A reliability engineering test evaluates two independent components, Component X and Compo… — GRE General Test

A reliability engineering test evaluates two independent components, Component X and Component Y, in a machine. The probability that Component X fails during operation is $0.20$ , and the probability that Component Y fails during operation is $0.30$ . What is the probability that at least one of the two components operates successfully during operation?

A
$0.44$
B
$0.50$
C
$0.56$
D
$0.70$
$0.94$ Answer

Answer

The probability that at least one of the two components operates successfully is

0.94

The correct answer is

0.94

. The complement of the event 'at least one component operates successfully' is the event 'both components fail'. Because Component X and Component Y fail independently,

P(\text{both fail}) = P(\text{X fails}) \times P(\text{Y fails}) = 0.20 \times 0.30 = 0.06

. Subtracting this complementary probability from

1

yields

1 - 0.06 = 0.94

Step-by-Step Solution

Determine the probability that each component fails.

P(\text{X fails}) = 0.20

and

P(\text{Y fails}) = 0.30

These probabilities are explicitly given in the problem statement.

Calculate the joint probability that BOTH components fail simultaneously using the multiplication rule for independent events.

P(\text{both fail}) = P(\text{X fails}) \times P(\text{Y fails}) = 0.20 \times 0.30 = 0.06

Since the components fail independently, their joint failure probability is the product of their individual failure probabilities.

Apply the complement rule to find the probability that at least one component operates successfully.

P(\text{at least one succeeds}) = 1 - P(\text{both fail}) = 1 - 0.06 = 0.94

The event 'at least one component succeeds' is the exact complement of 'both components fail'.

Key Concept

Independent Events and Complement Probability Rule

Question

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Step-by-Step Solution

Key Concept

Common Mistakes

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