Patients given drug therapy either improve, remain the same, or degrade with probabilities 0.5, 0.4, and 0.1, respectively. Suppose that 20 patients (assumed to be independent) are given the therapy. Let X1, X2, and X3 denote the number of patients who improved, stayed the same, or became degraded, respectively. Determine the following:

a. Are X1, X2, X3 independent?

b. P(X1 = 10)

c. P(X1 = 10, X2 = 8, X3 = 2)

d. P(X1 = 5|X2 = 12)

e. E(X1)