1. 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?
Solution:
$$
1 / 2^{2002}
$$

There are $2^{2003}$ equally likely starting configurations. All switches end up ON if and only if switches $1,3,5,7, \ldots, 2003,2,4, \ldots, 2002$ - i.e. all 2003 of them - were initially in the same position. This initial position can be ON or OFF, so this situation occurs with probability $2 / 2^{2003}=1 / 2^{2002}$.