Consider the following scenario: Let P (C) = 0.4. Let P (D) = 0.5. Let P (C|D) = 0.6. a. Find P (C AND D). b. Are C and D mutually exclusive? Why or why not? c. Are C and D independent events? Why or why not? d. Find P (C OR D). e. Find P (D|C).

a. By definition of conditional probability,

P (C | D) = P (C and D) / P (D) = => P (C and D) = 0.3

b. C and D are mutually exclusive if P (C and D) = 0, but this is clearly not the case, so no.

c. C and D are independent if P (C and D) = P (C) P (D). But P (C) P (D) = 0.2 ≠ 0.3, so no.

d. Using the inclusion/exclusion principle, we have

P (C or D) = P (C) + P (D) - P (C and D) = => P (C or D) = 0.6

e. Using the definition of conditional probability again, we have

P (D | C) = P (C and D) / P (C) = => P (D | C) = 0.75