Is a coin flip really 50/50? Almost. Heads and tails come up equally often, but a coin that is flipped by hand and caught tends to land on the side that faced up when it left the thumb. In a 2023 experiment with 350,757 flips by 48 people, that same-side share was 50.8%, a figure many headlines rounded to 51%.
The gap is small enough that it changes nothing when you flip a coin to settle a choice. It only matters in one narrow case, covered below, when the caller can see which side starts face up. The numbers here come from the study's latest arXiv version (April 2025), which was later published in the Journal of the American Statistical Association.
Why 50/50 is the textbook answer
An ideal coin has two equally likely faces, so each has a probability of 1/2. That is an assumption about the coin (balanced) and about the flip (not favoring either face). Whether real flips meet it is an empirical question.
Large historical experiments landed very close to 50%. The 2023 paper cites John Kerrich, who got 5,067 heads in 10,000 flips, and Karl Pearson, who got 12,012 tails in 24,000. Those records did not note which side the coin started on, so they could not test a small same-side effect.
Landing on the edge is also possible, but rare. Diaconis, Holmes and Montgomery cite an estimate, by Murray and Teare, that a US nickel lands on its edge about once in 6,000 tosses.
Is flipping a coin really random, or just physics?
A coin that is flipped and caught in the hand follows ordinary mechanics. In their 2007 SIAM Review paper, Persi Diaconis, Susan Holmes and Richard Montgomery describe a coin-tossing machine that, carefully adjusted, lands a coin heads up every time it starts heads up. Their conclusion is that coin tossing is "physics" and not "random". The randomness in a human flip comes from tiny differences in how the coin is launched.
Real flips also wobble. The spin axis changes direction during the flight, which physicists call precession. In the model, that wobble keeps the starting face up a little longer, so a vigorously flipped coin is slightly more likely to land the way it started.
The authors measured the wobble with high-speed photography. Of 50 flips, 27 gave usable results, and the landing probabilities computed from those flips averaged 0.508. The paper rounds that up to "about 0.51". So is a coin flip 51/49? Only as a model estimate based on 27 flips, not as a measured rate. The measurement came in 2023.
Spinning a coin on its edge is a different method, and the same paper finds it far from fair. In a class of 103 Berkeley students, tossed pennies clustered near 50 heads out of 100. Spun pennies showed many more tails, and several students got fewer than 10% heads.
What 350,757 flips showed
The experiment was led by František Bartoš and Eric-Jan Wagenmakers at the University of Amsterdam with 48 co-authors. It was designed to test the 2007 prediction directly. The authors say it dwarfs all previous efforts they know of.
How the experiment worked
Forty-eight people flipped coins of 44 different currency and denomination combinations. Five bachelor students contributed 75,036 flips, 35 people at on-site "marathons" of up to 12 hours contributed 203,440, and seven people who answered a Twitter call contributed 72,281. Each person collected sequences of 100 flips:
- The starting side of the first flip is chosen at random or by an algorithm.
- The coin is flipped and caught in the hand, and the landing side is recorded.
- The next flip starts from the side the coin just landed on.
- A flip that is not caught is discarded and repeated from the same starting side.
Participants filmed their flips and uploaded the videos, and the authors audited them. The audit found nothing suspicious.
What it found
The table summarizes the main measurements. Intervals are the 95% credible intervals from the authors' Bayesian analysis, which roughly show how uncertain each estimate is.
| Measure | Result | What it means |
|---|---|---|
| Flips that landed on the starting side | 178,079 of 350,757 = 50.77% (interval 50.60% to 50.94%) | Coins land on the starting side slightly more than half the time |
| After allowing for differences between people | 50.98% (interval 50.50% to 51.47%) | Slightly higher average, with a wider interval |
| Flips that landed heads | 175,421 of 350,757 = 50.01% (interval 49.85% to 50.18%) | No difference between heads and tails |
| Raw rate per person | 48.7% to 60.1% | The effect varies a lot from person to person |
| Practice | The effect shrank as people flipped more | The authors suggest about 10,000 flips might almost remove it |
An effect this size is very unlikely to appear by chance in 350,757 flips, and the authors call the evidence strong. The per-person rates come from small samples, so some of the spread is noise.
Do heads or tails come up more often?
Neither. The starting side of each sequence was randomized, so the same-side tendency favors heads and tails equally. That is why the heads share came out at 50.01%, and the data gave no sign that it differs between coins. So is a coin flip actually 50/50 for heads versus tails? Yes.
What the study does not show
- It does not show that one face of the coin is heavier or that heads comes up more than tails. The bias is tied to the starting side.
