Is a Coin Flip Really 50/50?
Almost. A hand-flipped coin that is caught lands on the side that started facing up about 50.8% of the time, according to a 350,757-flip study. Heads against tails is an even 50/50, and a digital coin flip, which has no starting side, is a true 50/50.
What did the 350,757-flip coin study find?
In 2007 the Stanford statisticians Persi Diaconis, Susan Holmes and Richard Montgomery published a physics model of the coin toss, Dynamical Bias in the Coin Toss in SIAM Review. Real coins wobble as they spin, so they spend slightly longer in the air with their starting face on top. The model predicted that a coin tossed by hand and caught lands on the side that started facing up about 51% of the time.
In 2023 a team led by František Bartoš at the University of Amsterdam tested that prediction with a very large experiment. Forty-eight people flipped coins from 46 currencies and recorded 350,757 flips, which the researchers posted on arXiv and later published in the Journal of the American Statistical Association. The coins landed on the same side they started 50.8% of the time, with a 95% credible interval of 50.6% to 50.9%. That is very close to the 51% prediction.
The size of the effect varied from person to person. Some tossers showed a stronger same-side bias than others, and the bias tended to shrink as individuals flipped more coins, which fits the idea that practice makes the toss less wobbly.
So a hand-flipped coin is not exactly 50/50. For choosing who pays for lunch, the difference does not matter. It matters only when someone knows which side started up and can use that.
Does heads or tails come up more often?
No. The same data showed heads and tails equally likely, with the chance of heads at 0.500. The tiny bias is about which side was facing up at the start, not about which side is heads. If the start is chosen at random, heads and tails still come out even over many flips.
How big is a 0.8-point bias, and could you notice it?
A 50.8% same-side chance means about 508 same-side landings in every 1,000 flips instead of 500. Luck alone moves a count of 1,000 flips by about 3 percentage points either way, so the bias is far too small to see in a few hundred flips. The chart shows how the margin of error shrinks as flips pile up.
View the chart data as a table
| Flips | 95% margin of error |
|---|---|
| 25 | +/- 19.60 points |
| 50 | +/- 13.86 points |
| 100 | +/- 9.80 points |
| 250 | +/- 6.20 points |
| 500 | +/- 4.38 points |
| 1,000 | +/- 3.10 points |
| 2,500 | +/- 1.96 points |
| 5,000 | +/- 1.39 points |
| 10,000 | +/- 0.98 points |
| 25,000 | +/- 0.62 points |
| 50,000 | +/- 0.44 points |
| 100,000 | +/- 0.31 points |
| 350,757 | +/- 0.17 points |
The 350,757 flips in the study bring the margin down to about 0.17 points. The observed 0.8-point gap is then about 9.5 standard errors above 50%, which is far too large to be luck. That is why the researchers could say the effect is real, and why you would need roughly 30,658 flips to detect it in your own experiment.
How many flips would it take?
Change the numbers to see how much data you need. Both answers use a 95% confidence level, and the first assumes an 80% chance of catching a real bias.
How this is calculated: flips = ((1.96 x 0.5 + 0.8416 x sqrt(p x (1 - p))) / (p - 0.5)) squared for the first answer, and (1.96 x 0.5 / margin) squared for the second. Both are rounded up.
Can you see the bias in your own flips?
Not with a digital coin, and not with a handful of real ones. Use the simulator below to feel how noisy small samples are. At 10 flips, a heads share of 70% or more turns up about 1 time in 6 (17%), and the share only settles near 50% after hundreds of flips.
Flip it yourself
Each flip reads one random bit from your browser's secure random source. This digital coin has no starting side, so the true chance is exactly 50%. Watch how far the heads share wanders at 10 flips compared with 1,000.
Press a flip button to start. The 95% margin appears from 30 flips. You can run up to 10,000 flips before resetting.
Which coin flip myths are wrong?
Heads comes up more often than tails.
In the 350,757-flip data the chance of heads was 0.500, with a 95% interval of 0.498 to 0.502. Heads and tails were equally likely. The bias is toward the side that started face up, whichever that is.
After a run of heads, tails is due.
A coin has no memory. Every flip of a fair coin is a fresh 50/50, so a streak does not change the next result. Streaks are normal: in 1,000 fair flips, a run of 8 or more of the same side turns up about 98% of the time.
A digital coin flip has the same bias as a real coin.
The bias comes from the physics of a wobbling coin in the air. A digital flip has no coin, no wobble and no starting side, so it is a true 50/50 as long as the random source is unbiased.
Spinning a coin is just another way to flip it.
Spinning a coin on its edge is a different process, and it can be far from 50/50. Persi Diaconis told Science News that a spinning penny lands tails about 80% of the time, because the two faces are not perfectly balanced.
How do you make a coin flip fair?
- Keep the starting side secret. Bartoš says the only prerequisite is that neither person knows which side started facing up. Then the small bias cannot help either of them.
- Agree what each side means first. Decide who gets what before the coin leaves your hand, so nobody argues with the result afterwards.
- Flip, do not spin. A spun coin can be heavily biased, as in the myth above.
- Use a digital flip when it matters. A digital coin has no starting side, so it removes the effect completely. For a straight yes or no question, the Yes or no generator does the same job.
Is an online coin flip fair?
It can be. Our coin flip tool asks your browser for a random value from crypto.getRandomValues, the same secure source browsers use for encryption, and reads a single bit. Two outcomes divide the range of possible values exactly evenly, so there is no modulo bias, and heads and tails each have a 50% chance. There is no coin to wobble and no starting side to favour.
Flip a digital coin now, as many times as you like, with a running tally.
Open the coin flip toolTest your coin flip knowledge
Quick quiz: what do you remember?
Pick an answer to see the explanation. Nothing is saved or sent anywhere.
Questions people ask
Is a coin flip really 50/50?
Almost. A hand-flipped coin that is caught lands on the side that started facing up about 50.8% of the time, according to a study of 350,757 flips. Heads against tails is an even 50/50, and a digital coin flip, which has no starting side, is a true 50/50.
Which side should I call in a coin flip?
If you can see which side starts facing up, calling that side gives a very small edge of about 0.8 percentage points. If the starting side is hidden or the flip is digital, there is no edge, so call whichever you like.
Is an online coin flip fair?
A well-built one is. The coin flip on RandomlyHQ uses your browser's secure random source and reads one bit, so each side has a 50% chance and nothing can wobble or favour a starting side.
How many flips would show that a coin is biased?
To detect a true same-side chance of 50.8% with 95% confidence and 80% power, you would need about 30,658 flips. A bias of 51% needs about 19,620 flips.
Sources and method
- Diaconis, P., Holmes, S. and Montgomery, R. (2007). Dynamical Bias in the Coin Toss. SIAM Review, 49(2), 211 to 235.
- Bartoš, F. et al. Fair coins tend to land on the same side they started: Evidence from 350,757 flips. arXiv:2310.04153, later published in the Journal of the American Statistical Association (2025).
The margin-of-error chart, the sample-size calculator and the quiz figures are computed in code and checked against an independent power calculation. The study figures are quoted from the authors' paper.