Use a voltmeter and see if you have power at the connection shown in this picture. It is the power feed for the ball in play lights that is attached to the Ball Count Unit. If you do, take a jumper wire and attach one end to this connection, and touch each of the connections at the top of the Ball Count Unit. As each is connected, the ball in play lamp should light as you move from one to the other.
Also make sure the wiper on the Ball Count Unit makes perfect contact with each contact as it advances.