Nick and Penny are independently performing independent Bernoulli trials. For concreteness, assume that Nick is ?ipping a nickel with probability p1 of Heads and Penny is ?ipping a penny with...

Nick and Penny are independently performing independent Bernoulli trials. For concreteness, assume that Nick is ?ipping a nickel with probability p1 of Heads and Penny is ?ipping a penny with probability p2 of Heads. Let X1,X2,... be Nick’s results and Y1,Y2,... be Penny’s results, with Xi ~ Bern(p1) and Yj ~ Bern(p2). (a) Find the distribution and expected value of the ?rst time at which they are simultaneously successful, i.e., the smallest n such that Xn = Yn = 1. Hint: De?ne 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: De?ne a new sequence of Bernoulli trials and use the story of the Geometric.
(c) For p1 = p2, ?nd the probability that their ?rst successes are simultaneous, and use this to ?nd the probability that Nick’s ?rst success precedes Penny’s.