NECO 2002 Mathematics · Question 3
The probability that a student will pass a mathematics test is ⅗. If 3 students are chosen at random, what is…
Question
The probability that a student will pass a mathematics test is ⅗. If 3 students are chosen at random, what is the probability that at least 2 of them will pass the test?
- A.
- B.
- C.
- D.
- E.
Answer: E.
Explanation written with AI and checked against the answer key
Analyze the question: We have 3 students, each with probability of passing p = 3/5 (and failing q = 2/5). "At least 2 pass" means exactly 2 pass or exactly 3 pass. This is a binomial probability problem.
Concept: P(exactly r successes in n trials) = nCr · p^r · q^(n−r)
Solution:
Exactly 2 pass: 3C2 × (3/5)² × (2/5) = 3 × 9/25 × 2/5 = 54/125
Exactly 3 pass: (3/5)³ = 27/125
Add them: P(at least 2) = 54/125 + 27/125 = 81/125
Check using the complement:
- P(0 pass) = (2/5)³ = 8/125
- P(exactly 1 pass) = 3 × (3/5) × (2/5)² = 36/125
- 1 − (8/125 + 36/125) = 1 − 44/125 = 81/125 ✓
Tip: For "at least" questions, either add the favourable cases or subtract the unfavourable ones from 1, whichever has fewer terms. Always check by doing it both ways. Also note that 27/125 (option A) is only the "all 3 pass" case, a common trap.
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