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C
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Difficulty:
65%
(hard)
Question Stats:
53% (01:31) correct
47%
(01:50)
wrong
based on 75
sessions
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On Halloween night, while trick-or-treating, Lucy is offered a bowl that contains only lemon candies and chocolate bars. If Lucy randomly takes 5 items from the bowl without replacement, what is the probability that at least n of the 5 items are lemon candies?
(1) The probability that all 5 items Lucy takes are chocolate bars is 1/21.
(2) n = 1.
(1) The probability that all 5 items Lucy takes are chocolate bars is 1/21.
(2) n = 1.
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14 Mar 2026, 02:15
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GMAT Club Official Solution:
On Halloween night, while trick-or-treating, Lucy is offered a bowl that contains only lemon candies and chocolate bars. If Lucy randomly takes 5 items from the bowl without replacement, what is the probability that at least n of the 5 items are lemon candies?
(1) The probability that all 5 items Lucy takes are chocolate bars is 1/21.
This statement gives only the probability that Lucy gets 0 lemon candies (all 5 items are chocolate bars). The question, however, asks for the probability that Lucy gets at least n lemon candies, and n is unknown. Not sufficient.
(2) n = 1.
This statement tells us that n = 1, so the question becomes: what is the probability that Lucy gets at least 1 lemon candy? But we still do not know how many candies of each kind are in the bowl. Not sufficient.
(1)+(2) With n = 1, we want P(at least 1 lemon) = 1 - P(0 lemons), and (1) gives P(0 lemons) = 1/21, so the required probability is 1 - 1/21 = 20/21. Sufficient.
Answer: C
On Halloween night, while trick-or-treating, Lucy is offered a bowl that contains only lemon candies and chocolate bars. If Lucy randomly takes 5 items from the bowl without replacement, what is the probability that at least n of the 5 items are lemon candies?
(1) The probability that all 5 items Lucy takes are chocolate bars is 1/21.
This statement gives only the probability that Lucy gets 0 lemon candies (all 5 items are chocolate bars). The question, however, asks for the probability that Lucy gets at least n lemon candies, and n is unknown. Not sufficient.
(2) n = 1.
This statement tells us that n = 1, so the question becomes: what is the probability that Lucy gets at least 1 lemon candy? But we still do not know how many candies of each kind are in the bowl. Not sufficient.
(1)+(2) With n = 1, we want P(at least 1 lemon) = 1 - P(0 lemons), and (1) gives P(0 lemons) = 1/21, so the required probability is 1 - 1/21 = 20/21. Sufficient.
Answer: C
General Discussion
06 Mar 2026, 09:18
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Key Concept Being Tested: Data Sufficiency with Probability — specifically whether you have enough information to determine both the composition of the sample space AND the threshold value n.
Common Trap: Most students see Statement 1, do some mental math on the probability, and think "I can figure out the bowl contents, so this must be sufficient." They forget that without knowing n, the question has no single answer. Separately, students accept Statement 2 alone without noticing they have zero information about how many of each candy type are in the bowl.
Step 1 — Understand what the question actually needs.
To answer "what is P(at least n of 5 draws are lemon)?" we need two things: (a) the composition of the bowl (how many lemon, how many chocolate), and (b) the value of n.
Step 2 — Test Statement 1 alone.
"The probability that all 5 items are chocolate bars is 1/21."
This lets us solve for the bowl composition. If C = number of chocolates and T = total items:
C(C,5) / C(T,5) = 1/21
Testing values: when C = 6 and T = 9, we get C(6,5)/C(9,5) = 6/126 = 1/21. ✓
So the bowl has 6 chocolates and 3 lemons (total 9 items).
But we still don't know n. If n = 1, the answer is one number. If n = 3, it's completely different. Statement 1 alone: INSUFFICIENT.
Step 3 — Test Statement 2 alone.
"n = 1." Now we know the threshold, but we have no idea how many lemon candies or chocolate bars are in the bowl. The probability changes entirely depending on bowl composition. Statement 2 alone: INSUFFICIENT.
Step 4 — Combine both statements.
From S1: bowl has 6 chocolates, 3 lemons (9 total).
From S2: n = 1.
P(at least 1 lemon) = 1 − P(0 lemons) = 1 − C(6,5)/C(9,5) = 1 − 6/126 = 1 − 1/21 = 20/21
We get a single definitive answer. Both together: SUFFICIENT.
Answer: C
Takeaway: Whenever a DS probability question has an unknown in the question stem itself (here, n), always flag that as a second piece of missing information — you need both the probability space AND the threshold to be fully determined before either statement can be sufficient alone.
Common Trap: Most students see Statement 1, do some mental math on the probability, and think "I can figure out the bowl contents, so this must be sufficient." They forget that without knowing n, the question has no single answer. Separately, students accept Statement 2 alone without noticing they have zero information about how many of each candy type are in the bowl.
Step 1 — Understand what the question actually needs.
To answer "what is P(at least n of 5 draws are lemon)?" we need two things: (a) the composition of the bowl (how many lemon, how many chocolate), and (b) the value of n.
Step 2 — Test Statement 1 alone.
"The probability that all 5 items are chocolate bars is 1/21."
This lets us solve for the bowl composition. If C = number of chocolates and T = total items:
C(C,5) / C(T,5) = 1/21
Testing values: when C = 6 and T = 9, we get C(6,5)/C(9,5) = 6/126 = 1/21. ✓
So the bowl has 6 chocolates and 3 lemons (total 9 items).
But we still don't know n. If n = 1, the answer is one number. If n = 3, it's completely different. Statement 1 alone: INSUFFICIENT.
Step 3 — Test Statement 2 alone.
"n = 1." Now we know the threshold, but we have no idea how many lemon candies or chocolate bars are in the bowl. The probability changes entirely depending on bowl composition. Statement 2 alone: INSUFFICIENT.
Step 4 — Combine both statements.
From S1: bowl has 6 chocolates, 3 lemons (9 total).
From S2: n = 1.
P(at least 1 lemon) = 1 − P(0 lemons) = 1 − C(6,5)/C(9,5) = 1 − 6/126 = 1 − 1/21 = 20/21
We get a single definitive answer. Both together: SUFFICIENT.
Answer: C
Takeaway: Whenever a DS probability question has an unknown in the question stem itself (here, n), always flag that as a second piece of missing information — you need both the probability space AND the threshold to be fully determined before either statement can be sufficient alone.
