Events (Formal) Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Events (Formal).

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A formal event is a subset of the sample space — a collection of outcomes to which a probability is assigned; events can be simple (one outcome) or compound (many outcomes).

An event is a question like 'Did I roll higher than 3?' that has yes/no answer.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Events can be combined with AND, OR, NOT to form compound events.

Common stuck point: Simple event = one outcome. Compound event = multiple outcomes.

Sense of Study hint: Write out the sample space as a set, then highlight or circle the outcomes that match your event. The event is that subset.

Worked Examples

Example 1

easy
Rolling a fair die: Event A = rolling an even number. Find P(A) and P(A^c), and verify the complement rule.

Solution

  1. 1
    Sample space: S = \{1,2,3,4,5,6\}
  2. 2
    Event A (even): \{2,4,6\}; P(A) = \frac{3}{6} = \frac{1}{2}
  3. 3
    Complement A^c (odd): \{1,3,5\}; P(A^c) = \frac{3}{6} = \frac{1}{2}
  4. 4
    Verify: P(A) + P(A^c) = \frac{1}{2} + \frac{1}{2} = 1 ✓

Answer

P(A) = \frac{1}{2}; P(A^c) = \frac{1}{2}; sum = 1. ✓
The complement rule states P(A^c) = 1 - P(A). An event and its complement are mutually exclusive and exhaustive — together they cover all possible outcomes. Often it's easier to compute P(A) = 1 - P(A^c) if the complement is simpler.

Example 2

medium
At least one approach: Find P(\text{at least one head in 3 coin flips}) using the complement rule.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
If P(\text{rain tomorrow}) = 0.35, find P(\text{no rain tomorrow}) and explain the complement rule.

Example 2

hard
A system has 3 independent components, each failing with probability 0.1. The system fails if at least one component fails. Find P(\text{system fails}) using the complement rule.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

sample space