SUPPLEMENTARY EXERCISES for ... - WH Freeman

Section 3 33example, 0.321 is the probability that a patient is under 65 years of age and the tests were done.The four probabilities in the table have sum 1 because the table lists all possible outcomes.Tests done?Yes NoAge under 65 0.321 0.124Age 65 or over 0.365 0.190(a) What is the probability that a patient in this study is under 65? That a patient is 65 orover?(b) What is the probability that the tests were done for a patient? That they were not done?(c) Are the events A = {the patient was 65 or older} and B = {the tests were done} independent?Were the tests omitted on older patients more or less frequently than would be the case if testingwere independent of age?S4.19 The most popular game of chance in Roman times was tossing four astragali. An astragalusis a small six-sided bone from the heel of an animal that comes to rest on one of foursides when tossed. (The other two sides are rounded.) The table gives the probabilities of theoutcomes for a single astragalus based on modern experiments. The names “broad convex”etc. describe the four sides of the heel bone. The best throw was the “Venus,” with all fouruppermost sides different. What is the probability of rolling a Venus? (From Florence N. David,Games, Gods and Gambling, Charles Griffin, London, 1962, p. 7.)Side broad convex broad concave narrow flat narrow hollowProbability 0.4 0.4 0.1 0.1Section 3S4.20 A study of social mobility in England looked at the social class reached by the sons oflower-class fathers. Social classes are numbered from 1 (low) to 5 (high). Take the randomvariable X to be the class of a randomly chosen son of a father in Class 1. The study foundthat the distribution of X isSon’s class 1 2 3 4 5Probability 0.48 0.38 0.08 0.05 0.01(a) What percent of the sons of lower-class fathers reach the highest class, Class 5?(b) Check that this distribution satisfies the two requirements for a discrete probability distribution.(c) What is P (X ≤ 3)? (Be careful: the event “X ≤ 3” includes the value 3.)(d) What is P (X < 3)?(e) Write the event “a son of a lower-class father reaches one of the two highest classes” in termsof values of X. What is the probability of this event?S4.21 A study of education followed a large group of fifth-grade children to see how many yearsof school they eventually completed. Let X be the highest year of school that a randomly chosenfifth grader completes. (Students who go on to college are included in the outcome X = 12.)The study found this probability distribution for X: