Q25 from " Introduction to probability by Joseph k. blitzstein" page 182.

Nick and Penny are independently performing independent Bernoulli trials. For concreteness, assume that Nick is flipping a nickel with probability p1 of Heads and Penny is flipping a penny with probability p2 of Heads. Let X1,X2, . . . be Nicks results and Y1, Y2, . . . be Pennys results, with Xi ~ Bern(p1) and Yj ~ Bern(p2). (a) Find the distribution and expected value of the first time at which they are simultaneously successful, i.e., the smallest n such that Xn = Yn = 1. Hint: Define a new sequence of Bernoulli trials and use the story of the Geometric. (b) Find the expected time until at least one has a success (including the success). Hint: Define a new sequence of Bernoulli trials and use the story of the Geometric. (c) For p1 = p2, find the probability that their first successes are simultaneous, and usethis to find the probability that Nicks first success precedes Pennys.