From now on, I'll assume that the coin is a fair one. It's easy to use such a coin to obtain probabilities of the form m/2^n (1/4, 7/8, 5/16, etcetera) by flipping a coin n times. But how can we obtain a probability of exactly 1/3. Von Neumann's trick can be modified to give us an answer.
Flip the fair coin twice. As mentioned, the four equally likely outcomes are H-H, H-T, T-H, T-T, where once again the order indicates the order of the two outcomes. Count H-H as a success, H-T and T-H as failures, and T-T as calling for another two flips. In this way one out of the three outcomes counts as a success, so the probability of success is 1/3, and the probability of failure is 2/3. Comparable tricks can be devised to yield any probability m/n we might want.
Sequences of coin flips can also be used to illustrate many ideas from statistics, but these last two puzzles about sequences are less well-known. Imagine that you continue to flip a fair coin and generate a sequence of Hs and Ts.
The problem is that you must bet on whether T-H is more, less, or equally likely to come up before H-H in the sequence of coin flips. Note in the particular string H-T-H-T-T-H-H-T-T-..., that T-H comes up after 3 flips and H-H after 7 flips.
A little thought makes clear that T-H will usually come before H-H since the only way H-H can come first is for the first two flips to be H-H. This occurs with probability 1/2 x 1/2 = 1/4. If this doesn't happen, the next H must be preceded by an T and so T-H will come up first. Thus T-H comes up before H-H 3/4 of the time.
Finally, a couple of much trickier questions of the same type for math aficionados: Continue with a string of fair coin flips. Which sequence is likely to come up first: T-H-T or T-H-H? How many times, on average, must you flip a coin before T-H-T comes up. And how many times, on average, must you flip a coin before T-H-H comes up? The answers are below.
Although coin flips may appear abstract and simplistic, they can be used to clarify almost any scenario in which probability plays a role. One can do worse than adopt a flip view of life.
Answers: T-H-T and T-H-H are equally likely to come first in a sequence of fair coin flips (they both begin with a T-H). Nevertheless, the average number of flips before T-H-H comes up is 8, whereas the average number of flips before T-H-T comes up is 10. (The counterintuitive result may be found by using conditional expectations and deriving a recursion formula for them or by simulation.)
John Allen Paulos, a professor of mathematics at Temple University in Philadelphia, is the author of the best-sellers "Innumeracy" and "A Mathematician Reads the Newspaper," as well as, most recently, "Irreligion." He's on Twitter and his "Who's Counting?" column on ABCNews.com usually appears the first weekend of every month.