4.43 From the probabilities shown in this Venn diagram, determine the probabilities of the following events.
(a) A does not occur.
(b) A occurs and B does not occur.
(c) Exactly one of the events A and B occurs.
4.62 Of 20 rats in a cage, 12 are males and 9 are infected with a virus that causes hemorrhagic fever. Of the 12 male rats, 7 are infected with the virus. One rat is randomly selected from the cage.
(a) If the selected rat is found to be infected, what is the probability that it is a female?
(b) If the selected rat is found to be a male, what is the probability that it is infected?
(c) Are the events "the selected rat is infected" and "the selected rat is male" independent? Why or why not?
4.73 In a county, men constitute 60% of the labor force. The rates of unemployment are 5.1% and 4.3% among males and females, respectively.
(a) In the context, of selecting a worker at random from the country labor force, state what probabilities the foregoing percentages represent. (Use symbols such as M for male, E for employed.)
(b) What is the overall rate of unemployment in the county?
(c) If a worker selected at random is found to be unemployed, what is the probability that the worker is a woman?
4.127 In an optical sensory experiment, a subject shows a fast response (F), a delayed response (D), or no response at all (N). The experiment will be performed on two subjects.
(a) Using a tree diagram, list the sample space.
(b) Suppose, for each subject, P(F) = .4, P(D) = .3, P(N) = .3, and the responses of different subjects are independent.
(i) Assign probabilities to the elementary outcomes.
(ii) Find the probability that at least one of the subjects shows a fast response.
(iii) Find the probability that both of the subjects respond.