You are in a city with streets on a perfect grid – every street is north-south or east-west. You are are driving north, and decide to randomly turn left or right with equal probability at the next 10 intersections.
After these 10 random turns, what is the probability you are still driving north?
Solution
50%
After your first turn, there is a 50% chance you are driving east and a 50% chance of driving west.
After your second turn, regardless of whether you turned east or west, there is a 50% chance you are driving north and a 50% chance of driving south.
So you can see that after every even number of turns (such as 10 turns), there is a 50% chance you are driving east and a 50% chance of driving west.