Sunday, 7 March 2004

Coin-toss bias

Robert Garcia Tagorda, Christopher Genovese, and Alex Tabarrok today take note of this article in Science News, which indicates that 3 researchers have found that coins, when tossed, land the same way up they started about 51% of the time.

Why hasn’t this been discovered in practice before? Interestingly, the article discusses a previous experiment with coin tossing that didn’t discover any bias:

During World War II, South African mathematician John Kerrich carried out 10,000 coin tosses while interned in a German prison camp. However, he didn’t record which side the coin started on, so he couldn’t have discovered the kind of bias the new analysis brings out.

Kerrich most likely didn’t discover the bias because some other part of his coin-tossing procedure ensured randomness. And, indeed, in a large number of trials, if there’s no bias in the starting condition (approximately equal numbers of coins are “heads” or “tails” when tossed), there will be no bias in the aggregate result—even given this finding.*

More to the point, the practical value of this finding seems minimal. The most obvious application—wagering—is precluded because no casino game that I’m aware of uses coin flips, though it’s possible that the ball in roulette and dice in craps may be similarly biased—again, given a known starting position, something that is rare in roulette at least (as the ball is under the control of the casino staff rather than the wagerers).

Proof: assume 2000 trials, 1000 starting heads and 1000 starting tails, and a .51 probability of getting the same side you started with. Of the “heads” trials, 510 will end up heads and 490 will end up tails. Of the “tails” trials, 490 will end up heads and 510 will end up tails. Thus, out of 2000 trials, you will have 1000 that result in heads and 1000 that result in tails.