EE 302 Division 1.
Homework 3, due Wednesday, 1/30/2002 at 5pm.
Reading assignment: Last subsection of 1.5; sections 1.6-2.2 (pages 39-49 of Chapter 1 and pages
1-9 of Chapter 2); accompanying practice problems with solutions to be distributed in class.
Problem 1. In blackjack, the objective is to get as close as possible to 21 without going over 21. The
cards are valued as follows:
•An Ace can count as either 1 or 11, whichever makes the best hand.
•The cards from 2 through 10 are valued as indicated.
•Jacks, Queens, and Kings are all valued at 10.
Suppose we use a 52-card deck.
(a) We draw two cards at random from a full deck. What is the probability to get a blackjack (i.e.,
a total of 21)?
(b) Suppose we started with a full deck and drew the 9 of clubs and the 9 of diamonds. What is the
conditional probability that, if we draw another card, we bust (i.e., go over 21)? What is the
conditional probability that this third card we draw will be a 3?
(c) Suppose we started with a full deck and drew a 7 of clubs and an 8 of spades. What is the
conditional probability that the next card leads to a bust?
(d) Suppose we started with a full deck and drewa5ofhearts and a 10 of clubs. What is the
conditional probability that the next card leads to a bust?
Problem 2. An important milestone in the development of the theory of probability was the exchange
of letters between Blaise Pascal (1623-1662) and Pierre Fermat (1601-1665), two French mathemati-
cians. One motivation for these letters were problems proposed to Pascal by his friend Chevalier de
M´er´e, in particular the one described in Problem 35 of Chapter 1. Another of de M´er´e’s problems is
Problem 51, and here is yet another one.
De M´er´e bet that at least one 6 would appear during a total of four rolls of a fair six-sided die. From
past experience, he knew that he was more successful than not with this game of chance. Tired of his
approach, he decided to change the game. He bet that a double 6 would appear at least once during
twenty-four rolls of two dice. Soon he realized that his old approach to the game resulted in more
money. Please help him figure out why his new approach was not as profitable. (Hint. First, count
the total number of outcomes of the first game. How many of them result in no 6’s?)
Problem 3. A drawer contains red socks and black socks. When two socks are drawn at random, the
probability that both are red is 1/2. What is the smallest possible number of socks in the drawer?
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