Refer to the Journal of Engineering for Industry (Aug. 1993) study of an automated drilling machine, Exercise 3.67. The eight machining conditions used in the study are reproduced here.

Suppose that two of the machining conditions listed will detect a flaw in the automated system. Define X as the number of these two conditions with steel material, and define Y as the number of these two conditions with .25-inch drill size.

a. Find the bivariate probability distribution p(x, y).
b. Find the marginal probability distribution p₂(y).
c. Find the conditional probability distribution p₁(x Ι y).

Data from Exercise 3.67

In high-volume machining centers, cutting tools are replaced at regular, heuristically chosen intervals. These intervals are generally untimely, i.e., either the tool is replaced too early or too late. The Journal of Engineering for Industry (Aug. 1993) reported on an automated real-time diagnostic system designed to replace the cutting tool of a drilling machine at optimum times. To test the system, data were collected over a broad range of machining conditions.

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Workpiece Dril Size Drill Speed Feed Rate Experiment Material (in.) (rpm) (ipr) 1 Cast iron .25 1,250 .011 2 Cast iron .25 1,800 .005 3 Steel .25 3,750 .003 4 Steel .25 2,500 .003 5 Steel .25 2,500 .008 6. Steel .125 4,000 .0065 Steel .125 4,000 .009 Steel .125 3,000 .010