Problem 3.(Combinations) Consider the following problem: "From an ordinary deck of 52 cards, seven cards are drawn at random and without replacement. What is the probability that at least one of the cards is a King?" A student in CS 237 solves this problem as follows:To make sure there is a least one King among the seven cards drawn, first choose a King; there are C(4,1) possibilities; then choose the other six cards from the 51 cards remaining in the deck, for which there are C(51,6) possibilities. Thus, the solution is C(4,1)*C(51,6) / C(52,7) = 0.5385.

However, upon testing the problem experimentally, the student finds that the correct answer is somewhat less, around 0.45.

(a) Calculate the correct answer using the techniques presented in class;

(b) Explain carefully why the student's solution is incorrect.

Problem 4.In this problem, we have a deck of 52 cards and we shuffle them randomly and deal the whole deck out to 4 people, so that each player has 13 cards. As usual, when dealing cards, it is without replacement and the "hand" that each player has is a set.

(a) What is the probability that each player will receive exactly three face cards?

(b) What is the probability that each player will receive all cards of one suit (i.e., one gets all clubs, another all hearts, another all diamonds, and another all spades)?