CS6015: Linear Algebra and Random Processes
Lecture 27: Sets, Experiments, Outcomes and Events
Learning Objectives
What are sets and some of their properties
What are experiments, sample spaces, outcomes and events?
Why do we care about Probability Theory?
Todo: augment the discussion on countable and uncountable infinities
The element of chance (Nothing in life is certain)
Randomness everywhere!
What is the chance that he would get infected if he went to the market?
No definite answer!
Why?
Due to the random nature of the world around us
Randomness everywhere!
What is the mode of transport ?
Is private car always safer than public transport?
How good is his immune system?
Does he have co-morbidities?
How many infections in the neighbourhood?
Randomness everywhere!
How many infections in the neighbourhood?
Randomness everywhere!
How many infections in the neighbourhood?
The study of this chance is the subject matter of Probability Theory!
Set theory
Experiments, sample spaces, events
Axioms of Probability
Random Variables
Distributions
Expectation
A brief overview of Set Theory
Set: A collection of elements
S = \{a,e,i,o,u\}
E = \{0,2,4,\dots,96,98,100\}
E = \{x: 0 \leq x \leq 100, x\%2 = 0\}
Compact notation: Useful for large sets
x \in S
mean x belongs to the set S
2 \in E,~~3 \notin E
Subsets and equal sets
\mathbb{I}: set~of~all~integers
S = \{x: x \in \mathbb{I}, x < 0 \}
Every element of in contained in
S
\mathbb{I}
S \subset \mathbb{I}
A = B~~iff~~A \subset B~and~B \subset A
subset
equal sets
Universal Set
A: set~of~all~aces
\Omega = set~of~all~52~cards
Every set of interest is a subset of the universal set
H: set~of~all~hearts
B: set~of~all~black~cards
F: set~of~all~face~cards
A \subset \Omega
H \subset \Omega
B \subset \Omega
F \subset \Omega
Empty Set
\phi = \{\}
Set with no elements (null set)
Set Operations
A^\mathsf{c} = \{x: x \in \Omega~and~x\notin A\}
Complement
Union (2 sets)
A \cup B = \{x: x \in A~or~x\in B\}
(black, gray and white in the image)
Intersection (2 sets)
A \cap B = \{x: x \in A~and~x\in B\}
(gray area in the image)
Set Operations (n sets)
Union (n sets)
x \in A_1 \cap A_2 \cap A_3 \cdots \cap A_n ~iff~x \in A_i \forall i
Intersection (n sets)
x \in A_1 \cup A_2 \cup A_3 \cdots \cup A_n ~iff~x \in A_i for~some~i
\Omega
Properties of Set operations
Commutativity
A \cup B = B \cup A
A \cap B = B \cap A
Associativity
A \cup (B \cup C) = (A \cup B) \cup C
A \cap (B \cap C) = (A \cap B) \cap C
\Omega
Properties of Set operations
Distributive Laws
A \cap (B \cup C) = (A \cap B) \cup (A \cap C)
Proof
A \cup (B \cap C) = (A \cup B) \cap (A \cup C)
x \in A\cap (B \cup C)
\implies x \in A~and~x \in (B \cup C)
\implies x \in A~and~B~or~x \in A~and~C
\Omega
Properties of Set operations
DeMorgan's Laws
(A \cup B)^\mathsf{c} = A^\mathsf{c} \cap B^\mathsf{c}
Proof
\implies x \in A^\mathsf{c} \cup B^\mathsf{c}
(A \cap B)^\mathsf{c} = A^\mathsf{c} \cup B^\mathsf{c}
x \in (A \cap B)^\mathsf{c}
\implies x \notin A \cap B
\implies x\notin A~or~x\notin B
\implies x \in A^\mathsf{c} ~or~x \in B^\mathsf{c}
\Omega
A
B
Countable v/s Uncountable Infinite Sets
: Set of all real numbers has infinite elements
: Set of all integers has infinite elements
\mathbb{I}
\mathbb{R}
(uncountable)
(countable)
An infinite set is said to be countable if there is a 1-1 correspondence between the elements of this set and the set of positive integers
Countable Infinite Sets
\mathbb{I} = \{-\infty, \dots, -3, -2, -1, 0, 1, 2, 3, \dots, \infty\}
= \{0,1,-1,2,-2,3,-3 \dots \}
: Set of all positive rational numbers
\mathbb{P}
\mathbb{P} = \{\frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \dots\}
1 2 3 4 5 6 7 ....
1 2 3 4 5 ....
