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A basket contains apples and oranges only. If 2 fruits are randomly selected from the basket one after another without replacement, what is the probability that at least one of them is an apple?
(1) The number of oranges in the basket is twice the number of apples.
(2) If 2 fruits are randomly selected from the basket one after another with replacement, the probability that at least one of them is an apple is \(\frac{5}{9}\).
(1) The number of oranges in the basket is twice the number of apples.
(2) If 2 fruits are randomly selected from the basket one after another with replacement, the probability that at least one of them is an apple is \(\frac{5}{9}\).
29 Jul 2024, 07:03
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Official Solution:
A basket contains apples and oranges only. If 2 fruits are randomly selected from the basket one after another without replacement, what is the probability that at least one of them is an apple?
(1) The number of oranges in the basket is twice the number of apples.
This implies \(O = 2A\). Different numbers of oranges and apples will give different probabilities. For example, if there are 2 oranges and 1 apple, then the probability of selecting at least one apple would be P(at least one apple) = 1 - P(no apples) = \(1 - (\frac{2}{3} * \frac{1}{2}) = \frac{2}{3}\). However, if there are 4 oranges and 2 apples, then the probability of selecting at least one apple would be P(at least one apple) = 1 - P(no apples) = \(1 - (\frac{4}{6} * \frac{3}{5}) = \frac{3}{5}\).
(2) If 2 fruits are randomly selected from the basket one after another with replacement, the probability that at least one of them is an apple is \(\frac{5}{9}\).
This implies \(1 - \frac{O}{O + A} * \frac{O}{O + A} = \frac{5}{9}\):
\((\frac{O}{O + A})^2 = \frac{4}{9}\)
\(\frac{O}{O + A} = \frac{2}{3}\)
\(3O = 2O + 2A\)
\(O = 2A\).
As discussed above, such a ratio is not sufficient to get a unique answer. Not sufficient.
(1) + (2) Both statements give the same exact information: \(O = 2A\). Thus, combining them does not add anything new. Not sufficient.
Answer: E
A basket contains apples and oranges only. If 2 fruits are randomly selected from the basket one after another without replacement, what is the probability that at least one of them is an apple?
(1) The number of oranges in the basket is twice the number of apples.
This implies \(O = 2A\). Different numbers of oranges and apples will give different probabilities. For example, if there are 2 oranges and 1 apple, then the probability of selecting at least one apple would be P(at least one apple) = 1 - P(no apples) = \(1 - (\frac{2}{3} * \frac{1}{2}) = \frac{2}{3}\). However, if there are 4 oranges and 2 apples, then the probability of selecting at least one apple would be P(at least one apple) = 1 - P(no apples) = \(1 - (\frac{4}{6} * \frac{3}{5}) = \frac{3}{5}\).
(2) If 2 fruits are randomly selected from the basket one after another with replacement, the probability that at least one of them is an apple is \(\frac{5}{9}\).
This implies \(1 - \frac{O}{O + A} * \frac{O}{O + A} = \frac{5}{9}\):
\((\frac{O}{O + A})^2 = \frac{4}{9}\)
\(\frac{O}{O + A} = \frac{2}{3}\)
\(3O = 2O + 2A\)
\(O = 2A\).
As discussed above, such a ratio is not sufficient to get a unique answer. Not sufficient.
(1) + (2) Both statements give the same exact information: \(O = 2A\). Thus, combining them does not add anything new. Not sufficient.
Answer: E
04 Sep 2024, 23:54
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usually DS can be done without actually solving, but in this case i feel because of second statement we will need to solve else we dont get to know that 1st and 2nd info is the same, right?
