# tablets AM probability PM probability

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1 Ismor Fischer, 8/21/2008 Stat 541 / Problems 3-1. In a certain population of males, the following longevity probabilities are determined. P(Live to age 60) = 0.90 P(Live to age 70, given live to age 60) = 0.80 P(Live to age 80, given live to age 70) = 0.75 From this information, calculate the following probabilities. P(Live to age 70) P(Live to age 80) P(Live to age 80, given live to age 60) 3-2. Patient noncompliance is one of many potential sources of bias in medical studies. Consider a study where patients are asked to take 2 tablets of a certain medication in the morning, and 2 tablets at bedtime. Suppose however, that patients do not always fully comply and take both tablets at both times; it can also occur that only 1 tablet, or even none, are taken at either of these times. (a) Explicitly construct the sample space S of all possible daily outcomes for a randomly selected patient. (b) Explicitly list the outcomes in the event that a patient takes at least one tablet at both times, and calculate its probability, assuming that the outcomes are equally likely. (c) Construct a probability table and corresponding probability histogram for the random variable X = the daily total number of tablets taken by a random patient. (d) Calculate the daily mean number of tablets taken. (e) Suppose that the outcomes are not equally likely, but vary as follows: # tablets AM probability PM probability Rework parts (b)-(d) using these probabilities. Assume independence between AM and PM A statistician s wife withdraws a certain amount of money X from an ATM every so often, using a method that is unknown to him: she randomly spins a circular wheel that is equally divided among four regions, each containing a specific dollar amount, as shown. $10 $20 Bank statements reveal that over the past n = 80 ATM transactions, $10 was withdrawn twelve times, $20 sixteen times, $30 twenty-eight times, and $40 twenty-four times. For this sample, construct a relative frequency table, and calculate the average amount x withdrawn per transaction, and the variance s 2. Suppose this process continues indefinitely. Construct a probability table, 2 and calculate the expected amount µ withdrawn per transaction, and the variance σ. $40 $30

2 Ismor Fischer, 8/21/2008 Stat 541 / A youngster finds a broken clock, on which the hour and minute hands can be randomly spun at the same time, independently of one another. Each hand can land in any one of the twelve equal areas below, resulting in elementary outcomes in the form of ordered pairs (hour hand, minute hand), e.g., (7, 11), as shown. Let the simple events A = hour hand lands on 7 and B = minute hand lands on 11. (a) Calculate each of the following probabilities. Show all work! P(A and B) P(A or B) (b) Let the discrete random variable X = the product of the two numbers spun. List all the elementary outcomes that belong to the event C = X = 36 and calculate its probability P(C). (c) After playing for a little while, some of the numbers fall off, creating new areas, as shown. For example, the configuration below corresponds to the ordered pair (9, 12). Now calculate P(C).

3 Ismor Fischer, 8/21/2008 Stat 541 / 3-36 C 3-5. For any event A, we know that P( A ) = 1 P( A). Now suppose B is any other event. Is it C true that P( A B) = 1 P( A B)? Prove in general, or find a counterexample Referring to the barking dogs problem in section 3.2, calculate each of the following. P(Angel barks OR Brutus barks) P(NEITHER Angel barks NOR Brutus barks), i.e., P(Angel does not bark AND Brutus does not bark) P(Angel barks AND Brutus does not bark) P(Angel does not bark AND Brutus barks) P(Exactly one dog barks) P(Brutus barks Angel barks) P(Brutus does not bark Angel barks) P(Angel barks Brutus does not bark) Also construct a Venn diagram, and a 2 2 probability table, including marginal sums Referring to the urn model in section 3.2, are the events A = First ball is red and B = Second ball is red independent in this sampling without replacement scenario? Does this agree with your intuition? Rework this problem in the sampling with replacement scenario After much teaching experience, Professor F has come up with a conjecture about office hours: There is a 75% probability that a random student arrives to a scheduled office hour within the first fifteen minutes (event A), from among those students who come at all (event B). Furthermore, there is an 80% probability that no students will come to the office hour, given that no students arrive within the first fifteen minutes. Assuming this conjecture is true, answer the following. (Note: Some algebra may be involved.) (a) Calculate P(B), the probability that any students come to the office hour. (b) Calculate P(A), the probability that any students arrive within the first fifteen minutes of the office hour. (c) Sketch a Venn diagram, and label all probabilities in it Suppose A and B are any two events, with P( A) = a, P( B) = b, and P( A B) = c. (a) Sketch a Venn diagram, and label all four probabilities in it. Also construct the corresponding 2 2 probability table, including the row and column marginal sums. (b) Imagine now that A and B are independent events. Repeat part (a). In particular, what relationship exists between each value in the probability table, and its corresponding row and column marginal probabilities? (c) Verify the conclusion in (b) for the example of independent events Lung cancer and Coffee drinker in section 3.2 of these notes.

