Chapter 1

Question 1

Set

Answer 1

A collection of some items
Ordering does not matter
Notion: A = {1, 2}

Question 2

Subset

Answer 2

Set A is a subset of set B if every element of A is also an element of B. We write A⊂B, where “⊂” indicates “subset.”

Question 3

Superset

Answer 3

B is a superset of A, or B ⊃ A, if every element of A is also in B

Question 4

Universal set

Answer 4

the set of all things that we could possibly consider in the context
we are studying. (denoted by S)

Question 5

Venn diagram

Answer 5

any set is depicted by a closed region, its drawn like a circle where the elements of A are within the circle of A and if elements are in both A and B the circles overlap

Question 6

Union

Answer 6

a set containing all elements that are in A or in B
{1,2} ∪ {2,3} = {1,2,3}

Question 7

Intersection

Answer 7

denoted by A ∩ B, consists of all elements thatare both in A and B

{1,2} ∩ {2,3} = {2}

Question 8

Complement

Answer 8

The complement of a set A, denoted by Ac or
¯
A, is the set of all elements that are in the universal set S but are not in A

Question 9

difference (subtraction)

Answer 9

The set A − B consists of elements that are in A but not in B

if A = {1,2,3} and B = {3,5}, then A−B={1,2}

A −B = A ∩ Bc

Question 10

mutually exclusive or disjoint

Answer 10

Two sets A and B are mutually exclusive or disjoint if they do not have any shared elements; i.e., their intersection is the empty set,
A ∩ B = ∅.

More generally, several
sets are called disjoint if they are pairwise disjoint, i.e., no two of them share a common elements

Question 11

Partition

Answer 11

a collection of nonempty sets A1,A2,⋯ is a partition of a set A if they are disjoint and their union is
A

Question 12

De Morgan’s law

Answer 12

(keep in mind the c is the complement)
For any sets A1, A2, ⋯ ,An, we have:

(A1 ∪ A2 ∪ A3 ∪ ⋯ An)c =Ac1 ∩ Ac2 ∩ Ac3 ⋯ ∩ Acn

(A1 ∩A2 ∩A3 ∩⋯An)c =Ac1 ∪ Ac2 ∪ Ac3 ⋯ ∪ Acn

Question 13

Distributive law

Answer 13

For any sets A, B, and C we have

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Question 14

cartesian product

Answer 14

A Cartesian product of two sets A and B, written as A × B, is the set containing ordered pairs from A and B. That is, if C = A ×B, then each element of C is of the form (x,y) , where x ∈ A and y ∈ B:

A×B={(x,y)|x ∈ A and y ∈ B}.

For example, if A = {1,2,3} and B = {H,T}, then A×B={(1,H),(1,T),(2,H),(2,T),(3,H),(3,T)}.

Note that here the pairs are ordered, so for example, (1,H) ≠ (H,1). Thus A × B is not the same as
B ×A

Question 15

multiplication principle

Answer 15

If you have two finite sets A and B, where A has M elements and B has N elements, then A ×B has M ×N elements.

Question 16

cardinality

Answer 16

the size of a set

The cardinality of a set is denoted by |A|

In a finite set the cardinality is how many elements it contains

Question 17

inclusion-exclusion principle (also counts for probability)

Answer 17

1. |A ∪ B| = |A|+|B|−|A∩B|,
2. |A ∪ B∪C| =|A|+|B|+|C| −|A∩B|−|A∩C|−|B∩C|+|A∩B∩C|

Question 18

countable vs uncountable set (2 types of infinite set)

Answer 18

you can list the elements of a countable set A, i.e., you can write
A = {a1,a2,⋯}, but you cannot list the elements in an uncountable
set.

Question 19

definition countable (and uncountable)

Answer 19

Set A is called countable if one of the following is true:

a. if it is a finite set, ∣A∣<∞ ; or

b. it can be put in one-to-one correspondence with natural numbers N, in which case the set is said to be countably infinite.

A set is called uncountable if it is not countable.

