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# Probability and Independence

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Age effects in medical care. The type of medical care a patient receives may vary with the age of the patient. A large study of women who had a breast lump investigated whether or not each woman received a mammogram and a biopsy when the lump was discovered. Here are some probabilities estimated by the study. The entries in the table are the probabilities that both of two events occur; for example, 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 No
Age under 65 0.321 0.124
Age 65 or over 0.365 0.190

(1) What is the probability that a patient in this study is under 65? That a patient is 65 or over?
(2) What is the probability that the tests were done for a patient? That they were not done?
(3) 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 testing were independent of age?