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.
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 Questions
- A die is rolled. Event A = rolling a number greater than 4. List the outcomes in A and find P(A).
- Using the same die, find P(A') where A is rolling greater than 4.
- Event B = rolling an odd number. Event C = rolling a multiple of 3. Are B and C mutually exclusive? Explain.
- A bag contains 6 red, 4 blue, and 2 yellow counters. Find P(not blue).
- P(event X) = 0.35. Find P(X') and express it as a percentage.