Medium

Probability

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. Find the expected number of $4-$circuits in our electrician's arrangement of light bulbs.

Notes

Hint