STAT 303

Question 1

define: Sample Space

Answer 1

The sample space of an experiment is the set of all possible outcomes of that experiment.

Question 2

define: Event

Answer 2

An event A of a sample space S is any subset of outcomes of S. An event A of a sample space S is denoted by A ⊆ S.

Question 3

define: Mutually Exclusive

Answer 3

A pair of events A and B in a sample space S are called mutually exclusive if A ∩ B = ∅.

Question 4

define: Discrete sample space

Answer 4

A sample space that is either finite or countably infinite

Question 5

define: Union of Events A and B

Answer 5

Let S be a sample space and let A, B ⊆ S.

The union of A and B, denoted A∪B, is the event that contains all outcomes that are contained in both A and B; that is,

A ∪ B = {x ∈ S : x ∈ A or x ∈ B} ⊆ S

Question 6

define: Intersection of Events A and B

Answer 6

Let S be a sample space and let A, B ⊆ S.

The intersection A and B, denoted A ∩ B, is the event that contains all outcome common to both A and B; that is,

A ∩ B = {x ∈ S : x ∈ A and x ∈ B} ⊆ S

Question 7

define: Complement

Answer 7

Let A ⊆ S be an event in a sample space S. The complement of A, denoted A’, is the event of S that consists of all outcomes in S that are not contained in A. That is,

A’ = {x ∈ S : x ∉ A}

Question 8

define: Probability

Answer 8

Let S be a sample space. A probability P on S is a set function that assigns each event A ⊆ S a real number value called the probability of A, denoted by P(A), that also satisfies the following three properties:
i) P (S) = 1
ii) P (A) ≥ 0 for all events A ⊆ S
iii) Countable Additivity

Question 9

define: Pairwise Mutually Exclusive Sqnc of Events

Answer 9

Let Ai ⊆ S be a sequence of events on sample space S, with i ∈ I some index set. Ai is called a pairwise mutually exclusive sequence if Ai ∩ Aj = ∅ whenever i ≠ j. So named because the events are mutually exclusive when taken in pairs.

Question 10

define: Countable additivity

Answer 10

Let P be a probability on a sample space S, where Ai is a pairwise mutually exclusive sequence of events. Then the probability of the union of Ai is equal to the sum of the probabilities of Ai

Question 11

define: Classical probability

Answer 11

Let S be a finite sample space. For any event A ⊆ S, let #A denote the number of outcomes contained in A. In particular, let N = #S. Define a set function P on the events of S by,

P(A) = #A / N

for all events A ⊆ S. The set function P defined as such is called classical probability on S.

Question 12

define: Boole’s Inequality

Answer 12

Let Ai be any sequence of events in a sample space S (with i ∈ I some index set). Then the probability of the union of Ai is ≤ the sum of the probabilities.

Question 13

What’s another name for Boole’s inequality?

Answer 13

Countable subadditivity.

Question 14

define: Bonferroni’s Inequality

Answer 14

Let Ai be a finite sequence of events in a sample space S. Then, the probability of the intersections of Ai is ≥ 1 - the sum of P(Ai’)

Question 15

List the properties of probability

Answer 15

Let P be a probability on a sample space S. Then,

1. P (∅) = 0
2. Finite Additivity
3. Complement law –> P(A’) = 1 - P(A) for all events A ⊆ S
4. P (A) ≤ 1 for all events A ⊆ S
5. P (A ∪ B) = P (A) + P (B) − P (A ∩ B) for any pair of events A, B ⊆ S
6. If A ⊆ B, then P (A) ≤ P (B)

Question 16

define: Conditional probability

Answer 16

Let S be a sample space equipped with a probability P. Let A, B ⊆ S be events with P(B) ≠ 0. The conditional probability of A given that B occurs is,

P(A|B) = P(A ∩ B) / P(B)

Question 17

List the properties of conditional probability

Answer 17

Let S be a sample space equipped with a probability function P. Let B ⊆ S be an event with P(B) ≠ 0.

1.