The Mathematics Behind the Coin Toss

Awareness of my proclivity for mathematical formulas and years spent as a child learning to game the system with countless how-to magic guides makes me wonder if I'm the only one who will find this interesting. I hope not. In his latest column for ABC, John Allen Paulos explains how to get a fair result from a biased coin toss:

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.

This isn't the first piece on coin tossing I've seen this month. There's a series of quick tricks you can learn when using a regular coin, points out Katharine Gammon in the new issue of Wired, if you're the one looking to do the cheating. If you don't want to be bothered with learning new tossing tricks, though, just always pick the side that's facing up: Because it spends more time face-up as the coin is spinning, the probability of that side winning the toss is 51 percent.

Read more at ABC News and Wired.

Presented by

Nicholas Jackson is a former associate editor at The Atlantic.

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