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31 Mar 2026, 00:02
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Difficulty:
85%
(hard)
Question Stats:
23% (01:55) correct
77%
(02:02)
wrong
based on 31
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Aaron and Bryce are both running for seats on city council. The probability that Aaron is elected is \(\frac{2}{3}\), and the probability that Bryce is elected is \(\frac{3}{5}\). Which of the following could be the probability that both are elected?
I. \(\frac{1}{2}\)
II. \(\frac{1}{5}\)
III. \(\frac{13}{20}\)
A) None
B) I only
C) II only
D) I and II only
E) I, II, and III
I. \(\frac{1}{2}\)
II. \(\frac{1}{5}\)
III. \(\frac{13}{20}\)
A) None
B) I only
C) II only
D) I and II only
E) I, II, and III
01 Apr 2026, 02:30
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This problem is tricky because it says 'could be,' not 'what is.' That word signals that the two events are NOT necessarily independent — so we can't just multiply the probabilities.
Since we don't know the exact relationship between Aaron and Bryce's elections, the probability of BOTH being elected (P(A and B)) must fall within a specific range.
Here's the key principle:
The MAXIMUM P(A and B) can be is the smaller of the two individual probabilities. Both can't win together more often than either one wins alone. So the max = min(2/3, 3/5) = 3/5.
The MINIMUM P(A and B) can be comes from the idea that their combined probability can't exceed 1. Using the inclusion-exclusion formula: P(A) + P(B) - 1 = 2/3 + 3/5 - 1 = 10/15 + 9/15 - 15/15 = 4/15. (If this were negative, the minimum would be 0.)
So P(A and B) must be between 4/15 and 3/5, which is roughly between 0.267 and 0.600.
Now let's check each option:
I. 1/2 = 0.500 — This falls between 0.267 and 0.600. YES, this works.
II. 1/5 = 0.200 — This is BELOW the minimum of 0.267. NO, this is too small. It would mean they win together less often than is mathematically possible.
III. 13/20 = 0.650 — This is ABOVE the maximum of 0.600. NO, this is too large. They can't both win more often than Bryce wins alone (3/5).
Only I works, so the answer is B.
The common mistake here is assuming the events are independent and calculating 2/3 × 3/5 = 2/5 as the only possible answer. When a problem says 'could be,' always think about the full range of possible values.
Answer: B
Since we don't know the exact relationship between Aaron and Bryce's elections, the probability of BOTH being elected (P(A and B)) must fall within a specific range.
Here's the key principle:
The MAXIMUM P(A and B) can be is the smaller of the two individual probabilities. Both can't win together more often than either one wins alone. So the max = min(2/3, 3/5) = 3/5.
The MINIMUM P(A and B) can be comes from the idea that their combined probability can't exceed 1. Using the inclusion-exclusion formula: P(A) + P(B) - 1 = 2/3 + 3/5 - 1 = 10/15 + 9/15 - 15/15 = 4/15. (If this were negative, the minimum would be 0.)
So P(A and B) must be between 4/15 and 3/5, which is roughly between 0.267 and 0.600.
Now let's check each option:
I. 1/2 = 0.500 — This falls between 0.267 and 0.600. YES, this works.
II. 1/5 = 0.200 — This is BELOW the minimum of 0.267. NO, this is too small. It would mean they win together less often than is mathematically possible.
III. 13/20 = 0.650 — This is ABOVE the maximum of 0.600. NO, this is too large. They can't both win more often than Bryce wins alone (3/5).
Only I works, so the answer is B.
The common mistake here is assuming the events are independent and calculating 2/3 × 3/5 = 2/5 as the only possible answer. When a problem says 'could be,' always think about the full range of possible values.
Answer: B
