1. Allen, Baker, Cabot, and Dean are to speak at a dinner. They will draw lots to determine the order in which they will speak. Please answer the following questions:

a.) List all the elements of a sample space Ω associated to the experiment of recording the order in which these four individuals speak at the dinner.

b.) Mark with a check the simple events in part a.) contained in the event A = {Allen speaks before Cabot} ⊂ Ω

c.) Mark with a cross the elements of the event, A = {Cabot′s speech is between those of Allen and Baker} 

d.) Mark with a star the elements of the event A = {The four persons speak in alphabetical order}

2. Consider the experiment ”A coin is tossed five times.”

a.) Determine the sample space Ω associated to this experiment. Furthermore, count the number of simple events in the same. In set theoretic notation, this means to compute |Ω| i.e. the cardinality of the sample space as a set.

Please use the fact that the discrete probability space (Ω,B,P) determined by the experiment in a.) is an equally likely sample space to compute the following probabilities of the events named in the following questions:

b.) Heads never occurs twice in a row.

c.) Neither heads nor tails ever occurs twice in a row. d.) Both heads and tails occur at least twice in a row.

3. Consider the experiment ”Two dice are thrown.” the sample space Ω in our class notes. Let

A = {The total is two}

B = {The total is seven}

C = {The number shown on the first die is odd}

D = {The number shown on the second die is odd} E = {The total is odd}

be events in this experiment. Given that (Ω,B,P) is an equally likely sample space, compute the following probabilities:

a.) P(A)

b.) P(B)

c.) P(C)

d.) P(D)

e.) P(E)

f.) P(A∪B)

g.) P(A∩B)

h.) P(A∪C)

i.) P(C∩D∩E) j.) P(B∪Dc)

4. Prove, by the aid of Venn diagrams, that for any probability space (Ω, B, P ) and events A, B, C ∈ B that P (A ∩ B) + P ((A ? B) ∪ (B ? A)) + P (A ∩ B) = 1

Recall the following definition in set theory. Definition: Let A and B be sets, then

A ? B = {x ∈ A|x ∈ A, x ∈/ B}

rmk: ”by aid of Venn diagrams” means drawing a set of Venn diagrams that exhibits the underlying set theoretic relation in conjunction with a remark to that effect that, ”because P is a probability distribution function it does this by the axioms of a probability space” is sufficient. TLDR: you can just use pictures.

5. Recall the defintion of the power set of an arbitrary set, A: We say the set of all subsets of A, denoted 2A, is the power set of A.

Let A = {a, b, c} be a set of three elements. Consider the set function

f : 2A → Z

defined by f(B) = |B| that is, the image of a subset of A under f is the number of its elements.

Draw a diagram that illustrates this function that includes both its domain and its image. Use this diagram in conjunction with the definition to illustrate why f is not an injection i.e. is not 1-1.

6. Recall, a rule assigning elements of a set A to a set B is said to be well-defined if the assignment of elements in A to those of B is unique. Furthermore, such a rule is said to be a function when it is well-defined. Consider the rule

f:Q→Z

where f ?m? = m − n for any q = m ∈ Q. Explain why this rule is not a function. nn

7. Suppose A,B,C partition the set Ω and that (Ω,B,P) is a probability space. Suppose further P(A) = 0.3 and P(B) = 0.5. Compute both P(A ∪ B) and P(C).

8. Suppose A,B ⊂ Ω are events in the probability space (Ω,B,P) and P(A) = 0.8,P(B) = 0.7, and P(A∩B) = 0.6. What is P(A∪B)?