Coin Toss: Is It Really Fair To Flip?

Why the '50/50 coin toss' is not always true.

coin toss coin flip

New England will take on the Rams in Monday’s SuperBowl, with both teams getting there courtesy of overtime wins in their respective Conference Championship games.

This once again ignited discussion about the NFL’s overtime rules. They do it a little differently to other sports, but the team which gets the ball first is considered to have a significant advantage.

The Patriots won the coin toss against the Chiefs and scored a touchdown, sending them straight into the Super Bowl amid some howls of protest about unfairness.

The humble coin toss is often used to illustrate a true 50:50 equation, but is that actually the case or is it actually 51:49?

The Science

Academics research many, many things, and believe it or not some have taken the time to submit this question to genuine, scientific rigour: is a coin toss really a 50:50 proposition?

There’s a 30+ page scientific paper on it, if you feel like putting yourself to sleep: Dynamical Bias in the Coin Toss, written about a decade ago by Persi Diaconis and Susan Holmes from Stanford University, and Richard Montgomery from the University of California.

So what did they find? Well firstly, pure coin tossing is physics, not random. They use the strictest of conditions and a perfectly weighted and balanced coin tossing machine (ignoring wind resistance) to demonstrate that properly placed, a coin that started heads-up will always land heads-up… 100 per cent of the time. Reliably control every other variable, and coin flipping is a certainty.

coin toss coin flip

That’s not usually the case of course. Your regular coin flip is subject to a huge amount of variables – how the flipper positions it and which side they start the flip, any imperfections and damage to the coin that causes the weight to be slightly off, how far the coin drops, etc.

But assuming all of that is appropriately random, as it would be in a regular coin flip that anybody might carry out, what did they find? From a mathematical standpoint, the results were surprising…

They found that if the coin is tossed and caught, it has about a 51% chance of landing the same way it was flipped (ie, starting “heads up” equals a 51% chance of a heads result, etc).

Yep, they concluded that the common toss of a coin is actually a 51:49 proportion, not 50:50.

That might not sound like much, but mathematically, that’s a significant edge. It’s a bigger edge than casinos rely on to turn a profit on some games.

So How Does It Work?

Basically, you have to think about what a coin is doing as it’s flipping through the air (slow it right down in your mind). If it starts on your thumb “heads-up”, it’ll then flip to tails-up, then heads-up, and so on…

Basically, you have to think about what a coin is doing as it’s flipping through the air (slow it right down in your mind). If it starts on your thumb “heads-up”, it’ll then flip to tails-up, then heads-up, and so on…

It’ll look like this:

H T H T H T H T H T H T H T H…

For as long as it’s in the air. The key point is that given it starts in the H position, at any point in the coin flip, one of two things will be true:

  • It will have spent more time in the H position (by a count of 1)
  • It will have spent an equal amount of time in both the H and T positions

At no point at all – no matter how many times it spins – will it have spent more time in the T position. It’s that point that leads to the 51:49 chance of it landing on the same side as it started: heads.

Other things they found:

If the coin is spun (rather than tossed) the weight of either side of the coin takes over and it’s far more likely to land heavier-side down. The heavier side of the coin is typically the side with more engraving detail on it (putting aside any built-up material on the coin). Coin “spins” are far from a fair proposition.

If the coin is allowed to hit the ground, the clattering and bouncing adds randomness.

A more robust coin toss (more revolutions) decreases the bias.

Our Maths Man

We asked Rod, the main behind the High-Low package and our nominated in-house maths wizard for his take on this…

“Great article! That’s pro-level coin flipping there, so meaningless to average Joe who pulls out a coin and flips it once, but definite edges there!

“51% on a flip is 2.0% P.O.T., assuming you get $2 per winner. I ran that through my simulator (100 flips x 10000 simulations) and you’re a 60/40 chance of winning level stakes, so small improvement, but not a lot. 1000 flips was 74/26, so that’s getting up there, if you can find someone who can be bothered flipping a coin that much and hasn’t worked out your trick.

“What really blew my mind though was the spinning coin. Have a read of the second paragraph on p. 8/figure 7a in the 31-page article. 96/103 students had 50/50 or worse for heads. 9/103 students had less than 10% heads from 100 spins, wow! Massive bias. That’s an American penny, so it must be heavier on the heads side (shows up tails). There’d be Aussie coins with that bias, not to mention if you wanted to shave a couple!”

Rod’s High Low has profited an amazing $91,000+ over the past 3 years, and is in the midst of a streak of 24 straight winning months.

You will struggle to find another membership on the market can claim the same sort of consistency and profitability.

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