Suppose you shuffle a standard 52-card deck of cards, and stack the deck face down on a table. Let c1 be the top card, and c2 the card directly under c1. If c1 and c2 are both aces, you stop. If not, you put both cards back, reshuffle the entire deck (all 52 cards) and repeat on this newly shuffled deck of 52 cards. You record the total number of repetitions until you stopped. Thus (for example) if on your first attempt both c1 and c2 are aces, you record the number 1.
a) What is the probability that the number of repetitions you had to perform is a multiple of 3? Simplify all geometric series.
b) What is the probability that the number of repetitions you had to perform is a multiple of 3 which is not a multiple of 12? Simplify all geometric series.