At least one solves it — Probability Aptitude Question
At least one solves it: A can solve a problem with probability 1/2 and B with probability 1/3, independently. Answer and worked solution on CodeKairo.
- Topic: Probability
- Section: Quantitative Aptitude
- Difficulty: Medium
- Time target: 75 seconds
Question
A can solve a problem with probability 1/2 and B with probability 1/3, independently. What is the probability that the problem gets solved?
Options
- 1/6
- 1/2
- 2/3
- 5/6
Show the answer and the worked solution
Correct answer: C. 2/3
Worked solution
- P(A fails) = 1/2, P(B fails) = 2/3.
- P(both fail) = 1/2 × 2/3 = 1/3.
- P(solved) = 1 − 1/3 = 2/3.
Answer: 2/3.
The idea
For independent events, P(at least one) = 1 − product of failure probabilities. Adding 1/2 and 1/3 double-counts the case where both succeed.
Shortcut. Add and remove the overlap: 1/2 + 1/3 − 1/6 = 4/6 = 2/3. It must beat the better solver's 1/2 and stay below 1/2 + 1/3 = 5/6, and 2/3 is the only option that does.
Common trap. "5/6" — adding 1/2 and 1/3 directly, which counts the case where both solve it twice; subtracting 1/6 gives 2/3.