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A training program has a roster that includes only advanced students and beginner students. If the probability of randomly selecting an advanced student from the roster is \(\frac{5}{6}\), how many advanced students must be added to the roster so that the probability of randomly selecting an advanced student becomes \(\frac{7}{8}\)?
(1) There are 44 more advanced students than beginner students.
(2) The ratio of advanced students to beginner students is \(5:1\).
(1) There are 44 more advanced students than beginner students.
(2) The ratio of advanced students to beginner students is \(5:1\).
06 Aug 2026, 10:56
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Official Solution:
A training program has a roster that includes only advanced students and beginner students. If the probability of randomly selecting an advanced student from the roster is \(\frac{5}{6}\), how many advanced students must be added to the roster so that the probability of randomly selecting an advanced student becomes \(\frac{7}{8}\)?
Since the probability of randomly selecting an advanced student is \(\frac{5}{6}\), the ratio of advanced students to beginner students is:
\(5:1\)
So let the numbers of advanced and beginner students be \(5x\) and \(x\), respectively.
(1) There are 44 more advanced students than beginner students.
So:
\(5x - x = 44\)
\(4x = 44\)
\(x = 11\)
Thus, there are 55 advanced students and 11 beginner students.
Let \(n\) advanced students be added. Then:
\(\frac{55 + n}{66 + n} = \frac{7}{8}\)
\(8(55 + n) = 7(66 + n)\)
\(440 + 8n = 462 + 7n\)
\(n = 22\)
Sufficient.
(2) The ratio of advanced students to beginner students is \(5:1\).
This is already given by the stem, so statement (2) provides no new information about the actual numbers of students.
Not sufficient.
Answer: A
A training program has a roster that includes only advanced students and beginner students. If the probability of randomly selecting an advanced student from the roster is \(\frac{5}{6}\), how many advanced students must be added to the roster so that the probability of randomly selecting an advanced student becomes \(\frac{7}{8}\)?
Since the probability of randomly selecting an advanced student is \(\frac{5}{6}\), the ratio of advanced students to beginner students is:
\(5:1\)
So let the numbers of advanced and beginner students be \(5x\) and \(x\), respectively.
(1) There are 44 more advanced students than beginner students.
So:
\(5x - x = 44\)
\(4x = 44\)
\(x = 11\)
Thus, there are 55 advanced students and 11 beginner students.
Let \(n\) advanced students be added. Then:
\(\frac{55 + n}{66 + n} = \frac{7}{8}\)
\(8(55 + n) = 7(66 + n)\)
\(440 + 8n = 462 + 7n\)
\(n = 22\)
Sufficient.
(2) The ratio of advanced students to beginner students is \(5:1\).
This is already given by the stem, so statement (2) provides no new information about the actual numbers of students.
Not sufficient.
Answer: A
