Homework 3

Looking closely, you will see that the Jack of Spades and the Jack of Hearts are revealing only one eye. These are the "one-eyed Jacks" and are used as wild cards in a popular home-version of poker, along with the four 2's (or "Deuces"). Also, the King of Diamonds is showing only one eye, so let us include that king among the wild cards! That will give us a total of7 wild cards: namely, the four 2's, the two one-eyed Jacks, and the single one-eyed King.

Let W denote the set of wild cards. Let J denote the set of Jacks, Q denote the set of Queens, K denote the set of kings, and H denote the set of hearts. Also, let F denote the face cards, which consist of the Jacks, Queens, and Kings.

1.) Assume a card will be selected at random from the full deck:

(a) Find the probability the card is a Queen or a Heart, denoted P(QU H).

(b) Find the probability the card is both a Queen and a Heart, denoted P(Q H).

(c) Find the probability that a card is a Heart given that it is a Queen. Show your work. Recall

P(A|B)= P(A B)/P(B).

(d) Is the event of drawing a Heart independent of drawing a Queen? Clearly state the criterion for the decision and how your answer findings answer that criterion.

(e) Find P(W), P(F), and P(W|F). (Note there are now 7 wild cards.)

(f) Is the event of drawing a Wild card independent of drawing a Face card? Show your work.

2. Consider a three-sided fair die, with sides numbered 1,2, and 3. If you roll it twice, your list of possible elementary outcomes is (1,1), (1,2), ... , (3,3).

(a) Complete the list of possible elementary outcomes. If you get lost in this, just have courage, and move ahead with what you have. There is no tragedy if you leave some out or you add something that doesn't belong.

(b) How many do you now have in total?

(c)Try tocompute the number of possible elementary outcomes, using the logic I employed to answer with 2^3 when that question was asked for three tosses of a coin.

(d) List the sample points that sum to four.

(e) Compute the probability of getting a sum of four with those dice. Explain it.