Coin flip
Heads or tails for one coin, a handful or 10,000, or a best-of-three series, with the odds of what came up.
How each flip is made
Each coin is one random bit from the Web Crypto generator built into your browser: 0 is heads and 1 is tails. The bit is unbiased, so each side has exactly a 1 in 2 chance, and each coin in a batch is drawn separately, so no flip affects another. Flipping 10,000 coins at once gives the same results as flipping one coin 10,000 times.
For a best of three or five, the tool flips one coin at a time and stops as soon as one side has a majority (two of three, or three of five), the same way a series is played by hand. The flips are shown in order, so you can see whether it went the distance.
How many heads to expect
In n flips, the number of ways to get exactly k heads is the binomial coefficient C(n, k), and every sequence of flips has the same chance, 1 in 2n. With ten flips there are 1,024 sequences:
exactly kC(n, k) ÷ 2n5 of 10: 252 ÷ 1,024
| Heads in 10 flips | Ways | Exactly | This many or more |
|---|---|---|---|
| 0 | 1 | 0.10% | 100% |
| 1 | 10 | 0.98% | 99.90% |
| 2 | 45 | 4.39% | 98.93% |
| 3 | 120 | 11.72% | 94.53% |
| 4 | 210 | 20.51% | 82.81% |
| 5 | 252 | 24.61% | 62.30% |
| 6 | 210 | 20.51% | 37.70% |
| 7 | 120 | 11.72% | 17.19% |
| 8 | 45 | 4.39% | 5.47% |
| 9 | 10 | 0.98% | 1.07% |
| 10 | 1 | 0.10% | 0.10% |
An exact half becomes less likely as the number of flips grows, even though the proportion of heads settles closer to 50%. Exactly 50 heads in 100 flips happens 7.96% of the time, exactly 500 in 1,000 2.52% and exactly 5,000 in 10,000 0.80%. Yet 40 to 60 heads in 100 flips covers 96.48% of outcomes, and 4,900 to 5,100 heads in 10,000 flips covers 95.56%.
Streaks are more common than they look
A run of heads seems suspicious, but in a long sequence there are many places for one to start. The chance of at least one run of a given length is worked out by tracking the length of the run in progress after each flip, which gives these exact figures:
| At least one run of | 10 flips | 20 flips | 50 flips | 100 flips |
|---|---|---|---|---|
| 4 heads | 24.51% | 47.80% | 82.74% | 97.27% |
| 5 heads | 10.94% | 24.99% | 55.19% | 81.01% |
| 6 heads | 4.69% | 12.23% | 31.46% | 54.61% |
| 7 heads | 1.95% | 5.82% | 16.53% | 31.75% |
| 5 of either side | 21.68% | 45.84% | 82.09% | 97.17% |
| 7 of either side | 3.91% | 11.51% | 30.89% | 54.23% |
In 100 flips the longest run of the same side averages 6.98, and it is more likely than not to be seven or longer. Ten heads in a row starting from a given flip is 1 in 1,024, but somewhere within 100 flips it happens 4.41% of the time. When you flip many coins, the tool shows the chance of a run at least as long as your longest one.
Best of three and best of five
Playing a series does not favour either side: each still wins half the time. It changes how many flips are needed.
| Series | Over in the fewest flips | One flip more | Goes the distance |
|---|---|---|---|
| Best of three | 50% (2 flips) | – | 50% (3 flips) |
| Best of five | 25% (3 flips) | 37.5% (4 flips) | 37.5% (5 flips) |
Do real coins land 50–50?
Not quite, and the bias is about which side was facing up at the start, not heads against tails. In 2007, Persi Diaconis, Susan Holmes and Richard Montgomery published a physical model of coin tossing in SIAM Review (“Dynamical bias in the coin toss”, vol. 49, pp. 211–235). A tossed coin does not simply spin end over end; it also wobbles (precesses), so it spends slightly more of its flight with the starting side up. Using high-speed photography of real flips, they estimated that a coin flipped and caught by hand lands the way it started about 51% of the time.
A team led by František Bartoš tested the prediction directly. Forty-eight people tossed coins from 44 different currencies and denominations a total of 350,757 times and recorded which side was up before and after. The coins landed on the same side they started 50.8% of the time (95% credible interval 50.6% to 50.9%), close to the model’s prediction. Heads and tails themselves came up evenly: 175,421 heads, or 50.0%. The study was posted as a preprint in October 2023 and published in the Journal of the American Statistical Association in 2025 (vol. 120, pp. 2118–2127).
So if you know which side is up before a hand toss, calling that side gives about a 50.8% chance instead of 50%. The edge is small, but it is real, and it disappears if the tosser hides the starting side. A software flip has no starting side at all. The guide to whether a coin toss is fair covers the experiments in more detail.
Questions
Is an online coin flip really 50–50?
Yes. Each flip is one random bit from your browser’s cryptographic generator, so heads and tails each have exactly a 1 in 2 chance and no flip depends on the one before. A real coin tossed by hand is very slightly less even, as the section on real coins explains.
After five heads in a row, is tails more likely?
No. The coin has no memory, so the sixth flip is still 1 in 2 either way. Believing otherwise is the gambler’s fallacy. Long runs are simply more common than most people expect; in 100 flips there is an 81% chance of at least one run of five or more heads.
Why use best of three?
It doesn’t change the odds, since each side still has a 50% chance of winning the series, but it makes a single unlucky flip less decisive and gives a result that feels more earned. A best of three needs a third flip half the time.
What is the longest run in the tally?
The longest stretch of the same side in a row across all your flips this visit, joined up from one flip or batch to the next. With 100 flips, the longest run of either side averages about seven.
Can I use this to settle a decision or start a game?
Yes. Agree who has heads before flipping, then flip once or use best of three. If you need to choose among more than two options, the dice roller or the random name picker gives each option the same chance.
Are the flips saved?
Only on this page while it stays open. The tally disappears when you leave, and a link copied from the address bar keeps the number of coins, not the results.