"Repealing the Law of Averages"

Mathematical Recreations column by Ian Stewart, Scientific American, April 1998, pp. 102-104.

This month's discussion focuses on coin flips. Stewart discusses the law of large numbers that says the frequencies observed should in the long run be very close to their probabilities. Some interesting remarks:

Random walk theory says that the balance of heads and tails will always return to zero (starting point).

In testing this, the longest observed run out-of-balance in my simulation was 10,810,830 flips.
Imagine a 2-dimensioned random-walk. This, too, will always return to the starting point.
Stanislaw M. Ulam (co-inventor of the hydrogen bomb) proved that in three dimensions, the probability of eventually returning to the origin is about .35.

"So if you get lost in a desert and wander around at random, you'll eventually get (back) to the oasis; however, if you're lost in space, there is only a chance of one in three that you'll wander back to earth.

He notes another counter-intuitive situation. In a large, predetermined number of trials, what is the proportion of time that the number of heads is larger than the number of tails? Many persons suppose that either coin face would lead about half of the time. Actually, this would be extremely rare. Almost always, one coin side will lead most of the time. The most likely outcome is that one side leads all of the time!

—John Schuyler, March 1998.