Exam 5: Probability

Understanding attitudes of humans towards wildlife is an important step in learning how to work with people on wildlife issues. Coyotes have expanded their range throughout the continental United States, even in the Washington, DC area. The data below are from a survey of George Mason University undergraduate students. "How much do you like coyotes?" Gender Dislike Very much Dislike Somewhat Neutral Like Somewhat Like Very much Total Male 8 12 186 49 25 280 Female 27 45 330 52 26 480 Total 35 57 516 101 51 760 Suppose a newspaper decides to select one of these students at random for an interview. -What is the probability that the selected student dislikes coyotes somewhat? 2. What is the probability that the selected student is male? 3. What is the probability that the selected student is male, given that he likes coyotes very much? 4. What is the probability that the selected student dislikes coyotes very much, given that she is a female?

Two events are said to be mutually exclusive if they can't occur at the same time.

The adult diamond python (Morelia spilota), an Australian snake, is about 3 feet long. In a multi-year study of the habitats of these creatures, 997 were captured. The following table displays the capture locations of these snakes by season of the year and habitat. The "other" category includes trees, logs, rocks, open ground, and under filtering cover such as shrubs. Diamond Python Habitat Buildings Other Total Spring 10 343 353 Summer 36 273 309 Autumn 17 157 174 Winter 0 161 161 Total 63 934 997 Suppose one of these diamond pythons is selected at random. Calculate the probability that: a) The selected diamond python was captured in a building, given that it was captured in the spring. b) The selected diamond python was captured somewhere other than in a building given that it was captured in the spring or summer. c) The selected diamond python was captured in a building in the summer.

Black bears (Ursus americanus) have a tendency to wander looking for food, and they have a high level of curiosity. These characteristics will sometimes get them into trouble when they travel through national parks. When they become "nuisances," the Park Service transplants them to other areas if possible. Data on the gender of transplanted bears and the outcome of the transplant for bears transplanted in Glacier National Park over a 10-year period are given in the table below. Male Female Totals Successful 32 17 49 Returned to capture area 34 45 79 Nuisance in another area 14 4 18 Killed outside of park 3 4 7 Totals 83 70 153 a) If a bear is randomly selected from the 153 bears in the sample, what is the probability it is male and became a nuisance in another area after relocation? b) If a bear is randomly selected from the 153 bears in the sample, what is the probability that it is female or was successfully transplanted? c) If a bear is randomly selected from the bears in the sample, what is the probability that it returned to the capture area, given that it is a female? d) After combining the above data with other National Parks, officials estimated that only about 22% of black bears in all parks become enough of a nuisance to be transplanted. They further estimate that 84% of nuisance bears are male, and fifty percent of non-nuisance bears are females. If a randomly selected bear were observed to be a male, what is the probability it would be enough of a nuisance to be transplanted?

Forty-five percent of donors have type O blood. The blood bank needs 4 donors with type O blood to restock their reserves. The director has the option of waiting to get 4 type O donors during the day, or ordering type O blood from a neighboring blood bank. Your task is to design and conduct a simulation to estimate the probability it takes 12 or more donors to get 4 with type O blood. a) To simulate the arrival of blood donors from this population, assign digits to the blood types that will result in a probability of success (Type O donor) of $0.45$ . Type O Digits:            Other than Type O Digits: b) Describe how you would use a random digit table to conduct one run of your simulation, where one run consists of observing the blood type of donors until 4 type $\mathrm { O }$ donors have visited the blood bank. c) Using the following lines from a random digit table, perform three runs of your simulation. Based on your results, what is your estimate of the probability of getting 4 Type O donors before 12 donors arrive at the blood bank? (You may mark above the digits to help explain your procedure.) 68417 35013 15529 72765 85089 57067 50211 47487 82739 57890 20807 81676 55300 94383 14893 60940 72024 17868 24943 61790 90656 87964 73311 12190 06628 71683 12285 39814 29103 81733 73035 57446 99209

Experimental studies in psychology use blinding to prevent researchers from biasing their measurements of subjects in the study. In a study of a psychotherapeutic intervention, a blinded clinician was asked to guess what treatment each subject received. Data from that experiment are shown below. Suppose a subject is to be chosen at random from the subjects in this study. Frequencies for Blinding Experiment Correct Guess Incorrect Guess Therapy 30 6 Placebo 26 11 a) What is the probability that the clinician made a correct guess for the selected subject? b) What is the probability the selected subject will have had therapy? c) What is the probability the selected subject will be one for whom the clinician correctly guessed had therapy? d) What is the probability the selected subject will be one for whom the clinician guessed correctly or who received the placebo treatment? e) What is the probability the selected subject will be one for whom the clinician guessed correctly and who received the placebo treatment? f) In a few sentences, explain why the probabilities calculated in parts (d) and (e) differ.

