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There are two boxes each containing two components. Each component is defective with probability ¼, independent of all other components. The probability that exactly one box contains exactly one defective component equals?
- a)3/8
- b)5/8
- c)15/32
- d)17/32
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
There are two boxes each containing two components. Each component is ...
Probability (1 faulty in box) 
The chances of box having other than 1 defective is 5/8
The probability exactly 1 one fault in exactly 1 box
Alternate:
The given situation is possible if
(1) 1 faulty among 4 components
(2) 3 faulty among 4 components
For case (1)
For case (2) 
Most Upvoted Answer
There are two boxes each containing two components. Each component is ...
Let's call the two components in the first box A and B, and the two components in the second box C and D. We know that each component is defective with probability 0.2, so the probability that a component is not defective is 0.8.
To find the probability that both components in a box are defective, we need to multiply the probability that the first component is defective by the probability that the second component is defective, assuming that the events are independent. So for box 1, the probability that both components are defective is:
P(A and B are defective) = P(A is defective) * P(B is defective) = 0.2 * 0.2 = 0.04
Similarly, for box 2, the probability that both components are defective is:
P(C and D are defective) = P(C is defective) * P(D is defective) = 0.2 * 0.2 = 0.04
Now, to find the probability that at least one box has both components defective, we can use the complement rule. The complement of "at least one box has both components defective" is "neither box has both components defective." The probability of this event is the product of the probabilities that neither box has both components defective:
P(neither box has both components defective) = P(A and B are not defective) * P(C and D are not defective)
Since each component has a probability of 0.8 of not being defective, we have:
P(neither box has both components defective) = 0.8 * 0.8 * 0.8 * 0.8 = 0.4096
So the probability that at least one box has both components defective is:
P(at least one box has both components defective) = 1 - P(neither box has both components defective) = 1 - 0.4096 = 0.5904
Therefore, the probability that at least one box has both components defective is 0.5904.
To find the probability that both components in a box are defective, we need to multiply the probability that the first component is defective by the probability that the second component is defective, assuming that the events are independent. So for box 1, the probability that both components are defective is:
P(A and B are defective) = P(A is defective) * P(B is defective) = 0.2 * 0.2 = 0.04
Similarly, for box 2, the probability that both components are defective is:
P(C and D are defective) = P(C is defective) * P(D is defective) = 0.2 * 0.2 = 0.04
Now, to find the probability that at least one box has both components defective, we can use the complement rule. The complement of "at least one box has both components defective" is "neither box has both components defective." The probability of this event is the product of the probabilities that neither box has both components defective:
P(neither box has both components defective) = P(A and B are not defective) * P(C and D are not defective)
Since each component has a probability of 0.8 of not being defective, we have:
P(neither box has both components defective) = 0.8 * 0.8 * 0.8 * 0.8 = 0.4096
So the probability that at least one box has both components defective is:
P(at least one box has both components defective) = 1 - P(neither box has both components defective) = 1 - 0.4096 = 0.5904
Therefore, the probability that at least one box has both components defective is 0.5904.
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