Probability
Last modified — 23 May 2026
By the end of this lecture, students are anticipated to be able to
A probability is the formal treatment of randomness
The key to this are models
Defining a probability requires the following:
Experiment: an action undertaken to make a discovery, test a hypothesis, or confirm a known fact.
Example: Release your pen from 4.9 meters above the ground
Predicted outcomes:
The pen will fall to the ground.
It will take about 1 sec to reach the ground.
Actual observations:
The pen hit the ground
Unsure if it took exactly one second….
The outcome of some experiments cannot be determined beforehand. They are uncertain.
If I roll a die, which number will show?
How many times will the R4 fly pass me with the “SORRY, BUS FULL” sign on?
If I walk to school without my rain jacket, will it rain?
If I randomly choose one sweater and one pair of pants from my closet, will they match?
Even though die rolls are random, patterns emerge when we repeat the experiment many times. We can study these using probability theory.
Probability Theory: The study of uncertainty, random phenomena, and patterns via mathematical models
Based on a set of axioms (statements or propositions accepted to hold true) and theorems (propositions which are established to hold true using sound logical reasoning)
This is what we will study in STAT 302!
In order to think about probabilities, we need to consider the possible outcomes of an experiment.
Sample space: the set of all possible outcomes of a random experiment.
Denoted by \(\Omega\) (or \(S\) in the textbook)
We also can consider a generic outcome, also called sample point, by \(\omega\) (i.e. \(\omega \in \Omega\)). Note that the textbook uses \(s \in S\).
Event: a subset of the sample space \(\omega\).
Notation: We commonly use upper case letters (\(A\), \(B\), \(C\), …) for events.
Events are sets!
\(\omega \in A\) means “\(\omega\) is an element of \(A\)”.
\(C \subset D\) means “\(C\) is a subset of \(D\)”.
Roll a dice:
Bus wait time: \(H =\) “wait is less than half an hour” $= $
Max-wind-speed: \(G =\) “wind is over 80 km/hour” \(=\)
Consider flipping two coins in a row:
A system has 3 components, which can either work or fail.
The experiment consists of observing the status (W/F) of the 3 components.
Describe the sample space for this experiment without listing all of the possible outcomes.
What is \(|\Omega|\)?
Let \(A= \{ \text{component 3 fails} \}\). What is \(|A|\)?
Even though random outcomes cannot be predicted, in some cases we have an idea about the chance that an outcome occurs.
If you toss a fair coin, the chance of observing a head is the same as that of observing a tail.
If you buy a lottery ticket, the chance of winning is very small.
A probability function \(\mathbb{P}\) quantifies these chances.
Probability functions are computed on events \(A \in \mathcal{B}\). We calculate \(\mathbb{P}(A)\). Mathematically / formally, we have: \[ \mathbb{P}\, : \, \mathcal{B} \to [0, 1] \] where \(\mathcal{B}\) is a collection of possible events.
Let \(\Omega\) be a sample space and \({\cal B}\) be a collection of events (i.e. subsets of \(\Omega\)).
Probability measure (or probability function): Any function \(\mathbb{P}\) with domain \({\cal B}\) that satisfies:
Axiom 1: \(\mathbb{P}( \Omega ) = 1\);
Axiom 2: \(\mathbb{P}( A ) \geq 0\) for any \(A \in {\cal B}\);
Axiom 3: Additivity If \(\{ A_{n}\}_{n \ge 1}\) is a sequence of disjoint events, then \[\mathbb{P}\left( \bigcup_{n=1}^{\infty }A_{n}\right) \, = \sum_{n=1}^{\infty }\mathbb{P}( A_{n})\]
Note: \(\{ A_{n}\}_{n \ge 1}\) is a sequence of disjoint events when \(A_i \cap A_j = \varnothing\) if \(i \ne j\)
We also assume \(\mathbb{P}(\varnothing) = 0\).
In a nutshell, additivity says that so long as events \(A, B, C\) are disjoint, the probability of the union \(\mathbb{P}(A \cup B \cup B) = \mathbb{P}(A) + \mathbb{P}(B) + \mathbb{P}(C)\).
Suppose you encounter a bowl of candies. There are 13 red candies, 20 blue candies, and 17 orange candies. What is the probability that you randomly select a red or blue candy from the bowl?
Let \(A\) and \(B\) denote arbitrary events, where \(\Omega\) is the sample space.
Prove the probability of the complement: \(\mathbb{P}( A^{c} ) =1-\mathbb{P}( A )\).
To do this, show that if \(\mathbb{P}\) satisfies Axioms 1, 2, and 3, and \(A\) is an arbitrary event, then necessarily \(\mathbb{P}( A^{c} ) \ = \ 1-\mathbb{P}( A )\).
Hint: What is \(A \cup A^c\)?
Prove monotonicity: \(A\subset B\Rightarrow \mathbb{P}( A ) \leq \mathbb{P}(B )\)
Prove the probability of the union: \(\mathbb{P}( A\cup B ) =\mathbb{P}( A ) +\mathbb{P}( B ) - \mathbb{P}( A\cap B )\)
Hint: First prove that \(A\cup B = A\cup ( B\cap A^{c} )\)
Prove Boole’s inequality: \(\mathbb{P}( \bigcup _{i=1}^{m}A_{i} ) \leq \sum_{i=1}^{m}\mathbb{P}( A_{i} )\)
No reading for next class: most of Section 1.4 we are de-emphasizing.
Start working on your Assignment due May 20th, 11:59pm
Stat 302 - Winter 2025/26