Dr. Samir Orujov
School of Business (ADA)
September 2025
Understand sample spaces and events as fundamental building blocks
Master Kolmogorov's three axioms that define probability
Derive and apply key properties from the axioms
Apply the addition rule for unions of events
Calculate probabilities using equally likely outcomes
Understand probability given additional information
Probability is a mathematical framework for quantifying uncertainty and randomness.
Key Insight: Probability assigns numbers between 0 and 1 to events, where:
Sample Space: The set of all possible outcomes of a random experiment
Denoted by Ω (omega) or S
Rolling a die: Ω = {1, 2, 3, 4, 5, 6}
Flipping a coin: Ω = {H, T}
Measuring temperature: Ω = ℝ (all real numbers)
Event: A subset of the sample space
An event is a collection of outcomes we're interested in
Event A: "Rolling an even number" = {2, 4, 6}
Event B: "Rolling less than 3" = {1, 2}
Event C: "Rolling a 7" = ∅ (impossible event)
Event D: "Rolling any number" = {1, 2, 3, 4, 5, 6} = Ω
"A or B occurs"
At least one happens
"A and B occur"
Both happen
"A does not occur"
Everything except A
"A but not B"
A occurs, B doesn't
Andrey Kolmogorov established three simple axioms that define all of probability theory
Just THREE rules create the entire framework!
• Provide mathematical rigor
• Ensure consistency
• Allow derivation of all other properties
• Unite different interpretations of probability
Meaning: Probabilities cannot be negative
• You can't have "negative chance" of something happening
• Worst case: probability is 0 (impossible)
• This axiom ensures probabilities make logical sense
Meaning: The probability of the entire sample space is 1
• Something must happen when we perform an experiment
• The probability that "some outcome occurs" is certain (100%)
• This normalizes our probability scale from 0 to 1
Meaning: For mutually exclusive events, add their probabilities
P(rolling 1 or 2) = P(rolling 1) + P(rolling 2) = 1/6 + 1/6 = 1/3
Note: This only works when events can't happen together!
1. $P(\emptyset) = 0$ (empty set has probability 0)
2. $P(A') = 1 - P(A)$ (complement rule)
3. $P(A) \leq 1$ for any event A
4. If $A \subseteq B$, then $P(A) \leq P(B)$
Step 1: Note that A and A' are disjoint
Step 2: A ∪ A' = Ω (covers everything)
Step 3: By Axiom 2: P(A ∪ A') = P(Ω) = 1
Step 4: By Axiom 3: P(A ∪ A') = P(A) + P(A')
Step 5: Therefore: P(A) + P(A') = 1
Result: P(A') = 1 - P(A) ✓
Why subtract P(A∩B)? We counted it twice!
If P(rain) = 0.3, P(cold) = 0.4, P(rain AND cold) = 0.15, what's P(rain OR cold)?
Answer: P(rain ∪ cold) = 0.3 + 0.4 - 0.15 = 0.55
Assumption: All outcomes are equally likely
Drawing a heart from a standard deck:
P(heart) = 13/52 = 1/4
Read as: "Probability of A given B"
Meaning: The probability of A occurring, knowing that B has occurred
Given: The roll is even
Find: P(roll > 4 | even)
Solution:
Even outcomes: {2, 4, 6}
Even AND > 4: {6}
P(>4 | even) = 1/3
A test is positive for 90% of sick patients and 5% of healthy patients.
If 1% of the population is sick, what's P(sick|positive)?
Events A and B are independent if:
Equivalent conditions:
• P(A|B) = P(A)
• P(B|A) = P(B)
Independent: Two coin flips
Dependent: Drawing cards without replacement
Components:
• P(A): Prior probability
• P(B|A): Likelihood
• P(A|B): Posterior probability
Correct! P(A') = 1 - P(A) is a property that can be derived from the axioms, but it is not one of the three fundamental axioms itself.
Correct! Since A and B are disjoint (P(A∩B) = 0), P(A∪B) = P(A) + P(B) = 0.4 + 0.3 = 0.7
Correct! For independent events, P(A∩B) = P(A) × P(B) = 0.5 × 0.4 = 0.2
The mathematical foundation that defines probability as a measure on events
Complement rule, addition rule, and other essential probability rules
From coin flips to conditional probability and Bayes' theorem
How probability theory applies to statistics, finance, and decision-making
Think about: How will you use these probability concepts in your future coursework and career?
Practice Problems: Apply the axioms to solve real-world probability questions involving cards, dice, and everyday scenarios.
"Probability is not just about chance—it's the language of uncertainty that helps us make informed decisions."