A particular birth defect of the heart is rare; a ne,vborn

Chapter 2, Problem 2.13

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A particular birth defect of the heart is rare; a ne,vborn infant will have t he defect D 'vith probability P[D) = 10- 4 . In the general exa1n of a ne,vborn, a particular heart arrhythmia A occurs with probability 0. 99 in infants 'vi th the defect. However, the arrhythmia also appears ,;vith probability 0.1 in infants without the defect. \!\!hen the arrhythmia is present, a lab test for the defect is performed. The result of the lab test is either positive (event r+) or negative (event T - ). In a newborn 'vith the defect, the lab test is positive 'vith probability p = 0.999 independent from test to test. In a ne,vborn ,;vithout the defect, the lab test is negative 'vith probability p = 0.999. If the arrhythmia is present and the test is positive, then heart surgery (event H) is performed. (a) Given the arryth1nia A is present, 'vhat is the probability the infant has the defect D? (b) Given that an infant has the defect, what is the probability P [H IDJ that heart surgery is performed? ( c) Given that the infant does not have the defect, what is t he probability q = P [HIDc) t hat an unnecessary heart surgery is performed? (d) F ind the probability P[H) that an inf ant has heart surge1y performed for the arrythmia. ( e) Given that heart surgery is performed, what is the probability that the newborn does not have the defect?