A release-checked medium problem for training Expected Value, Combinatorics.
Question
An electrician sets up $12$ light bulbs on a circle. Each pair of light bulbs has a wire connecting them. Each individual wire is active with probability $\dfrac{1}{2}$, independent of all other wires. A collection of $2 \leq k \leq 12$ light bulbs is said to form a complete $k-$circuit if between any 2 light bulbs in the circuit, we have an active wire. A lightbulb can be part of multiple circuits. Find the expected number of $4-$circuits in our electrician's arrangement of light bulbs.
Practice focus
This Probability problem is tagged Expected Value, Combinatorics. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.