In a large online introductory sociology course with thousan

Question

In an online sociology course with thousands of students, each student received a grade of 7, 8, 9, or 10. 50% of the students received a score of at least 8, 30% received a score of at least 9, and 10% received a score of 10. Two students are chosen at random from those who received scores of at least 8. 
Which of the following is closest to the probability that one of the selected students received a score of 9 and the other received a score of 10?

(A) 0.04
(B) 0.05
(C) 0.10
(D) 0.12
(E) 0.16

Mandy1098 · Answer

grade 9=200
grade10=100
so probability of one grade 9 and one 10 out of 500=200/500*100/499*2=400/2500(appo.)
Ans = .16

kevincan · Answer

Welcome Mandy1098!

Dereno · Answer

Assume the total population to be 100.

( even though the question mentions as 1000s, we do it for convenience. If not assume it as 100x, where x can assume any value)

Each student received a grade of 7,8,9,10.

Given that ATLEAST 8 ( grade of 8,9,10) = 50% (100) = 50

So, grade 7 = 50

If atleast 9 (9 or 10) equals 30%, then exactly 8 = 50%-30% = 20% = 20.

Exactly 10 = 10% = 10.

So, exactly 9 = 20.

Among the 50 students, we select 2 = 50 C 2.

We need to select 1 from score of 9 and another from Score of 10.

20 C1 * 10 C1 = 20*10

Probability = (20*10) / (50C2)

= (20*10*2)/(50*49)

= 8/49

= approx 8/50

= approx 16/100

=  0.16

Option E

Edskore · Answer

Key concept being tested: Conditional probability with overlapping percentage ranges — specifically calculating probabilities within a subgroup, not the full population.

The trap that gets most people: the problem says two students are chosen from the group who scored at least 8. Many students compute the probability using all students as the denominator (1,000 or 100), which gives the wrong answer. The denominator must be only the at-least-8 subgroup.

Step 1 — Map out the grade distribution using 100 students (cleaner numbers).
- At least 8 (grades 8, 9, or 10): 50 students
- At least 9 (grades 9 or 10): 30 students
- Exactly 10: 10 students
- Exactly 9: 30 - 10 = 20 students
- Exactly 8: 50 - 30 = 20 students

Step 2 — Identify the selection pool.
Two students are chosen from the at-least-8 group: 50 students total.

Step 3 — Count the favorable outcomes.
We want exactly one score-9 student AND one score-10 student.
Ways to pick 1 from the 20 score-9 students x 1 from the 10 score-10 students:
20 x 10 = 200 favorable pairs

Step 4 — Count total possible outcomes.
Choose 2 from 50: C(50, 2) = (50 x 49)/2 = 1,225

Step 5 — Calculate probability.
P = 200/1,225 = 8/49 = approx 0.163 = approx 0.16

Answer: E

Why C(50,2) and not 50 x 49? Because the order doesn't matter — "Student A scores 9, Student B scores 10" is the same selection as the reverse. Using permutations here (50 x 49 = 2,450) gives 200/2,450 = 0.08, which is option C — a classic trap answer.

Takeaway: When a probability problem restricts the selection to a subgroup, always reset your denominator to that subgroup's size, and ask yourself whether order matters before deciding between combinations and permutations.

— Kavya | 725 (99th percentile), GMAT Focus Edition

Kinshook · Answer

In an online sociology course with thousands of students, each student received a grade of 7, 8, 9, or 10. 50% of the students received a score of at least 8, 30% received a score of at least 9, and 10% received a score of 10. Two students are chosen at random from those who received scores of at least 8. 
Which of the following is closest to the probability that one of the selected students received a score of 9 and the other received a score of 10?

% of students who got at least 8 = 50%
% of students who got score of 9 = 30% - 10% = 20%
% of students who got score of 10 = 10%
Total ways to select 2 students = 50%*50% = .25
Favorable ways (one of the selected students received a score of 9 and the other received a score of 10) = 2*20%*10% = .04

The probability that one of the selected students received a score of 9 and the other received a score of 10 = .04/.25 = .16

IMO E

In a large online introductory sociology course with thousan

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