Patients given drug therapy either improve, remain the same, or degrade with probabilities 0.5, 0.4, 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. 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)