4 Ismor Fischer, 8/21/2008 Stat 541 / A certain medical syndrome is usually associated with two overlapping sets of symptoms, A and B. Suppose it is known that: If B occurs, then A occurs with probability If A occurs, then B occurs with probability If A does not occur, then B does not occur with probability Find the probability that A does not occur if B does not occur. (Hint: Use a Venn diagram; some algebra may also be involved.) The progression of a certain disease is typically characterized by the onset of up to three distinct symptoms, with the following properties: Each symptom occurs with 60% probability, regardless of the others. If a single symptom occurs, there is a 45% probability that the two other symptoms will also occur. If any two symptoms occur, there is a 75% probability that the remaining symptom will also occur. Answer each of the following. (Hint: Use a Venn diagram.) (a) (b) (c) (d) (e) (f) What is the probability that all three symptoms will occur? What is the probability that at least two symptoms occur? What is the probability that exactly two symptoms occur? What is the probability that exactly one symptom occurs? What is the probability that none of the symptoms occurs? Is the event that a symptom occurs statistically independent of the event that any other symptom occurs?

5 Ismor Fischer, 8/21/2008 Stat 541 / An amateur game player throws darts at the dartboard shown below, with each target area worth the number of points indicated. However, because of the player s inexperience, all of the darts hit random points that are uniformly distributed on the dartboard (a) Let X = points obtained per throw. What is the sample space S of this experiment? 2 (b) Calculate the probability of each outcome in S. (Hint: The area of a circle is π r.) (c) What is the expected value of X, as darts are repeatedly thrown at the dartboard at random? (d) What is the standard deviation of X? (e) Suppose that, if the total number of points in three independent random throws is exactly 100, the player wins a prize. With what probability does this occur? (Hint: For the random variable T = total points in three throws, calculate the probability of each ordered triple outcome ( X1, X2, X 3) in the event T = 100. )

6 Ismor Fischer, 8/21/2008 Stat 541 / The Monty Hall Problem (simplest version) Between 1963 and 1976, a popular game show called Let s Make A Deal aired on network television, starring charismatic host Monty Hall, who would engage in deals small games of chance with randomly chosen studio audience members (usually dressed in outrageous costumes) for cash and prizes. One of these games consisted of first having a contestant pick one of three closed doors, behind one of which was a big prize (such as a car), and behind the other two were zonk prizes (often a goat, or some other farm animal). Once a selection was made, Hall who knew what was behind each door would open one of the other doors that contained a zonk. At this point, Hall would then offer the contestant a chance to switch their choice to the other closed door, or stay with their original choice, before finally revealing the contestant s chosen prize. Question: In order to avoid getting zonked, should the optimal strategy for the contestant be to switch, stay, or does it not make a difference?

