Events - Collections of Outcomes We Care About
An event is a specific collection of outcomes from a probability experiment that we are interested in. Where an outcome is a single result, an event can consist of one outcome or several outcomes grouped together.
The formal mathematical treatment of events as sets of outcomes was cemented in 1933, when the Soviet mathematician Andrey Kolmogorov published a short but hugely influential book that placed the whole of probability theory on rigorous set-theoretic foundations – the same axioms taught in every introductory probability course today. Before Kolmogorov, mathematicians had spent nearly three centuries using probability successfully without ever agreeing on a fully rigorous definition of what an "event" actually was. His framework, where an event really is just a subset of the sample space, is exactly the idea behind P(A) = (outcomes in A) ÷ (total outcomes) used throughout this section.
Simple vs. Compound Events
A simple event contains exactly one outcome. Example: rolling a 4 on a die – only one outcome qualifies.
A compound event contains more than one outcome. Example: rolling an even number on a die – outcomes 2, 4, and 6 all qualify.
Notation
Events are usually labelled with capital letters. P(A) means "the probability of event A". If A = rolling an even number on a fair die, then A = {2, 4, 6} and P(A) = 3/6 = 1/2.
Probability of an Event
P(event) = Number of outcomes in the event ÷ Total number of equally likely outcomes
Complementary Events
The complement of event A, written A' (or A-complement), is the event that A does NOT happen. Every outcome that is not in A belongs to A'.
P(A') = 1 − P(A)
Mutually Exclusive Events
Two events are mutually exclusive if they cannot both occur at the same time – they share no outcomes. Rolling a 2 and rolling a 5 on the same die throw are mutually exclusive.
For mutually exclusive events A and B: P(A or B) = P(A) + P(B)
Worked Examples
Primes on a die: 2, 3, 5 – so A = {2, 3, 5}. Total outcomes = 6.
P(A) = 3/6 = 1/2. P(A') = 1 − 1/2 = 1/2.
A King cannot also be a Queen, so yes – mutually exclusive.
P(A) = 4/52 = 1/13. P(B) = 4/52 = 1/13.
P(A or B) = 1/13 + 1/13 = 2/13.
Red outcomes = 3, Green outcomes = 5. Total = 12.
P(C) = (3 + 5) / 12 = 8/12 = 2/3. (Red and green are mutually exclusive, so probabilities add.)
Key Takeaways
- An event is a set of outcomes – it can be simple (one outcome) or compound (several).
- P(event) = favourable outcomes ÷ total equally likely outcomes.
- P(A') = 1 − P(A). The event and its complement always add up to 1.
- Mutually exclusive events cannot both occur: P(A or B) = P(A) + P(B).
Practice: Events & Complements
Related Topics
Continue exploring related topics:
- Conditional Probability - Probability Given Prior Knowledge
- Probability Experiments - Trials and Random Results
- Independent Events - When One Does Not Affect the Other
- Probability Overview - The Mathematics of Chance
- Probability Scale - Measuring Likelihood from 0 to 1
- Tree Diagrams - Visualising Multi-Step Probability