Uncountable Infinite Sets
: Set of all real numbers
\mathbb{R}
\mathbb{Q} = [0,1]
Experiments and Sample Spaces
Experiment: Bowling a bowl
Outcome: {0, 1, 2, 3, 4, 5, 6} runs
Experiment: Going to the mall
Outcome: {infected, not_infected}
Experiment: Blood Test
Outcome: {positive, negative}
Experiment: Writing an exam
Outcome: {A, B, C, D, E, F}
The outcome in every trial is uncertain but the set of outcomes is certain
An experiment or trial is any procedurethat can be repeated infinite times and has a well-defined set of outcomes
The set of all possible outcomes of an experiment is called the sample space. The elements in a sample space are mutually exclusive and collectively exhaustive
Experiments involving coin tosses
\Omega = \{H, T\}
\Omega = \{HH,
HT,
TH,
TT\}
\Omega = \{HHH,
HHT,
HTH, HTT,
THH, THT,
TTH, TTT\}
|\Omega| = 2
|\Omega| = 4
|\Omega| = 8
\dots n~coins
|\Omega| = 2^n
Experiments involving fair dice
\Omega = \{1, 2, 3, 4, 5, 6\}
|\Omega| = 6
|\Omega| = 36
\dots n~dice
|\Omega| = 6^n
\Omega = \{ (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), \newline (2,1), (2,2), (2,3), (2,4), (2,5), (2,6),\newline(3,1), (3,2), (3,3), (3,4), (3,5), (3,6),\newline (4,1), (4,2), (4,3), (4,4), (4,5), (4,6),\newline (5,1), (5,2), (5,3), (5,4), (5,5), (5,6),\newline (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) \}
Experiments involving cards
|\Omega| = 52
|\Omega| = 52^3
|\Omega| = 52^2
|\Omega| = 52^4
More the number of outcomes, less the probability of any single outcome (assuming all outcomes are equally likely)
Experiments:continuous outcomes
\Omega = \{(x,y)~s.t.~0\leq x,y\leq 1\}
Events of an experiment
\Omega = \{HH, HT, TH, TT\}
An event is a set of outcomes of an experiment. This set is a subset of the sample space
A = \{HH, HT\}
(the event that the first toss results in a head)
B = \{TT\}
(the event that both the tosses result in tails)
(the event that there are exactly 2 aces)
|C| = {4 \choose 2} * {48 \choose 1} = 288
We say that event A has occurred if the outcome of the experiment lies in the set A
A = \{HH, HT\}
(the event that the first toss results in a head)
B = \{TT\}
(the event that both the tosses result in tails)
(the event that there are exactly 2 aces)
|C| = {4 \choose 2} * {48 \choose 1} = 288
\Omega = \{HH, HT, TH, TT\}
Union of events
A = \{(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)\}
B = \{(1,4), (2,4), (3,4), (4,4), (5,4), (6,4)\}
C = A\cup B
(the event that the first die shows a 2)
(the event that the second die shows a 4)
= \{(2,1), (2,2), (2,3), (2,4), (2,5), (2,6),\\~~~~~~~~ (1,4), (2,4), (3,4), (4,4), (5,4), (6,5)\}
(the event that the first die shows a 2 or the second die shows a 4)
Intersection of events
A = \{(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)\}
B = \{(1,4), (2,4), (3,4), (4,4), (5,4), (6,4)\}
D = A \cap B
(the event that the first die shows a 2)
(the event that the second die shows a 4)
(the event that the first die shows a 2 and the second die shows a 4)
= \{(2,4)\}
Complement of events
A = \{(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)\}
B = \{(1,4), (2,4), (3,4), (4,4), (5,4), (6,4)\}
E = A^\mathsf{c}
(the event that the first die shows a 2)
(the event that the second die shows a 4)
(the event that the first die does not show a 2)
Multiple events
A
: the hand contains the ace of spade
B
: the hand contains the ace of clubs
C
: the hand contains the ace of hearts
A \cup B \cup C
A \cap B \cap C
\Omega
A
B
C
Disjoint events
A
: the event that the first die shows a 1
B
: the event that the first die shows a 2
A~and~A^\mathsf{c}
Two events A and B are said to be disjoint if they cannot occur simultaneously, i.e.,
A\cap A^\mathsf{c}= \phi
A\cap B= \phi
A\cap B= \phi
A\cup A^\mathsf{c}= \Omega
A\cup B \neq \Omega
(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), \newline (2,1), (2,2), (2,3), (2,4), (2,5), (2,6),\newline(3,1), (3,2), (3,3), (3,4), (3,5), (3,6),\newline (4,1), (4,2), (4,3), (4,4), (4,5), (4,6),\newline (5,1), (5,2), (5,3), (5,4), (5,5), (5,6),\newline (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)
\Omega
A
B
Disjoint events
A = \{HH\}
B=\{TT\}
The events are said to be mutually disjoint or pairwise disjoint if
A\cap B= \phi
A_i\cap A_j= \phi~\forall~i,j~s.t.~i\neq j
B\cap C = \phi
A_1, A_2, \dots, A_n
C=\{HT,TH\}
A\cap C = \phi
Partition of the sample space
A = \{HH\}
B=\{TT\}
If the events are mutually disjoint and then are said to partition the sample space
A\cap B= \phi
B\cap C = \phi
A_1, A_2, \dots, A_n
C=\{HT,TH\}
A\cap C = \phi
A_1 \cup A_2 \cup \dots \cup A_n = \Omega
A_1, A_2, \dots, A_n
A\cup B\cup C = \Omega
A_1
A_5
A_4
A_3
A_2
A_6
A_7