- It does not give a rate that applies to everyone. 50.8% is an average, and some people show almost no bias.
- It covers only flips caught in the hand. Coins that bounce on a floor or spin on a table are outside it.
- The participants were not randomly selected. They were students, marathon participants and Twitter volunteers.
- Participants knew the hypothesis. The authors consider deliberate manipulation unlikely but cannot rule it out.
- The practice effect came from an exploratory analysis. Fatigue and waning attention are not excluded as explanations.
What a 50.8% bias means in real flips
Here is a hypothetical example. Suppose one person always starts the coin heads up and gets heads at the study's average same-side rate of 50.77%. The table compares that person with a fair coin at 50%. The standard deviation of a fair coin is √(N × 0.5 × 0.5), and the last column shows how many standard deviations the extra heads amount to.
| Flips | Expected heads, fair coin | Expected heads at 50.77% | Difference | Fair-coin standard deviation | Difference / standard deviation |
|---|---|---|---|---|---|
| 100 | 50 | 50.8 | 0.8 | 5.0 | 0.15 |
| 1,000 | 500 | 507.7 | 7.7 | 15.8 | 0.49 |
| 10,000 | 5,000 | 5,077 | 77 | 50.0 | 1.54 |
| 100,000 | 50,000 | 50,770 | 770 | 158.1 | 4.87 |
These values are calculated, not measured, and assume every flip is made by the same person with the same bias. Real tossers differ.
In 1,000 flips the extra heads amount to about half a standard deviation. Even a fair coin gives 508 or more heads 31.8% of the time, so one run of 1,000 flips cannot reveal the effect. Under a normal approximation, it takes about 16,000 flips just to shrink the margin of error to the size of the effect. Earlier records did not note the starting side, and an effect this small gets lost in the noise of short series.
The authors also translate the effect into betting terms. If you bet $1 on a flip that pays $2 for a win and $0 for a loss, and repeat it 1,000 times, knowing the starting side would earn you about $15 on average at 50.77%. The authors, using a slightly higher estimate, give $19. They compare that to the casino edge in six-deck blackjack ($5 per 1,000 comparable bets) and single-zero roulette ($27), and they suggest hiding the starting side when the stakes are high.
Is an online coin flip 50/50 too?
Yes, by design, but not because it is "fairer" than a real coin. An online coin flip uses a pseudo-random number generator. In Chrome and other V8-based browsers, JavaScript's `Math.random()` has used the xorshift128+ algorithm since Chrome 49, and the V8 team reports that it passes the TestU01 test suite. MDN notes that it is not cryptographically secure and should not be used for security purposes.
A simulator has no starting side and no wobble, so the physical bias measured in the experiment has nothing to act on. What replaces it is the algorithm's own pseudo-randomness, where each result comes from a number sequence and the browser picks the seed. For everyday choices the difference does not matter. Our coin flip tool explains the simulator versus a real coin in more detail, and the random number generator covers choices with more than two options.
Law of large numbers and the gambler's fallacy
The law of large numbers says that the share of heads gets closer to 1/2 as the number of flips grows. It does not say that the number of heads and tails becomes equal. As Wikipedia summarizes it, the absolute gap between the two counts tends to grow with the number of flips, and only the gap's share of all flips shrinks.
Kerrich's 10,000 flips show it. His 5,067 heads are a 50.67% share, but the heads-minus-tails gap is 134. In 1,000,000 flips the typical gap between heads and tails would be around 800, yet the heads share would still stay within about 0.04 percentage points of 50%. You can see the spread for your own numbers with the standard deviation calculator.
The gambler's fallacy is the belief that an independent event that has come up less often than expected is due soon. As Wikipedia explains, after four heads in a row, the next flip of a fair coin is still 1/2. Streaks are normal. A script run for this article gives an 81% chance of at least 5 heads in a row somewhere in 100 fair flips, and 97% for at least 5 of the same face. The same-side bias in the study depends on which side faces up before the flip, not on earlier results, so it does not make tails due either.
Is a coin toss fair for everyday decisions?
Yes. In a coin toss, the measured gap stayed under one percentage point across 350,757 flips, and heads and tails came out even. The bias gives a small edge only if the person flipping knows the starting side and the person calling can choose accordingly. The authors recommend hiding the starting side when the stakes are high.
In practice, that means the caller should not see which side faces up as the flip begins. Calling before the coin is shown, or having the flipper pick the starting side at random each time, covers it. Spinning the coin on a table is not a fair method. For everyday decisions none of this is needed, because a 0.8-point gap does not make a single flip predictable.