Question

The probability that certain electronic component fails when first used is 0.10. If it does not fail immediately, the probability that it lasts for one year is 0.99. The probability that a new component will last for one year is

• A. 0.9
• B. 0.01
• C. 0.119
• D. 0.891

Hint:

When dealing with conditional probabilities, always ensure that the total probability theorem is used for scenarios with multiple possible outcomes.

Answer 1

The Correct Option is D

Let \(P(F)\) be the event that the electronic component fails when first used. So, \[ P(F) = 0.10 \quad {and} \quad P(F') = 1 - P(F) = 0.90 \] Let \(E\) be the event that a new component will last for one year, then \[ P(E) = P(F) \cdot P(E | F) + P(F') \cdot P(E | F') \] Using the total probability theorem, we have: \[ P(E) = 0.10 \times 0 + 0.90 \times 0.99 = 0.891 \] Thus, the probability that the new component will last for one year is \(0.891\).

Answer 2

Step 1: Understanding the Problem
We are given two scenarios for the electronic component:
1. The probability that the component fails when first used is 0.10. Therefore, the probability of the component not failing immediately is \( 1 - 0.10 = 0.90 \).
2. If the component does not fail immediately, the probability that it lasts for one year is 0.99.
We need to calculate the overall probability that the component lasts for one year.

Step 2: Using the Law of Total Probability
The total probability that the component lasts for one year, \( P(\text{lasting for 1 year}) \), can be found using the law of total probability, which considers both possible outcomes (failure and non-failure immediately):
\[ P(\text{lasting for 1 year}) = P(\text{no failure immediately}) \times P(\text{lasting for 1 year} | \text{no failure immediately}) + P(\text{failure immediately}) \times P(\text{lasting for 1 year} | \text{failure immediately}) \]
Since the probability of failure immediately is 0.10, the probability that it lasts for one year if it fails immediately is 0 (i.e., no chance of lasting after failure).
Thus, the formula simplifies to:
\[ P(\text{lasting for 1 year}) = (0.90) \times (0.99) + (0.10) \times (0) \]

Step 3: Calculation
Substituting the values:
\[ P(\text{lasting for 1 year}) = 0.90 \times 0.99 = 0.891 \]

Step 4: Conclusion
Therefore, the probability that a new component will last for one year is 0.891.