Random wire joins forming one 10-mile loop

Ten parallel wires, each $1$ mile long, have their left ends on one side of a wall and their right ends on the other. One electrician, who sees only the left ends, joins them up into $5$ random pairs; another, who sees only the right ends, independently joins those into $5$ random pairs. What is the probability that the result is a single closed loop $10$ miles long?

$128/315$. Follow the circuit from one wire. Each time you cross to the far side, the end you arrive at must be joined to a wire not yet visited: the chances are $\tfrac89,\tfrac67,\tfrac45,\tfrac23$ on successive steps (the right-side pairing can be taken as fixed). The product is $\tfrac{128}{315}$.