Two friends seated together

Question

Eight people, including Alice and Bob, are seated uniformly at random. What is the probability that Alice and Bob sit next to each other (a) in a row of 8 seats, (b) around a circular table of 8?

Accepted Answer

(a) Row. Condition on Alice's seat. Bob is equally likely to occupy any of the other 7 seats. If Alice is at an end (2 of 8 seats), 1 of the 7 is adjacent; otherwise 2 of 7 are: P = \frac{2}{8}\cdot\frac{1}{7} + \frac{6}{8}\cdot\frac{2}{7} = \frac{2 + 12}{56} = \frac{14}{56} = \frac{1}{4}. Block check: glue Alice and Bob into one block, 7! \cdot 2 arrangements out of 8!, giving \frac{2 \cdot 7!}{8!} = \frac{2}{8} = \frac14. In general, for a row of n: P = \frac{2}{n}. (b) Circle. By rotational symmetry, fix Alice anywhere. Every seat at a round table has exactly 2 neighbors, and Bob lands on …

Two friends seated together

Your answer

More Probability questions