PSTAT 5A: Introduction to Probability
Lecture 4: Fundamentals of Probability, Rules , Axioms & Properties
2025-07-01
Probability Range
Ranges from 0 to 1 (or 0% to 100%)
0: Event will never occur (impossible)
1: Event will certainly occur (certain)
0.5: Event has equal chance of occurring or not
Example
When we roll a die, there are six possible outcomes:
1, 2, 3, 4, 5, 6.
The probability of any of them turning up is 1/6 or 16%.
Sample Space
🎯 Definition The sample space (denoted \(S\) or \(\Omega\)) is the set of all possible outcomes of a random experiment
Sample Space Examples
- Coin flip: \(S = \{H, T\}\)
- Two coin flips: \(S = \{HH, HT, TH, TT\}\)
- 🎲 Die roll: \(S = \{1, 2, 3, 4, 5, 6\}\)
- Two die rolls
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- \(S = \{A\heartsuit,\ 2\heartsuit,\ \dots,\ K\heartsuit,\\ \phantom{S = \{}A\diamondsuit,\ 2\diamondsuit,\ \dots,\ K\diamondsuit,\\ \phantom{S = \{}A\clubsuit,\ 2\clubsuit,\ \dots,\ K\clubsuit,\\ \phantom{S = \{}A\spadesuit,\ 2\spadesuit,\ \dots,\ K\spadesuit\}\)
Types of Sample Spaces
Finite Sample Space
- Limited number of outcomes
- Example: Rolling a die
Infinite Sample Space
- Unlimited outcomes (countable or uncountable)
- Example: Measuring exact height of students
Events
🎯 Definition An event is a subset of the sample space
Simple event: Contains exactly one outcome (Ex: \(A = \{3\}\) (rolling a 3))
Compound event: Contains multiple outcomes (Ex: \(B = \{2, 4, 6\}\) (rolling an even number))
Event Notation
For a die roll with \(S = \{1, 2, 3, 4, 5, 6\}\):
- \(A = \{1, 3, 5\}\) (rolling an odd number)
- \(B = \{4, 5, 6\}\) (rolling 4 or higher)
- \(C = \{6\}\) (rolling a six)
We can describe events in words or using set notation
What is a Set?
🎯 Definition: A collection of things that share common characteristics. They can be elements, members, objects or similar terms.
Examples:
- Set of even numbers:
- {2, 4, 6, 8, …}
- Set of vowels: {a, e, i, o, u}
Union: A ∪ B
🎯 Definition: Contains all set elements, including intersections.
In Probability: The event that A OR B occurs (or both).
\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]
Intersection: A ∩ B
🎯 Definition: Area where two or more sets overlap.
In Probability: The event that A AND B occurs.
Properties:
Always smaller than or equal to individual sets
Can be empty (disjoint sets)
Absolute Complement: \(A^c\)
🎯 Definition: All elements that do not belong to the set.
In Probability: The event that A does NOT occur.
- \(P(A^c) = 1 - P(A)\)
Key Property:
\(A \cup A^c = S\) (Sample Space)
Example: Set Operations
For die roll \(S = \{1, 2, 3, 4, 5, 6\}\):
\(A = \{1, 3, 5\}\) (odd numbers)
\(B = \{4, 5, 6\}\) (4 or higher)
Find:
\(A \cup B\)
\(A \cap B\)
\(A^c\)
Solution.
\(A \cup B = \{1, 3, 4, 5, 6\}\)
\(A \cap B = \{5\}\)
\(A^c = \{2, 4, 6\}\)
Mutually Exclusive Events
Events \(A\) and \(B\) are mutually exclusive (or disjoint) if they cannot occur simultaneously
\[A \cap B = \emptyset\]
When rolling a die
\(A = \{1, 3, 5\}\) (odd)
\(B = \{2, 4, 6\}\) (even)
\(A\) and \(B\) are mutually exclusive
