Reference no: EM13845118

Problem 1: We shall prove in class next time that An ↑ A implies limn→∞ P(An) = P(A). Use this fact to prove that An ↓ A implies limn→∞ P(An) = P(A). [Hint: De Morgan might be of help.]

Problem 2: For any sets A, B, C, prove that

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(C ∩ A) + P(A ∩ B ∩ C)

If the events A, B, C are independent, show that the above expression can be simplified to

P(A ∪ B ∪ C) = 1 - [1 - P(A)][1 - P(B)][1 - P(C)].

Problem 3: Suppose I have a biased coin with probability of H = 0.25. I make 5 independent flips of that coin. What is the probability that I see more heads than tails?

Problem 4: Suppose A and B are two independent events. Show that (i) A and Bc are independent, (ii) Ac and Bc are independent.