A fishing rod has 7 guides, 4 of which are defective. If 2 are selected at random with replacement, find the probability that both are defective. Round your answer to three decimal places.
Question
A fishing rod has 7 guides, 4 of which are defective. If 2 are selected at random with replacement, find the probability that both are defective. Round your answer to three decimal places.
Solution
When we select with replacement, it means after selecting a guide, we put it back before selecting again. Therefore, the probability does not change between selections.
The total number of guides is 7, and 4 of them are defective. So, the probability of selecting a defective guide is 4/7.
Since we are selecting with replacement, the probability of selecting a defective guide the second time is also 4/7.
The probability of both guides being defective is the product of the probabilities of each individual selection, because these are independent events.
So, the probability that both are defective is (4/7) * (4/7) = 16/49 = 0.327 (rounded to three decimal places).
So, the probability that both guides are defective is approximately 0.327, or 32.7%.
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