a. Suppose E and F are events in a probability space (S, F, P) . Find the simplest possible condition(s) on E and F for which it is true that EnF and EUF are independent if and only if your condition (s) is/ are satisfied. State the condition(s) clearly, and prove that En F and EUF are independent if and only if your condition(s) is/are satisfied. Remark 1 Because (S, F, P) is arbitrary / unspecified, your answer must be valid for all probability spaces, not just for certain ones, or it does not answer the question. This comment applies to any problems in which you are working with an unspecified probability space. b. Suppose E and F are events in a probability space (, F, P) . Find the simplest possible condition(s) on E and F for which it is true that E\\ F and F \\ E are independent if and only if your condition (s) is/ are satisfied. State the condition(s) clearly, and prove that E \\ F and F \\ E are independent if and only if your condition (s) is/are satisfied. Remark 2 Because (S, F, P) is arbitrary / unspecified, your answer must be valid for all probability spaces, not just for certain ones, or it does not answer the question. This comment applies to any problems in which you are working with an unspecified probability space