Question

1. Consider a standard card deck with 52 cards. There are 13 ranks, 1 to 13 and each rank has 4 suits. We say that a set of 4 cards is winning if the respective ranks can be put in a consecutive order a₁,a₂,a₃,a₄ so that a_i+1 = a_i + 1. For example, the set of cards {3hearts, 4spades, 5diamonds, 6hearts} is winning because the cards have consecutive ranks 3,4,5,6. (The cards can have any suit.) Suppose that you pick uniformly at random a set of 4 cards out of all possible sets with 4 cards. What is the probability that the set of 4 cards you picked is winning? Explain your answer.

Hint: First calculate how many total sets of 4 cards there are, by counting the subsets using the choose function. Then calculate how many consecutive sequences of 4 cards there are, and with that also consider the 4 suits per card. Be careful that the cards are picked in sets and not in ordered tuples.

Answer 1