Events (Formal) Math Example 1

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Example 1

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

Solution

  1. 1
    Sample space: S={1,2,3,4,5,6}S = \{1,2,3,4,5,6\}
  2. 2
    Event AA (even): {2,4,6}\{2,4,6\}; P(A)=36=12P(A) = \frac{3}{6} = \frac{1}{2}
  3. 3
    Complement AcA^c (odd): {1,3,5}\{1,3,5\}; P(Ac)=36=12P(A^c) = \frac{3}{6} = \frac{1}{2}
  4. 4
    Verify: P(A)+P(Ac)=12+12=1P(A) + P(A^c) = \frac{1}{2} + \frac{1}{2} = 1 ✓

Answer

P(A)=12P(A) = \frac{1}{2}; P(Ac)=12P(A^c) = \frac{1}{2}; sum = 1. ✓
The complement rule states P(Ac)=1−P(A)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(Ac)P(A) = 1 - P(A^c) if the complement is simpler.

About Events (Formal)

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).

Learn more about Events (Formal) →

More Events (Formal) Examples

Example 2 medium

At least one approach: Find [formula] using the complement rule.

Example 3 easy

If [formula], find [formula] and explain the complement rule.

Example 4 hard

A system has 3 independent components, each failing with probability 0.1. The system fails if at lea

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