Events (Formal) Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
At least one approach: Find P(atĀ leastĀ oneĀ headĀ inĀ 3Ā coinĀ flips)P(\text{at least one head in 3 coin flips}) using the complement rule.

Solution

  1. 1
    Direct counting is tedious; use complement: P(atĀ leastĀ oneĀ head)=1āˆ’P(noĀ heads)P(\text{at least one head}) = 1 - P(\text{no heads})
  2. 2
    P(noĀ heads)=P(TTT)=(12)3=18P(\text{no heads}) = P(TTT) = \left(\frac{1}{2}\right)^3 = \frac{1}{8}
  3. 3
    P(atĀ leastĀ oneĀ head)=1āˆ’18=78P(\text{at least one head}) = 1 - \frac{1}{8} = \frac{7}{8}
  4. 4
    Verify by listing: all outcomes except TTT = 7 out of 8 āœ“

Answer

P(atĀ leastĀ oneĀ head)=78P(\text{at least one head}) = \frac{7}{8}
The complement is especially powerful for 'at least one' problems. Instead of adding probabilities for 1, 2, or 3 heads, we compute 1āˆ’P(0Ā heads)1 - P(0 \text{ heads}). This strategy reduces complex unions to a simple subtraction.

About Events (Formal)

A formal event is a subset of the sample space ā€” 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) ā†’

More Events (Formal) Examples

Example 1 easy

Rolling a fair die: Event [formula] = rolling an even number. Find [formula] and [formula], and veri

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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