The Mathematics Behind the Coin Toss

More

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.

Jump to comments
Presented by

Nicholas Jackson is a former associate editor at The Atlantic.

Get Today's Top Stories in Your Inbox (preview)

Adventures in Legal Weed

Colorado is now well into its first year as the first state to legalize recreational marijuana. How's it going? James Hamblin visits Aspen.


Elsewhere on the web

Join the Discussion

After you comment, click Post. If you’re not already logged in you will be asked to log in or register. blog comments powered by Disqus

Video

Adventures in Legal Weed

Colorado is now well into its first year as the first state to legalize recreational marijuana. How's it going? James Hamblin visits Aspen.

Video

What Makes a Story Great?

The storytellers behind House of CardsandThis American Life reflect on the creative process.

Video

Tracing Sriracha's Origin to Thailand

Ever wonder how the wildly popular hot sauce got its name? It all started in Si Racha.

Video

Where Confiscated Wildlife Ends Up

A government facility outside of Denver houses more than a million products of the illegal wildlife trade, from tigers and bears to bald eagles.

Video

Is Wine Healthy?

James Hamblin prepares to impress his date with knowledge about the health benefits of wine.

Video

The World's Largest Balloon Festival

Nine days, more than 700 balloons, and a whole lot of hot air

Writers

Up
Down

More in Technology

Just In