Question: A standard deck of cards will be shuffled and then the cards will be turned over one at a time until the first ace is revealed. Let B be the event that the next card in the deck will also be an ace.
(a) Intuitively, how do you think P(B) compares in size with 1/13 (the overall proportion of aces in a deck of cards)? Explain your intuition. (Give an intuitive discussion rather than a mathematical calculation; the goal here is to describe your intuition explicitly.)
(b) Let Cj be the event that the first ace is at position j in the deck. Find P(B|Cj ) in terms of j, fully simplified.
(c) Using the law of total probability, find an expression for P(B) as a sum. (The sum can be left unsimplified, but it should be something that could easily be computed in software such as R that can calculate sums.)
(d) Find a fully simplified expression for P(B) using a symmetry argument. Hint: If you were deciding whether to bet on the next card after the first ace being an ace or to bet on the last card in the deck being an ace, would you have a preference?