A marketing company performed a risk analysis for a manufacturer of synthetic fibers and concluded that new com- petitors present no risk 13% of the time (due mostly to the diversity of fibers manufactured), moderate risk 72% of the time (some overlapping of products), and very high risk (com petitor manufactures the exact same products) 15% of the time. It is known that 12 international companies are planning to open new facilities for the manufacture of synthetic fibers within the next three years. Assume the companies are inde pendent. Let X, Y, and Z denote the number of new competi- tors that will pose no, moderate, and very high risk for the interested company, respectively.

(a) What is the range of the joint probability distribution of X, Y. and Z? Determine the following:

(b) P(X 1, 3,2=1)

(c) P(Z=2)

(d) P(Z-2y-1.X-10)

(e) P(Z1X-10)

(f) P(Y1,Z=1X=10) (g) E(ZX = 10)