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07 Aug 2025, 05:51
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There are 10 trees in a garden, and a gardener randomly selects 3 different trees to prune. Is the probability that all 3 selected trees are oaks greater than \(\frac{1}{20}\)?
(1) The probability that two randomly selected trees are both oaks is \(\frac{2}{15}\).
(2) There are 6 maple trees in the garden.
(1) The probability that two randomly selected trees are both oaks is \(\frac{2}{15}\).
(2) There are 6 maple trees in the garden.
07 Aug 2025, 05:51
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Official Solution:
There are 10 trees in a garden, and a gardener randomly selects 3 different trees to prune. Is the probability that all 3 selected trees are oaks greater than \(\frac{1}{20}\)?
(1) The probability that two randomly selected trees are both oaks is \(\frac{2}{15}\).
Let the number of oak trees be \(x\). Then:
\(\frac{x}{10} * \frac{x - 1}{9} = \frac{2}{15}\)
We can solve this equation to find the number of oak trees and use that to answer the question. Sufficient.
(Just for reference: solving the equation gives \(x = 4\), and the probability that all 3 selected trees are oaks is \(\frac{4}{10} * \frac{3}{9} * \frac{2}{8} = \frac{1}{30}\), which is less than \(\frac{1}{20}\).)
(2) There are 6 maple trees in the garden.
This means there can be at most 4 oak trees. So the probability that all 3 selected trees are oaks is at most \(\frac{4}{10} * \frac{3}{9} * \frac{2}{8} = \frac{24}{720} = \frac{1}{30}\), which is still less than \(\frac{1}{20}\). So again, the answer is NO. Sufficient.
Answer: D
There are 10 trees in a garden, and a gardener randomly selects 3 different trees to prune. Is the probability that all 3 selected trees are oaks greater than \(\frac{1}{20}\)?
(1) The probability that two randomly selected trees are both oaks is \(\frac{2}{15}\).
Let the number of oak trees be \(x\). Then:
\(\frac{x}{10} * \frac{x - 1}{9} = \frac{2}{15}\)
We can solve this equation to find the number of oak trees and use that to answer the question. Sufficient.
(Just for reference: solving the equation gives \(x = 4\), and the probability that all 3 selected trees are oaks is \(\frac{4}{10} * \frac{3}{9} * \frac{2}{8} = \frac{1}{30}\), which is less than \(\frac{1}{20}\).)
(2) There are 6 maple trees in the garden.
This means there can be at most 4 oak trees. So the probability that all 3 selected trees are oaks is at most \(\frac{4}{10} * \frac{3}{9} * \frac{2}{8} = \frac{24}{720} = \frac{1}{30}\), which is still less than \(\frac{1}{20}\). So again, the answer is NO. Sufficient.
Answer: D
14 Aug 2025, 13:47
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I don’t quite agree with the solution. "There are 6 maple trees in the garden"
This information does not give me how many different trees are in the garden; We only know maple trees = 6, but we can have a lot of different species of trees besides maple and oak trees.
This information does not give me how many different trees are in the garden; We only know maple trees = 6, but we can have a lot of different species of trees besides maple and oak trees.
