Set theory problems with solutions

See the attachment for the full problem description.

In the plain text below U denotes Union and n denotes intersection.Problem 1
Consider the experiment of flipping a coin three times.
a. What is the sample space?
b. Which sample outcomes make up the event A: at least two of the coins show tails?

Problem 2
Use a Venn diagram to illustrate the second of the DeMorgan's Laws: (A U B)^C = A^C n B^C
(the complement of the union is the intersection of the individual complements).
Hint: proceed as in the example solved in class: draw a Venn diagram of the LHS and of the RHS, and show they
are the same.

Problem 3
A supervisor needs to select two workers for a job (consisting of two identical tasks) out of a group of five workers. Suppose that the workers vary in competence, 1 being the best, 2 - the second best, and so on, and 5 - the worst. The supervisor does not wish to show any biases in his selection, so he decides to select the two workers at random.
(a) What are the possible outcomes of this experiment, i.e. what is the sample space?

Note: In answering this question, please, use the following notation: (k,l) meaning "the
supervisor selected workers k and l". E.g. (1,3) meaning "the supervisor selected workers 1 and
3"; (4,5) meaning "the supervisor selected workers 4 and 5", etc.)

(b) How many elements are there in the sample space? (Just count them). Verify that this number
is equal to the number of ways to select 2 out 5 elements where order does not matter, i.e. verify
this is the same as combinations C5/2.

(c) Now define events A as: the employer selects the best worker and one of the two poorest
workers (i.e. he selects workers 1 and 4 OR workers 1 and 5).
Find the probability of A, P(A).
Hint: In part (c) you may find it easier to denote:
A1: selecting (1,4)
A2: selecting (1,5).
Then you need to find P(A) = P(A1 U A2). This is easy as A1 and A2 are mutually exclusive.

Problem 4
Let A and B be two events defined on a sample space S such that P(A)=0.3, P(B)=0.5,
and P(AUB)=0.7. Find:
a. P(AnB)
Hint: express P(AnB) from the expression for P(AUB).
b. P(A^CUB^C)
Hint: use De Morgan's law: P(A^CUB^C)= P(A n B)^C.
c. P(A^C n B)
Hint: express P(A^C n B) from the total probability definition for B.

Problem 5
Let A and B be any two events defined on S. Suppose that P(A) = 0.4, P(B) = 0.5, and P(AnB)=0.1. What is the probability that either A or B occur, but not both?
Hint: A Venn diagram would help: we are looking for the probability of the event defined by the shaded area:
(see the attachment)

Problem 6
Let A and B be any two events defined on S. Suppose that P(A) = 0.4, P(B) = 0.3, and P(AnB)=0.05. What is the probability that neither of the events A and B occurs?
Hint: Most problems asking to find the probability that neither of several events are solved easiest by expressing this probability as: P(neither of them occurs) = 1 - P(at least one occurs). Or, in terms of notation, we need to find:
P(ACB^C) = P[(AUB)^ C] (using one of DeMorgan's Laws)
= 1 - P(AUB)
It is easy to see that P(A^CnB^C) = 1 - P(AUB) from a Venn diagram:
(see the attachment)

So, essentially, you just need to find P(AUB) using Theorem 4 from the lecture notes.

Problem 7
Let A and B be any two events defined on S. Using the definition of independence and the axioms of probability, prove that:
a. if A and B are independent, then so are A and B^C.
b. if A and B are independent, then so are A^C and B^C
.
Hint: Use the result from part a.

Problem 8
Show that for any two events A and B with P(A)>0 and P(B)>0 the following holds: if A and B are independent, they cannot be mutually exclusive. (This is the opposite of what you proved in sections).
Hint: suppose that A and B are mutually exclusive, and show that this leads to a contradiction.

Problem 9
Chevalier de Mere's Problem
(based on "Probability Demystified", by Allan Bluman, Ch. 1 "Brief History of Probability")

During the mid-1600s, a professional gambler named Chevalier de Mere made a considerable amount of money on a gambling game. He would bet that in four rolls of a die, he could obtain at least one 6. Chevalier de Mere was winning the game more than half of the time, so people suspected him of cheating and refused to play. He then decided to invent a new game - he would bet that if he rolled a pair of dice 24 times, he would get at least one double 6. However, to his dismay, he started losing more often than he won, and lost money. Unable to figure out why he was losing, he asked the renowned French mathematician Blaise Pascal (1623-1662) to study the game. Because of this problem, Pascal became interested in studying probability and began a correspondence with his French fellow-mathematician Pierre de Fermat (1601-1665). Together the two of them were able to solve Chevalier de Mere's dilemma and formulated the foundations of probability theory.

a. Consider the first game Chevalier de Mere invented. Find the probability of obtaining at least one 6 in four rolls of a fair 6-sided die.
b. Now consider Chevalier de Mere's second game. Find the probability of obtaining at least one pair of 6s when rolling a pair of fair 6-sided dice 24 times.