value for the probability of being accepted by both schools. Explain the reasoning behind your answer. Rossman/Chance, Workshop Statistics, 4/e 18

19 d. Determine the smallest possible value for the probability of being accepted by both schools. Explain the reasoning behind your answer. Exercise 30-7: Commuting to School Suppose you encounter two traffic lights on your commute to school. You estimate the probability is.5 that the first light will be red when you get to it,.4 that the second light will be red, and.6 that at least one of the lights will be red. a. Let R 1 denote that the first light is red and R 2 denote that the second light is red. Express the given probabilities in terms of R 1, R 2, and connector words such as or, and, and not. b. Express the complement of at least one of the lights will be red in words. Also calculate its probability. c. Produce a probability table to display the probabilities associated with the two lights being red or not. Fill in all of the probabilities in the table. d. Determine the probability that exactly one of the lights will be red. e. Determine the conditional probability that the second light will be red, given the first light is red. f. In terms of probabilities, do the two lights function independently? Justify your answer numerically. Exercise 30-8: Rolling Dice 30-8, 31-7, 31-8, 31-9, Suppose you roll two fair, six-sided dice. a. List the sample space of 36 equally likely outcomes. (Hint: For example, some outcomes are (1,1), (1,2), and (1,3). Note that (1,2) is a different outcome than (2,1).) b. The possible values for the sum of the two dice are 2, 3,, 11, 12. Are these values equally likely? Explain. Rossman/Chance, Workshop Statistics, 4/e 19

20 Let A denote the event that the first die lands on 5, B denote the event that the second die lands on 2, C denote the event that the sum equals 7, and D denote the event that the sum equals 11. c. Which two pairs of these events are mutually exclusive? Explain. d. Which two pairs of these events are independent? Justify your answer numerically. Rossman/Chance, Workshop Statistics, 4/e 20

21 Exercise 30-9: Women Senators 6-12, 18-4, 18-8, 30-9 The 2011 U.S. Senate consisted of 51 Democrats, 2 Independents, and 47 Republicans. Of the 17 women Senators, 12 were Democrats and 5 were Republicans. Suppose one of these 100 senators was chosen at random. a. Show how to use the addition rule to calculate the probability that the selected senator was a woman or a Democrat. b. Determine the conditional probability that the senator was a Democrat, given she was a woman. How does this compare to the (unconditional) probability that the senator was a Democrat? c. Determine the conditional probability that the senator was a Republican, given he was a man. How does this compare to the (unconditional) probability that the senator was a Republican? d. Is party independent of gender among these 100 senators? More formally, is the event selected senator is female independent of selected senator is a Democrat? Explain. Exercise 30-10: Spam Suppose 80% of the messages coming into your school s system are spam. Because of fairly effective spam filters, suppose only 10% of spam messages are actually delivered to you. Also suppose all non-spam messages are delivered to you. Let S denote the event that a message is spam, and let D denote the event that a message is delivered to you. a. Express the two given probabilities (.8 and.1) in terms of these symbols. (Hint: One of them is a conditional probability.) b. Use the multiplication rule to determine the probability that a message intended for you is spam and is delivered to you. c. Fill in a probability table like the following: Rossman/Chance, Workshop Statistics, 4/e 21

22 Message is Spam Message is Not Spam Total Message is Delivered to You Message is Not Delivered to You Total 1.0 d. Use this table and the definition of conditional probability to determine the probability that a message is spam given it is delivered to you. Exercise 30-11: California Demographics The 2000 U.S. Census revealed that 49.8% of California residents were male and 50.2% were female. It also found that 27.3% were younger than age 18, whereas 10.6% were age 65 or older. Moreover, females younger than age 18 comprised 13.3% of this population, and females aged 65 or older comprised 6.1%. a. Organize this information into a probability table like the following: Female Male Total Under Age 18 Age Age 65 or Older Total 1.0 b. Determine the probability that a randomly selected Californian is female or younger than age 18 or age 65 or older. c. Identify (in words) the complement of the event in part b. Also report its probability. d. Is the event female independent of the event age 65 or older? Justify your answer numerically. e. Are the events female and age mutually exclusive? Explain. Exercise 30-12: Pet Ownership 13-9, 15-14, 15-15, 18-2, 20-21, According to the Statistical Abstract of the United States: 2006, 36.1% of sampled American households have a pet dog and 31.6% have a pet cat. Does it follow that 77.7% of the sampled households have a pet dog or a pet cat? Justify your answer, and explain how the addition rule applies to this question. Rossman/Chance, Workshop Statistics, 4/e 22

23 Exercise 30-13: Weighty Feelings 6-13, 25-15, 26-13, As reported in Activity 6-13, the following table classifies respondents to the National Health and Nutrition Examination Survey (NHANES) according to gender and how they feel about their current weight: Female Male Total Underweight About Right Overweight Total Suppose one of these 5876 people is selected at random. a. Determine the conditional probability that the person feels underweight, given he is male. b. Determine the conditional probability that the person is male, given he or she feels underweight. c. Are the events person is male and person feels underweight mutually exclusive? If so, explain why. If not, explain what would have to be true about the table for them to be mutually exclusive. d. Are the events person is male and person feels underweight independent? Justify your answer numerically. Exercise 30-14: Red Dye Experiments The following table presents the results of a study to assess whether red dye #2 causes cancer in laboratory rats (Fienberg, 1980). The three variables are dosage (high or low), presence of cancerous tumor (yes or no), and whether the rat died before the end of the study or survived to the end of the 131- week study: Rat Died Before End of Study Rat Survived to End of Study Low Dosage High Dosage Low Dosage High Dosage Cancerous Tumor No Cancerous Tumor Total Rossman/Chance, Workshop Statistics, 4/e 23

