Question

Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions?

$1.6$

$1.8$

$2.0$

$2.2$

$2.4$

Solution:

After Chris chooses an adjacent pair, Silva has $5$ equally likely adjacent pairs to choose.

If Silva chooses the same pair, all $5$ balls return to their original positions. This has probability $\frac15$ .

If Silva chooses a pair sharing exactly one ball with Chris's pair, then $2$ balls are in their original positions. There are $2$ such pairs, so this has probability $\frac25$ .

If Silva chooses a disjoint adjacent pair, then $1$ ball is in its original position. This also has probability $\frac25$ .

The expected number is $5\cdot\frac15+2\cdot\frac25+1\cdot\frac25=\frac{11}{5}=2.2.$

Thus, the answer is D .

2021 AMC 10B Fall Problem 16

Solution:

Problem 16 in Other Years