A small ferryboat transports vehicles from one island to another. Consider the chance experiment where the type of vehicle -- passenger (P) or recreational (R) vehicle -- is recorded for each of the next two vehicles that arrive at the dock. a) List all the outcomes in the sample space. b) Using the sample space in part (a), list the outcomes in each of the following events. $\mathrm { A } =$ the event that both vehicles are passenger cars $\mathrm { B } =$ the event that both vehicles are of the same type $\mathrm { C } =$ the event that there is at least one passenger car A: B: C:

In a few sentences, define the following terms: a) Simple event b) Sample space

In November 2002, Janet Napolitano, a Democrat, was elected Governor of Arizona, defeating Republican Matt Salmon and Independent Richard Mahoney. This was a somewhat surprising outcome, since there are more registered Republicans than Democrats in the state. The table below presents data from a survey of a sample of voters in the election. The candidate supported by the voter is represented by the rows, and the party affiliation of the voter is represented by the columns. Suppose that one of these voters is selected at random. Use the information in the table to a) What is the probability that the selected voter voted for Napolitano? b) What is the probability that the selected voter is a registered Democrat? c) What is the probability that the selected voter voted for Napolitano, given that the selected voter is a Democrat? d) A local reporter, commenting on this election, said, "Napolitano won because she attracted a larger share of crossover voters." (A crossover voter is one who votes differently than his or her registration category. For example, a Democrat party member voting Republican, or an Independent voting for a Democrat candidate would be crossover voters). What is the probability that the selected voter voted for Napolitano, given that he or she is a crossover voter?

In a survey of airline travelers, passengers traveling alone in the coach section were asked if they are bothered by a seatmate of the opposite gender using a shared armrest. The table below contains the data gathered in this study. Bothered Not bothered Females 19 26 Males 38 18 Suppose one of these passengers is to be randomly selected for a follow-up interview. Use the information in the table to answer the questions below. In showing your work, define and use appropriate notation. a) What is the probability that the selected passenger is female? b) What is the probability that the selected passenger is female or is bothered? c) What is the probability that the selected passenger is male and is not bothered?

Like many professionals, the clergy in mainline Protestant churches have pension plans. Due to the nature of the ministry, investment strategies may involve what are known as "screens." Screens are rules that prevent a pension fund administrator from investing in corporations that are involved with, for example, alcohol, gambling, tobacco and weapons of mass destruction. Ministers may elect to invest in two broad categories: "regular" and "social purpose" funds, which would typically use screens in their investment strategy. The use of screens may reduce their monthly benefit at retirement. The data below are from a survey of ministers about their support in principle for the use of such screens. Each minister asked if the screens should be applied to the regular funds, the social purpose funds, both, or neither. The ministers were also classified by the current percentage of their investments in the social purpose funds: 0%, 10 - 59%, 60% or greater. Use screens for: 0\% group 10-59\% group 60+\% group Total Social purpose funds only 70 94 58 Both social purpose and regular funds 83 235 266 Neither 47 11 3 Uncertain 23 15 6 Total a) What is the probability that a minister selected at random from those who participated in the survey was uncertain about the use of screens? b) What is the probability that a minister selected at random from those who participated in the survey was in the 60+% group and supported the use of screens for social purpose funds only? c) What is the probability that a minister selected at random from those who participated in the survey felt the screens should be used for social purpose funds only or for both social purpose and regular funds, given that they were in the 10- 59% group? d) What is the probability that a randomly selected "uncertain" minister would be in the 0% group?