24 a. Is this an observational study or a randomized experiment? b. Suppose one of these 88 rats was selected at random. Determine the conditional probability that the rat had a cancerous tumor, given it received a high dosage of the dye. c. Determine the conditional probability that the rat had a cancerous tumor, given it received a low dosage of the dye. d. Compare the values of these two conditional probabilities, and comment on what they reveal about the question of whether a higher dosage leads to a greater chance of developing cancer. e. Determine the conditional probability that the rat had a cancerous tumor, given it died before the study was completed. f. Determine the conditional probability that the rat had a cancerous tumor, given it survived until the end of the study. g. Consider the alternative explanation: If the study had continued longer, perhaps more of the rats would have developed cancer. In particular, maybe more of the low dosage group would contract cancer eventually. Explain how the calculations in parts e and f help to refute this argument. Exercise 30-15: Daily Lottery Suppose you play a daily number lottery game in which three digits from 0 9 are selected at random, so your probability of winning is 1/1000. Also suppose lottery results are independent from day to day. a. If you play every day for a 7-day week, what is the probability that you lose every day? b. If you play every day for a 7-day week, what is the probability that you win at least once? (Hint: Make use of your answer to part a.) c. Repeat parts a and b if you play every day for a 30-day month. d. Repeat parts a and b if you play every day for a 365-day year. Rossman/Chance, Workshop Statistics, 4/e 24

25 e. For how many days must you play in order for your probability of winning at least once to exceed.5? Explain your answer. f. For how many days must you play in order for your probability of winning at least once to exceed.9? Explain your answer. Exercise 30-16: Multiple Tests of Significance Many statistical tests are conducted with a.05 probability of making a Type I error (which means, as you may recall from Topic 23, that a null hypothesis is rejected when it really is true). If ten statistical tests are performed independently, determine the probability that a Type I error is made for at least one of the tests. (Hint: First find the probability of the complement of this event.) Exercise 30-17: Sports Series 30-17, Suppose the Domestic Shorthairs have a.7 probability of beating the Cache Cows in any one game, and assume results are independent from game to game. In a best-of-three series, the first team to win two games is the winner of the series. a. Determine the probability that the Domestic Shorthairs win the series by winning the first two games. b. Determine the probability that the Domestic Shorthairs win the series by losing one of the first two games and then winning the third game. c. Use your answers to parts a and b to determine the probability that the Domestic Shorthairs win the series, regardless of how many games it takes. Also identify and justify the probability rule you use to combine your answers to parts a and b. d. Determine the probability that the series ends in two games, regardless of who wins. Be sure to explain how you are arriving at your solution. Rossman/Chance, Workshop Statistics, 4/e 25

26 Exercise 30-18: Game Show Contestants Suppose you apply to be a contestant on the television game show Jeopardy. You believe you have a.4 probability of being invited to an audition. Given you are invited to an audition, you believe you have a.8 probability of being selected to appear on the show. a. Determine the probability that you are invited to audition and selected to appear on the show. b. Determine the probability that you are invited to audition and not selected to appear on the show. c. Determine the probability that you do not appear on the show. (Hint: There are two ways that this could happen.) Exercise 30-19: Family Births 11-3, 11-12, 30-19, 30-20, 31-19, Suppose a family has two children, each of whom is equally likely to be a boy or girl, independently from child to child. a. What is the probability that the second child is a boy, given the first child is a boy? Explain how you know. b. What is the probability that both children are boys? c. What is the probability that at least one child is a boy? Explain how you can use the addition rule to calculate this probability. d. What is the conditional probability that both children are boys, given at least one child is a boy? Rossman/Chance, Workshop Statistics, 4/e 26

27 Exercise 30-20: Family Births 11-3, 11-12, 30-19, 30-20, 31-19, Reconsider the previous activity. Continue to suppose the first child is equally likely to be a boy or a girl. But now suppose that conditional on the first child s sex, the second child has a.75 probability of being the same sex as the first. a. For each of the four possible outcomes {B 1 B 2, B 1 G 2, G 1 B 2, G 1 G 2 }, use the multiplication rule to determine its probability. b. Determine the probability that both children are the same sex. c. Determine the conditional probability that both children are boys, given at least one child is a boy. Exercise 30-21: Tennis Serves a. In the final match of the 2000 U.S. Open men s tennis championship, Pete Sampras won 76% of the points for which his first serve landed in and 43% of the points in which his first serve did not land in. His first serve landed in for 64% of the points that he served. Determine the percentage of points that Sampras won when he served (the overall probability of Sampras winning a point on his serve if picked at random). Also indicate which probability rule you use to do the calculation. b. In the final match of the 2000 U.S. Open women s tennis championship, Venus Williams won 58% of the points on which she served and 47% of the points on which her opponent Lindsay Davenport served. Williams served on 54% of the points they played. Use this information to determine the percentage of points Williams won in the match. Also indicate which probability rule you use to do the calculation. Rossman/Chance, Workshop Statistics, 4/e 27

28 Exercise 30-22: Watching Films 6-17, 6-18, 30-1, 30-3, Suppose Gustav and Francine keep track of how many films each has seen from the top 100 list described in Activity 30-1, with (partial) results as shown in this table: Gustav Has Seen Gustav Has Not Seen Total Francine Has Seen 80 Francine Has Not Seen 20 Total a. What is the smallest value that could go in the upper-left cell (number of films that both Gustav and Francine have seen)? Explain your answer. b. Fill in the table so the conditional probability that Francine has seen the film given Gustav has seen it is less than the (unconditional) probability that Francine has seen the film. In other words, complete the table so Pr(F G) < Pr(F). Rossman/Chance, Workshop Statistics, 4/e 28