A machine operates using two independent components, Component A and Component B. The prob… — GRE General Test

A machine operates using two independent components, Component $A$ and Component $B$ . The probability that Component $A$ functions properly on a given day is $0.90$ , and the probability that Component $B$ functions properly on that same day is $0.80$ . What is the probability that at least one of the components functions properly on a given day?

Answer: 0.98

Answer

The probability that at least one component functions properly is

0.98

To determine the probability that at least one component functions properly, use the complement rule:

P(\text{at least one}) = 1 - P(\text{neither})

. Since Component

A

and Component

B

operate independently, the probability that

A

fails is

1 - 0.90 = 0.10

and the probability that

B

fails is

1 - 0.80 = 0.20

. The probability of both components failing simultaneously is

0.10 \times 0.20 = 0.02

. Subtracting this probability from

1

gives

1 - 0.02 = 0.98

Step-by-Step Solution

Find the probability of failure for each component.

P(A^c) = 1 - 0.90 = 0.10

and

P(B^c) = 1 - 0.80 = 0.20

The event that a component fails is the complement of the event that it functions properly.

Calculate the joint probability of both components failing.

P(A^c \text{ and } B^c) = 0.10 \times 0.20 = 0.02

Because the components operate independently, their failure events are independent, so their individual probabilities are multiplied.

Calculate the probability that at least one component functions properly.

P(\text{at least one functions}) = 1 - 0.02 = 0.98

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

Key Concept

Probability of Independent Events and Complement Rule

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