UPSC Prelims 2005 Q119: Science & Technology – Quantitative Aptitude
- (a) 1/2
- (b) 1/5
- (c) 2/11
- (d) 1/512
Answer: (d) 1/512
Explanation
> Note for students: Printed copies of this paper show different option lists (some show 2/9 and 2/11, the stored list shows 1/512). The correct value is 2 x (1/2)^10 = 1/512. If your paper has no 1/512, skip the question.
Ten particles in one half of a box
| Quantity | Value |
|---|---|
| Chance one particle is in the left half | 1/2 |
| All ten in the left half | (1/2)^10 = 1/1024 |
| All ten in the right half | (1/2)^10 = 1/1024 |
| All ten in the same half | 2 x 1/1024 = 1/512 |
Ten identical particles are moving randomly inside a closed box, so each particle is in a given half with probability 1/2 at any given point of time, independent of the others. The probability that all ten particles will be lying in the same half of the box is 2 x (1/2)^10 = 1/512.
Answer: (d) 1/512.
Why each option is right or wrong
- (a) Incorrect. 1/2 is the chance for one particle.
- (b) Incorrect. 1/5 has no basis in the problem.
- (c) Incorrect. 2/11 has no basis in the problem.
- (d) CORRECT. 2 x (1/2)^10 = 1/512.
Key point
Independent events: multiply; count both halves.
Remember
All ten particles are in one half with probability 1/512.
Common mistakes
- Forgetting the two possible halves.
- Using 1/10 or 1/2 for all ten directly.
- Mixing up 2^10 = 1024 with 2^9 = 512.
Exam tip
Probability that n independent particles are all in one half: 2 x (1/2)^n.
Sources
- NCERT Class XI Mathematics — Probability
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Probability)