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?

Answer 1

a) The event that is equal to since the probabilities first card is and ace and the last card is an ace are both equal . All card have equal chances of being withdrawn next.

b) Let be the event that the first ace is at position   in the deck.

Now,

c) The probability of observing two aces in  the and the and th position is  

Using conditional probability,

Using total probability theorem,

d)The above sum is (note the sum of fisrt 49 natural numbers and sum of squares of first 49 natural numbers)

Our intuition in part (a) is right.