Math

# A Coin Toss Is Not As Random Or Fair As It Seems

Who gets the last slice of pizza? Flip a coin. On second thought...maybe not. The coin toss is a time-honored solution to indecision, but, weirdly, it's not a truly fair method. Better just split that slice.

## Trick Question: Heads Or Tails?

Imagine all the qualms in history ever settled by the flip of a coin. We bet few of those involved knew the odds weren't quite even. However, the odds do change with the way you go about using the coin: a spin or a flip. A spinning penny will land tails-up a staggering 80 percent of the time, according to Persi Diaconis from Stanford University's Department of Statistics, who performed a mathematical analysis on the matter in 2004. That's because a penny is slightly heavier on the heads side, skewing the center of mass and making the coin more likely to turn up tails. Coins with unserrated edges (like a nickel) tend to be slightly more biased too. (Insider info: Magicians will often shave the tails side down so the weight discrepancy even greater, making the likelihood of a tails spin even greater.)

Flipping a coin could get you a similar bias, too. If you flick a coin into the air and let it fall onto a hard surface to reveal to result, the coin often ends up spinning before settling anyway. And, as we learned earlier, a spinning coin is way more often than not a tails-up coin. Note that older pennies may not give you quite as pronounced a bias as newer ones. It's gross, but you have to account for all the dirt, grime, and other junk that can build up on coins over time and throw off the center of mass.

## Catch or Drop?

Forget heads or tails, let's ask a better question: Catch or drop? It may seem silly, but according to Diaconis, it is fairer to catch a coin that was flicked into the air than to let the coin bounce and spin on the ground until it lies flat. Your hand isn't a hard, flat surface like the ground, so it'll land wherever you stick your hand in its descending path. But, you guessed it, even that is not truly random. Diaconis holds that any flipped coin will still be slightly biased toward landing tails-up, about a 51-49 odds, to be precise. In 1986, mathematician Joseph Keller proved that one fair way to toss a coin is to throw it so that it spins perfectly around a horizontal axis through the coin's center. Doing that would require superhuman ability, so don't count on it. For us mere mortals, just stick to calling it tails. You'll probably be right.

– Numberphile

### Key Facts In This Video

1. Magicians often subtly shave down one side of a coin so it always lands on a certain side. 00:23

2. Coins with unserrated edges tend to be more biased toward a certain side when falling after spinning on its side. 01:18

3. Coins landing on the ground maybe be more biased toward landing on a certain side than coins caught in a hand. 02:39

Joanie Faletto