Three of the most common pets are cats, dogs, and fish. Many families have more than one type of pet, and some have all three! Define the following events, with the probabilities given. (The fish-and-cats combination doesn't seem too popular!) = a randomly selected family has at least one pet fish: ()=0.20 = a randomly selected family has at least one pet dog: ()=0.32 = a randomly selected family has at least one pet cat: ()=0.35 Also $P ( F \cap D ) = 0.18 ; \quad P ( F \cap C ) = 0.07 ; \quad P ( D \mid C ) = 0.30$ Suppose that a family is selected at random. Calculate each of the following (show your work): a) $P ( F \mid D )$ b) $P ( F \cup D )$ c) $P ( C \cap D )$

Students in two classes of upper-level mathematics were classified according to class standing and gender, resulting in the following table. Distribution of students: Advanced math Males 45 30 Females 30 20 One of these students will be selected at random. Define events A, B, and C as follows: A = the event that the selected student is a female B = the event that the selected student is a male C = the event that the selected student is a senior. For each pair of events in the following table, indicate whether the two events are disjoint and/or independent by putting a Y or N in each of the cells. =,= Disjoint Independent , , ,

At Beth & Mary's Ice Cream Emporium customers always choose one topping to sprinkle on their ice cream. The toppings are classified as either candy (C) or fruit (F) toppings. Consider the chance experiment where the choice of toppings -- (C) or (F) -- is recorded for each of the next two customers who order ice cream. a) List all the outcomes in the sample space. b) Using your sample space in part (a), list the outcomes in each of the following events. A = the event that both customers pick a candy topping B = the event that both customers pick the same type of topping C = the event that at least one customer picks a candy topping A: B: C:

At Thomas Jefferson High School, students are heavily involved in extra-curricular activities. Suppose that a student is to be selected at random from the students at this school. Let the events A, M, and S be defined as follows, with the probabilities listed: $\mathrm { A } =$ a randomly selected student is active in the performing arts: $\mathrm { P } ( \mathrm { A } ) = 0.20$ $\mathrm { M } =$ a randomly selected student is active music: $\quad \mathrm { P } ( \mathrm { M } ) = 0.32$ $\mathrm { S } =$ a randomly selected student is active in sports: $\mathrm { P } ( \mathrm { S } ) = 0.35$ Also $P ( A \cap M ) = 0.18 ; \quad P ( A \cap S ) = 0.07 ; \quad P ( M \mid S ) = 0.30$ Calculate each of the following (show your work): a) $P ( A \mid M )$ b) $P ( A \cup M )$ c) $P ( M \cap S )$

At the State Fair the rifle range offers 4 different hats as prizes for perfect scores, one hat for each of the State University campuses. The hats are in boxes. When someone gets a perfect score, a box is chosen at random and given to that person. No substitutions are allowed. A local football fan (an ace shot who never misses) wishes to collect all 4 hat designs. There are a very large number of hats on hand and there are equal numbers of each hat design, so each design has a probability of 0.25 of being the prize at any given time. You are to design a simulation that could be used to estimate the average number of perfect scores needed to get a complete the set of hats. a) To simulate this strategy, assign digits to the hat designs that will result in the probability of selection of each design being 0.25. Hat 1 Digits: $~~~~~~~~~~~~$ Hat 2 Digits: Hat 3 Digits: $~~~~~~~~~~~~$ Hat 4 Digits: b) Describe how you would use a random digit table to conduct one run of your simulation. Hint: one run continues until a complete set of the 4 hats is acquired. c) Using the following lines from a random number table, demonstrate how your assignment of digits in part (a) would be used to carry out one run of the simulation. (You may mark above the digits to help explain your procedure.) 61790 55300 05756 72765 96409 12531 35013 82853 73676 57890 99400 37754 42648 82425 36290 45467 71709 77558 00095 82363 29485 82226 d) Suppose that the rifle range manager decides to order different numbers of hats, consistent with the popularity of each campus's football team. Hat 1 will be put in 50% of the boxes, Hats 2 and 3 will each be put in 20% of the boxes, and Hat 4 will be put in 10% of the boxes. Assign digits to the hats in a way that will be consistent with these probabilities. Hat 1 Digits:            Hat 2 Digits: Hat 3 Digits:            Hat 4 Digits: e) Perform three runs, and use your results to estimate the probability that it would take more than 10 boxes to complete the set of 4 hats. (You may mark above the digits to help explain your procedure.) 19223 95734 05756 28713 96409 12531 42544 82853 73676 47150 99400 37754 42648 82425 36290 45467 71709 77558 00095 32863 29485 82226 68417 35013 15529 72765 85089 57067 50211 47487 82739 57890 20807 81676 55300 94383 14893 60940 72024 17868 24943 61790 90656 87964

As discussed in the text, the classical approach to probability has a serious limitation that is overcome by the relative frequency approach. What is the limitation?

The event "A or B" consists of all of the outcomes that are in both of the events.

The collection of possible outcomes of a chance experiment is called the sample space of the experiment.

An event, by definition, consists of exactly one outcome.