7 Ismor Fischer, 8/21/2008 Stat 541 / The following data are taken from a study investigating the use of a technique called radionuclide ventriculography as a diagnostic test for detecting coronary artery disease. [Source: Begg. C. B., and McNeil, B. J., Assessment of Radiologic Tests: Control of Bias and Other Design Considerations, Radiology, Volume 167, May 1988, ] Test Result Coronary Artery Disease Present Absent Total Positive Negative Total (a) Calculate the sensitivity and specificity of radionuclide ventriculography in this study. (b) For a population in which the prevalence of coronary artery disease is 0.10, calculate the predictive power of a positive test and the predictive power of a negative test, using radionuclide ventriculography Recall that in a prospective cohort study, exposure (E+ or E ) is given, so that the odds ratio is defined as odds of disease, given exposure P(D+ E+) P(D E+) OR = = odds of disease, given no exposure P(D+ E ) P(D E ). Recall that in a retrospective case-control study, disease status (D+ or D ) is given; in this case, the corresponding odds ratio is defined as OR = odds of exposure, given disease P(E+ D+) P(E D+) = odds of exposure, given no disease P(E+ D ) P(E D ). (a) Show algebraically that these two definitions are mathematically equivalent, so that the same cross product ratio calculation can be used in either a cohort or case-control study, as the following two problems demonstrate. (Recall the definition of conditional probability.) (b) Why is relative risk generally not appropriate to compute in a case-control study, unless the prevalence of disease is rare in the population? In a case-control study, investigators first identified women with breast cancer (cases) and those without (controls), in an effort to establish if there was any association with the previous use of oral contraceptives. [Source: Hennekens, C. H., Speizer, F. E., Lipnick, R. J., Rosner, B., Bain, C., Belanger, C., Stampfer, M. J., Willett, W., and Peto, R., A Case-Control Study of Oral Contraceptive Use and Breast Cancer, Journal of the National Cancer Institute, Volume 72, January 1984, ] Calculate the odds ratio for the resulting table of data, and interpret. Breast Cancer Yes No Oral Yes Contraceptive No

8 Ismor Fischer, 8/21/2008 Stat 541 / In a cohort study, the associations between risk factors for breast cancer were examined among women participating in the National Health and Nutrition Examination Survey. [Source: Carter, C. L., Jones, D. Y., Schatzkin, A., and Brinton, L. A., A Prospective Study of Reproductive, Familial, and Socioeconomic Risk Factors for Breast Cancer Using NHANES I Data, Public Health Reports, Volume 104, January-February 1989, ] At the onset of the study, each participant in a sample of 6165 women was measured for exposure (in this case, if she first gave birth at age 25 or older). They were then followed forward in time to look for subsequent occurrences of disease, versus those who remained disease-free for the duration of the study. Calculate the odds ratio and relative risk for the resulting table of data, and interpret. Breast Cancer Yes No Age at First 25 or over Birth under An observational study investigates the connection between aspirin use and three vascular conditions gastrointestinal bleeding, primary stroke, and cardiovascular disease using a group of patients exhibiting these disjoint conditions with the following prior probabilities: P(GI bleeding) = 0.2, P(Stroke) = 0.3, and P(CVD) = 0.5, as well as with the following conditional probabilities: P(Aspirin GI bleeding) = 0.09, P(Aspirin Stroke) = 0.04, and P(Aspirin CVD) = (a) Calculate the following posterior probabilities: P(GI bleeding Aspirin), P(Stroke Aspirin), and P(CVD Aspirin). (b) Interpret: Compare the prior probability of each category with its corresponding posterior probability. What conclusions can you draw? Be as specific as possible On the basis of a retrospective study, it is determined (from hospital records, tumor registries, and death certificates) that the overall five-year survival (event S) of a particular form of cancer in a population has a prior probability of P(S) = 0.4. Furthermore, the conditional probability of having received a certain treatment (event T) among the survivors is given by P(T S) = 0.8, while the conditional probability of treatment among the non-survivors is only P(T S C ) = 0.3. Treatment (T): P(T S) = 0.8 P(T S C ) = 0.3 PAST 5 years PRESENT Given: Survivors (S) vs. Non-survivors (S C ) P(S) = 0.4 (a) A cancer patient is uncertain about whether or not to undergo this treatment, and consults with her oncologist, who is familiar with this study. Compare the prior probability of overall survival given above with each of the following posterior probabilities, and interpret in context. Survival among treated individuals, P(S T) Survival among untreated individuals, P(S T C ), (b) Also calculate the following. Odds of survival, given treatment Odds of survival, given no treatment Odds ratio of survival for this disease

9 Ismor Fischer, 8/21/2008 Stat 541 / Compare this problem with 2-8! Consider the binary population variable Y 1, with probability π = 0, with probability 1 π (see figure). (a) Construct a probability table for this random variable. (b) Show that the population mean μy = π. POPULATION = 1 = 0 2 Y (c) Show that the population variance σ = π (1 π). Note that π controls both the mean and the variance!