Five persons, including A and B, sit at random in a row of five chairs. If it is known that A sits at one of the two ends of the row, then the probability that B sits next to A is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{1}{4}\)
Given that A sits at an end, B is equally likely to sit in any of the other 4 chairs. This is the reduced sample space.
A chair at an end of the row has only one neighbouring chair, so exactly 1 of the 4 chairs is next to A:
\( \displaystyle P = \frac{1}{4} \)
Check by counting: A at an end gives \(2\times 4! = 48\) arrangements; A at an end with B beside A gives \(2\times 3! = 12\) arrangements, and \(\frac{12}{48} = \frac{1}{4}\).
Why the other options are incorrect
Option (1) \(\displaystyle\frac{2}{5}\) — Incorrect. This is the probability that A and B sit together when nothing is known about A’s chair.
Option (2) \(\displaystyle\frac{1}{5}\) — Incorrect. This takes B’s chair from all 5 chairs, including the one A is sitting in.
Option (3) \(\displaystyle\frac{1}{2}\) — Incorrect. This counts two neighbouring chairs, which is true only for a chair in the middle of the row.