Answer Happy

2. (10 points) Consider an infinite sequence of trials. The probability of success at the ith trial is some positive number p₁, 0 < pi < 1. Let A be the event that there is no success, and let B be the event that there is an infinite number of successes. 1 (5 points) Assume that the trials are independent and that Pi = ∞o. Let An be the event that there were no successes in the first n trials. Show that P(A) ≤ P(A₂) and conclude that P(A) = 0 by taking logarithms and the limit as n tends to infinity. 2 (5 points) Assume that the trials are independent and that Σ1 Pi<∞. Let B₁ be the event that there is at least one success after time n and B, be the event that the first success after time n occurs at time i. Show that P(B) ≤ P(B₂) = Ein+1 P(Bn.) and conclude that P(B) = 0 by taking the limit as n tends to infinity.