Events (Formal) Math Example 4

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

Example 4

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

Solution

  1. 1
    Use complement: P(system fails)=1−P(no components fail)P(\text{system fails}) = 1 - P(\text{no components fail})
  2. 2
    P(component works)=0.9P(\text{component works}) = 0.9; components are independent
  3. 3
    P(all 3 work)=0.9×0.9×0.9=0.729P(\text{all 3 work}) = 0.9 \times 0.9 \times 0.9 = 0.729
  4. 4
    P(system fails)=1−0.729=0.271P(\text{system fails}) = 1 - 0.729 = 0.271

Answer

P(system fails)=1−0.93=0.271P(\text{system fails}) = 1 - 0.9^3 = 0.271
The complement strategy is essential for reliability engineering. 'At least one failure' is the complement of 'all work.' For independent components, multiply individual probabilities of working. This is widely used in engineering and risk analysis.

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 1 easy

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

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.

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