Question 12
You have $2003$ switches, numbered from $1$ to $2003$, arranged in a circle. Initially, each switch is either ON or OFF, and all configurations of switches are equally likely. You perform the following operation: for each switch $S$, if the two switches next to $S$ were initially in the same position, then you set $S$ to ON; otherwise, you set $S$ to OFF. What is the probability that all switches will now be ON?