More Coin Tosses

Suppose we are really ambitious about tossing a fair coin. Indeed, suppose we toss a fair coin 1000 times, and for each head we step one unit distance radially away from a lamppost, and for each tail we step back radially the same unit distance. About how many times would you expect to be at the lamppost?

Answer

Most people would expect to return to the lamppost quite often—20 or more times during the 1,000 tosses. However, returning more than 2 times is unlikely! There will be a long drift away from the lamppost for most of the coin-tossing time.

One can do the actual coin tossing to experience the drift away from the origin for long time periods, or one can run a computer simulation. The expected distance after N tosses will be  times the unit step distance, the random walk distance in one dimension.