Question

An aircraft has two engines, each having a reliability \( R \). The aircraft will crash only when both engines stop working. The reliability of the aircraft flying without crash is:

• A. \( R^2 \)
• B. \( 2R \)
• C. \( 2R - R^2 \)
• D. \( R^2 - 2R \)

Hint:

In problems involving multiple components working in parallel, the overall reliability is the complement of the probability that all components fail.

Answer 1

The Correct Option is C

In this problem, we are given two engines, each with reliability \( R \), and the aircraft will only crash if both engines fail. This implies that the aircraft will not crash as long as at least one engine is working. To find the reliability of the aircraft flying without crashing, we first need to calculate the probability of both engines failing, as the complement of this will give us the desired reliability. 
The probability of one engine failing is \( 1 - R \), and the probability that both engines fail (i.e., both engines stop working) is: \[ (1 - R)^2 \] Therefore, the probability that at least one engine is working (i.e., the aircraft does not crash) is the complement of this probability: \[ 1 - (1 - R)^2 \] Expanding this expression: \[ 1 - (1 - 2R + R^2) = 2R - R^2 \] Thus, the reliability of the aircraft flying without crashing is \( 2R - R^2 \), which corresponds to option (C).