Lecture 1: Outcomes, Events, and Probability

STA237: Probability, Statistics, and Data Analysis I

Michael Jongho Moon

PhD Student, DoSS, University of Toronto

May 9, 2022

Introduction to probability

At which station will there be the next subway delay?

Definitely Finch.
Probably Spadina..
Maybe Queen’s Park…
None of them. TTC is perfect.

In plain language, definitely, probably, and maybe express a degree of uncertainty or a degree of belief.

What probability is

A discipline

Probability is the science of uncertainty.

(Evans and Rosenthal)

An expression

a number between 0 and 1 that expresses hows likely [an] event is to occur…

(Dekking et al.)

Another way of thinking about probability is in terms of relative frequency.

(Evans and Rosenthal)

Example

Past frequency of delays at each stop provides a hint where the next delay may occur
You woud assign a high probability to a station with a relatively high frequency

Why we study probability

Probability is everywhere and understanding probability can help you…

plan your next subway trip
forget about the Lotto Max ticket you threw away by accident
understand that launching the space shuttle Challenger was a bad idea without launching it (Section 1.4 of Dekking et al.)

Random experiment, outcomes and events

Definitions

A (random) experiment is a mechanism/phenomenon that results in random or unpredictable outcomes.

The station where you experience your next TTC subway delay is the outcome.

A sample space is the collection of all possible outcomes from an experiment. It’s often denoted \(\Omega\) (Omega).

\[\Omega=\{\text{St Goerge}, \text{Spadina}, \cdots\}\] \[=\text{All stations}\]

An event is a subset of the sample space.

Some basic set theory

Events

Consider the following events.

\(A\): Your next delay is along Bloor-Danforth line.

\(B\): Your next delay is in downtown Toronto.

Union

\[A\cup B\]

Represents the event that includes outcomes from event \(A\) or \(B\)
That is, a delay along Bloor-Danforth line or in downtown

Intersection

\[A\cap B\]

Represents the event that includes outcomes from event \(A\) and \(B\)
That is, a delay along Bloor-Danforth line (and) in downtown

Complement

\[A^c\]

Represents the event that excludes outcomes from A
A delay that is NOT along Bloor-Danforth line

Example: Niether \(A\) nor \(B\)

How would you write an event that includes outcomes that belong to neither \(A\) nor \(B\) using set notation?

We could write…

a delay that is

NOT \((\cdot^c)\)

along Bloor-Danforth line \((A)\)

OR \((\cup)\)

in downtown \((B)\).

a delay that is

NOT \((\cdot^c)\) along Bloor-Danforth line \((A)\)

AND \((\cap)\)

NOT \((\cdot^c)\) in downtown \((B)\).

\[(A\cup B)^c\]

\[=\]

\[A^c \cap B^c\]

De Morgan’s Laws

For any two events \(A\) and \(B\), we have

\[(A\cup B)^c = A^c \cap B^c\]

and

\[(A\cap B)^c = A^c\cup B^c.\]

Example: Exactly one of \(A\) or \(B\)

An event that includes outcomes that belong to exactly one of \(A\) or \(B\), but not both.

\[A\cup B \cap (A \cap B)^c\]

\[A\cup B \cap (A^c \cup B^c)\]

Other useful notations

Disjoint \(A\) and \(B\)

(mutually exclusive)

\[A\cap B=\{\}=\emptyset\]

\(A\) implies \(B\)

\(A\) is a subset of \(B\)

\[A\cap B=A\]

\[A\subset B\]

Probability

Probability function

A probability function \(P\) defined on a finite sample space \(\Omega\) assigns each event \(A\) in \(\Omega\) a number \(P(A)\) such that

\(0\le P(A) \le 1\);
\(P(\Omega) = 1\); and
\(P(A\cup B) = P(A) + P(B)\)
if \(A\) and \(B\) are disjoint.

The number \(P(A)\) is called the probability that \(A\) occurs.

A probability function \(P\) defined on an infinite sample space \(\Omega\) assigns each event \(A\) in \(\Omega\) a number \(P(A)\) such that

\(0\le P(A) \le 1\);
\(P(\Omega) = 1\); and
\(P(A_1\cup A_2 \cup A_3 \cup \cdots) = P(A_1) + P(A_2) + P(A_3) + \cdots\)
if \(A_1\), \(A_2\), \(A_3\), … are disjoint.

The number \(P(A)\) is called the probability that \(A\) occurs.

Probability and set operations

Probability of a union

\[A\]

\[(A\cap B^c)\cup (A\cap B)\]

For any two events \(A\) and \(B\), we can decompose each into two disjoint subsets.

\[\implies P(A)=P(A\cap B^c) + P(A\cap B)\]

Probability of a union

\[A\cup B\]

\[=(A\cap B^c)\cup (A\cap B)\cup (A^c\cap B)\]

\[\implies P(A\cup B)=P(A\cap B^c) + P(A\cap B) + P(A^c \cap B)\]

Probability of a complement

\[\Omega\]

\[A\cup A^c\]

For any event \(A\), we can decompose the sample space \(\Omega\) into two disjoint subsets.

\[\implies P(\Omega)=P(A) + P(A^c)\]

\[\implies P(A^c)=1-P(A)\]

Calculating probability by counting

Applies only when

all outcomes of the sample space are equally likely; and
\(\Omega\) is finite.

For any event \(A\) of such sample space \(\Omega\),

\[P(A)=\frac{\text{number of outcomes that belong to }A}{\text{total number of outcomes in }\Omega}\]

Example: Rolling a die

Suppose you roll a fair die once.

\[A=\text{You roll an even number.}\] \[B=\text{You roll a number less than 3.}\]

Compute the following probabilities.

\[P(A)\]

\[P(A\cap B)\]

\[P(A\cup B)\]

Multiple experiments

Example: Rolling a die twice

Suppose you roll the die twice .

Let \(\Omega_1\) be the sample space for the first roll and \(\Omega_2\) the sample space for the second.

We will denote the sample space of rolling the die twice with \(\Omega\).

What is \(\Omega\)?

Product of sample space

In general,

\[\Omega=\Omega_1 \times \Omega_2=\left\{\left(\omega_1, \omega_2\right):\omega_1\in \Omega_1, \omega_2\in\Omega_2\right\}\]

That is, the sample space generated by observing multiple experiments is a product of the individual sample spaces consisting of all combinations of outcomes of individual experiments.

Example: Rolling a die twice

Compute \(P\left(\left\{\left(1,6\right)\right\}\right)\).

Practice questions

Exercises from Dekking et al. Chapter 2: 2.1, 2.2, 2.6, 2.7, 2.9-2.19

Simulation in R worksheet

Follow this link to open the worksheet