14EXTRA PROBLEMS85Estimate the relative risk of developing TB in relation to the use of intravenous drug use fromthis study and interpret the result.4. Suppose in a group of recent graduates, it was found that 28% had a law degree and 17% hada MBA. Five percent had both of these degrees.What is the probability that a randomlyselected graduate has a MBA if they do not have a law degree?5. A study was conducted to investigate a new procedure for detecting renal disease in patientswith hypertension.Using the new procedure, experimenters screened a random sample of200 hypertensive patients. Then the presence or absence of renal disease was determined byanother method. The data obtained are shown in the following table:Disease AbsentDisease PresentDisease Detected by the New Procedure1621Disease Not Detected by the New Procedure1576What are the false-positive rate and false-negative rates for the new procedure?6. We are interested in studying the survival rate of the marmot population on Mt. Washington.A marmot must survive last year to be encountered this year.Assume that survival andencounter probabilities are constant, the probability of surviving and being encountered is0.40, and we encounter a marmot with probability 0.58.What is the probability that amarmot survives the year?14.5Set 6 Mutually exclusive and independence1. Each member of a certain population of unhealthy patients has a risk of heart attack at levelslow, medium, or high; and a risk of stroke at levels low, medium or high. The proportions ofpatients within each of the 9 categories are given in the following table.Risk ofRisk of strokeheart attackLowMediumHighLow0.130.150.12Medium0.020.300.08High0.050.050.10(a) Suppose that one patient is randomly chosen from this population. If the chosen patienthas a medium or high risk of heart attack, what is the probability that the patient hasonly a low risk of a stroke?(b) Suppose that one patient is randomly chosen from this population. Which of the followingstatements are true?i. The events chosen patient has a low risk of heart attack and chosen patient has ahigh risk of stroke are independent.ii. The events chosen patient has a medium risk of heart attack and chosen patient hasa medium risk of a stroke are mutually exclusive.
14EXTRA PROBLEMS86iii. The events chosen patient has a low risk of stroke and chosen patient has a high riskof a stroke are mutually exclusive.(c) Suppose that two patients are randomly chosen from this population and that this pop-ulation is large. What is the probability that both patients have a high risk of stroke?2. The following table is a result of classifying 1000 randomly selected car accident vehicles bythe type of injury the driver sustained and the type of restraint used by the driver.