Question

The probability that a 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. \(0.99\)
• B. \(0.871\)
• C. \(0.891\)
• D. \(0.762\)

Hint:

Think of this problem like a path on a probability tree diagram. To reach the final destination ("lasts 1 year"), you must traverse the first branch ("survives immediate use", \(0.90\)) and then multiply it by the next branch down that path ("survives 1 year given it worked initially", \(0.99\)).

Answer 1

The Correct Option is C

Concept: This problem can be effectively solved using conditional probability and the multiplication rule of probability. Let us define two events:
• \(A\): The component does not fail immediately (it works when first used).
• \(B\): The component lasts for one year. For a new component to last for one year, it must satisfy two sequential criteria: it must first survive its initial usage, and then it must continue running for the remainder of the year. The intersection probability is given by: \[ P(A \cap B) = P(A) \times P(B \mid A) \] Step 1: Calculate the probability that the component survives the initial use, \(P(A)\).
We are given that the probability of immediate failure is \(0.10\). Since survival and immediate failure are complementary events: \[ P(A) = 1 - P(\text{Immediate Failure}) = 1 - 0.10 = 0.90 \]

Step 2: Identify the conditional probability, \(P(B \mid A)\).
The problem statement notes: "If it does not fail immediately, the probability that it lasts for one year is 0.99." This is a direct statement of conditional probability given that event \(A\) has occurred: \[ P(B \mid A) = 0.99 \]

Step 3: Compute the total probability that a new component lasts for one year.
Applying the multiplication rule to find the joint probability \(P(A \cap B)\): \[ P(\text{Lasts for one year}) = P(A) \times P(B \mid A) \] \[ P(\text{Lasts for one year}) = 0.90 \times 0.99 = 0.891 \]