FLIP A COIN: IS THE 50/50 PROBABILITY OF COIN TOSS A MYTH?

October 5, 2023 John Miller

It could be playing cricket or simply having a random bet; the first thing that comes to mind is tossing a coin. Well, our statistics classes in school taught us how tossing a coin was a 50/50 probability match. But is it really? Contrary to what we have been taught is Sir Persi Diaconis’s opinion. He states that with the physics involved and the uneven distribution of weight while crafting coins, there is never a 50/50 probability.

50/50? That’s just a myth!

Sir Persi Diaconis, a math and stats professor at Stanford University, used to be a professional magician. His years of research led him to the conclusion that a 50/50 probability of coin tosses was simply a myth. What did this exactly mean? Well, what he wanted to say is, supposedly, if you flip a coin 50 times, you will always get an uneven number of heads and tails. It’s always going to be 51/49!

Now, this does not happen randomly. He gave out his theory in his research paper, Dynamical Bias in the Coin Toss, to prove his point. 

• The experiment used a mechanical spring to toss coins into the air evenly each time. They tested the mechanism to make it mimic a real coin flip, and the results were not exactly 50/50; it was close, with a 51/49 split in favor of the side that started up. To prove this, he even trained himself to get heads up 10 out of 10 times he flipped a coin.
• The experiment included spinning the coins, which had a significant effect, especially on pennies. When spun, there was an 80% chance that pennies would land tails up because of the weight distribution between Lincoln’s head and the monument on the back.

The factors that act and affect the coin flip

• Coin Flip Physics: 

Factors like how hard you flick a coin? What angle was it flipped at? What does the coin look like? All affect the result when you flip a coin.

• Where It Lands Matters: 

Another thing that affects what the outcome would be after we flip a coin is what surface is it landing on. Soft surfaces sink in contact with the coin, while on hard surfaces, your coin will topple. 

• Coins Aren’t Perfect: 

A major factor is the design and weight distribution of a coin. Which side of the coin is heavy, whether the edges are nicely evened or not, all affect the end results.

• Big Numbers Matter: 

Even though individual flips can be all over the place due to these factors, when you flip a coin many, many times, the results tend to even out and get close to 50-50. This is a big idea in probability called the law of large numbers.

All these factors point out how coin tosses aren’t that reliable. This is what Diaconis proved.

Are coin tosses in sports like cricket fair?

Diaconis made his point clear that it is never a 50/50 probability. This leads us to another important question: are tosses at sports events fair? 

• Players might choose the side that’s facing up while tossing, as it would fall on that side only
• The side that’s heavier will always land facing downwards
• If a coin is caught in the air, there is a 51% chance that the side that was facing up will land upwards

But this is not at all fair! Then how does the sports council fight this bias? Thus came the concept of Unbias coins.

Bias and unbiased coins

A biased or unfair coin is one that gives us a probability of more than ½, which means there are more chances of one side coming up than the other. while an unbiased coin has an exact ½ probability of each side coming up when you flip a coin. Here’s a clear difference between these coins,

Characteristics	Biased Coin	Unbiased Coin
Also Called	Unfair Coin	Fair Coin
Probability	Two Sides Have Different Probabilities	Two Sides Have the Same Probability
Probability Value	Probability of One Side ≠ 1/2	Probability of One Side = 1/2
Construction	Uneven weight on sides and center	Even weight from all sides

In every sports meet where they have to flip a coin to decide the winner, an unbiased coin is used. These coins have an equal probability of each side falling. Moreover, the coin is not allowed to be caught in the air; it is allowed to drop on the ground so that there is only a 50% chance that the side that was tossed facing up is received at the end of the toss.