Who's Counting: Crooked Coins, Fair Probabilities and Strange Sequences

Nov. 7, 2010 — -- I've always loved the idea of coin flips, and hence this paean to a few of their properties.

One of the most basic tools in the study of probability and its applications, coin flips reflect or model many profound dichotomies, ranging from yin-yang and win-lose to yes-no and 0-1.

A more technical property of coin flips is, however, more useful. They are independent in the sense that the outcome of one flip has no influence on the outcome of any other. When two (or more) events are independent in this sense, the probability that they both (or all) occur is computed by multiplying the probabilities of the individual events.

Thus, for example, the probability of obtaining two heads in two flips of a fair coin is 1/2 x 1/2 = 1/4.

Of the four equally likely possibilities (first flip, second flip) -- T-T; T-H; H-T; H-H - only one of the outcomes is two heads.

For the same reason the probability of obtaining two heads followed by a tail and then another head (H-H-T-H) is 1/2 x 1/2 x 1/2 x 1/2 = 1/16, and the probability of five straight flips of a fair coin resulting in heads (H-H-H-H-H) is 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/32.

Alas, people sometimes use crooked or biased coins, either knowingly or in an attempt to cheat others.

To obtain a fair result from a biased coin, the mathematician John von Neumann devised the following trick. He advised the two parties involved to flip the coin twice. If it comes up heads both times or tails both times, they are to flip the coin two more times.

If it comes up H-T, the first party will be declared the winner, while if it comes up T-H, the second party is declared the winner. The probabilities of both these latter events (H-T and T-H) are the same because the coin flips are independent even if the coin is biased.

For example, if the coin lands heads 70 percent of the time and tails 30 percent of the time, an H-T sequence has probability .7 x .3 = .21 while a T-H sequence has probability .3 x .7 = .21. So 21 percent of the time the first party wins, 21 percent of the time the second party wins, and the other 58 percent of the time when H-H or T-T comes up, the coin is flipped two more times.