# “Assuming a fair coin toss”

Long the premise of every lesson on probability, coin tosses are taking a beating (thanks Boing! Boing! for the share). Some interesting research (Diaconis, Holmes, and Montgomery) uncovered these biases:

1. If the coin is tossed and caught, it has about a 51% chance of landing on the same face it was launched. (If it starts out as heads, there’s a 51% chance it will end as heads).
2. If the coin is spun, rather than tossed, it can have a much-larger-than-50% chance of ending with the heavier side down. Spun coins can exhibit “huge bias” (some spun coins will fall tails-up 80% of the time).
3. If the coin is tossed and allowed to clatter to the floor, this probably adds randomness.
4. If the coin is tossed and allowed to clatter to the floor where it spins, as will sometimes happen, the above spinning bias probably comes into play.
5. A coin will land on its edge around 1 in 6000 throws, creating a flipistic singularity.
6. The same initial coin-flipping conditions produce the same coin flip result. That is, there’s a certain amount of determinism to the coin flip.
7. A more robust coin toss (more revolutions) decreases the bias.

Always the hacker, I suggest following the link for strategies to winning a coin toss.

Is there a place for this research in my math classes?

Filed under: spherical cows-and-Other-Oversimplifications.

## 3 thoughts on ““Assuming a fair coin toss””

1. phil

Do you do game theory? If the fairness of a coin toss can be manipulated by the tosser, how can you structure the coin toss to restore fairness? Is there a dominant strategy? Is there a Nash equilibrium?

This could also be the start of a programming problem. Write an API for refereeing coin flips. (i.e., submit_coin(side), flip(num_flips), get_result()) and pit people against each other. To make things more interesting, make each contest a series of n thousand trials, each of which trial consists of n thousand flips of a single coin. Introduce random variations in coin weight, and let the flipper adjust parameters to indirectly control spin, inferring the effects from the results of previous flips. (Binomial theorem and other statistics.) The person who best figures out when they’re flipping an unfair coin will have a huge advantage.

2. phil

I should clarify. The “API” there was based on my notion of how a fair coin toss should be structured, given that the flipper can influence the outcome. I removed that because I hear that students sometimes use the Internet, and I didn’t want to spoil the pedagogy for you. If the context isn’t obvious (and you care), I can email it to you.

3. Liz

Radiolab had an interesting podcast titled Stochasticity (randomness) and it has an interesting bit about coin tosses. What it really gets at is the predictability of humans, but your blog made me think about it and I think you will find it interesting too. http://feeds.wnyc.org/radiolab