Question 20

guidelines to remember is a set is countable or uncountable

Answer 20

• N,Z,Q, and any of their subsets are countable.
• Any set containing an interval on the real line such as [a,b],(a,b],[a,b), or (a,b), where a<b is uncountable.
• Any subset of a countable set is countable.
• Any superset of an uncountable set is uncountable.
• If A1,A2,… is a list of countable sets, then the set UiAi = A1∪A2∪A3… is also countable
• if A and B are countable, then A x B is also countable

Question 21

domain and co-domain in functions

Answer 21

A function f is a rule that takes an input from a specific set, called the domain, and produces an output from another set, called co-domain. Thus, a function maps elements from the domain set to elements in the co-domain with the property that each input is mapped to exactly one output.

If A is the domain and B is the co-domain for the function f, we use the following notation:

f : A→B.

Question 22

range

Answer 22

the set containing all the possible values of f(x). Thus, the range of a function is always a subset of its co-domain.

Question 23

random experiment

Answer 23

a process by which we observe
something uncertain (you do not know what the result is gonna be)

Question 24

outcome

Answer 24

a result of a random experiment

Question 25

sample space

Answer 25

The set of all possible outcomes (the universal set of a random experiment)

Question 26

trial

Answer 26

a particular performance of a random experiment When we repeat a random experiment several times, we call each one of them a trial.

Question 27

event

Answer 27

an eventis a collection of possible outcomes. In other words, an event is a subset of the sample space to which we assign a probability

Question 28

probability

Answer 28

We assign a probability measure P(A) to an event A. This is a value between 0 and 1 that shows how likely the event is.

Question 29

axioms of probability theory

Answer 29

1. For any event A, P(A) >= 0 2. Probability of the sample space S is P(S) = 1 3. If A1,A2,A3,... are disjoint events, then P(A1 ∪ A2 ∪ A3, ⋯) = P(A1)+P(A2)+P(A3)+⋯

Question 30

discrete vs continuous sample set

Answer 30

in discrete probability models we can compute the probability of events by adding all the corresponding outcomes, while in continuous probability models we need to use integration instead of summation

Question 31

conditional probability

Answer 31

it's the probability of an event occurring given that another event has already occurred. It is denoted as P(A|B), which reads as "the probability of event A given event B"

Question 32

conditional probability formula

Answer 32

P(A|B) = (P(A ∩ B) / (P(B)), when P(B) > 0 also written as: P(A∩B) = P(A)P(B|A) = P(B)P(A|B)

Question 33

chain rule for conditional probability

Answer 33

P(A1 ∩ A2 ∩ ... ∩ An) = P(A1)P(A2|A1)P(A3|A2,A1)⋯P(An|An−1An−2⋯A1)

Question 34

independence

Answer 34

Two events are independent if one does not convey any information about the other Two events A and B are independent if P(A ∩ B) = P(A)P(B)

Question 35

Three events A, B and C are independent if all of the following conditions hold:

Answer 35

P(A∩B)=P(A)P(B), P(A∩C)=P(A)P(C), P(B∩C)=P(B)P(C), P(A∩B∩C)=P(A)P(B)P(C).

Question 36

independence when it comes to other operations (complements, unions etc)

Answer 36

If A and B are independent then: - A and Bc are independent, - Ac and B are independent, - Ac and Bc are independent If A1,A2,⋯,An are independent then: P(A1∪A2∪⋯∪An) = 1−(1−P(A1))(1−P(A2))⋯(1−P(An)) Consider two events A and B, with P(A) ≠ 0 and P(B) ≠ 0. If A and B are disjoint, then they are not independent.

Question 37

the law of total probability

Answer 37

If B1,B2,B3,⋯ is a partition of the sample space S, then for any event A we have P(A) =∑i P(A∩Bi) =∑i P(A|Bi)P(Bi). Because: P(A) =P(A∩B)+P(A∩Bc) and using the definition of conditional probability, P(A ∩ B) = P(A|B)P(B) , we can write P(A) =P(A|B)P(B)+P(A|Bc)P(Bc).

Question 38

Bayes' Rule

Answer 38

- For any two events A and B, where P(A) ≠ 0, we have P(B|A) = P(A|B)P(B) / P(A) - If B1, B2, B3, ... form a partition of the sample S, and A is any event with P(A) ≠ 0, we have P(Bj|A) = (P(A|Bj)P(Bj)) / (∑iP(A|Bi)P(Bi))

Question 39

conditional independence

Answer 39

if A and B are conditionally independent given C, then P(A|B,C) = P(A|C)