The textbook answer
For an idealised "fair" coin, the probability of heads is 0.5, the probability of tails is 0.5, and each flip is independent of every other flip. That's the model you meet in school and the model the Coin Toss Simulator implements faithfully — it draws a fresh random number for each flip and never looks at history.
Three things follow from that model, and almost all of the confusion about coin flips comes from missing one of them:
- Each flip is independent. The coin has no memory.
- Streaks happen. Long runs of one outcome are expected, not anomalous.
- Over many flips, the proportion of heads converges to 0.5, but the absolute count of heads does not need to equal the count of tails.
Are real coins really 50/50?
Not exactly. Two physical effects nudge real-coin flips slightly off the textbook number:
- Same-side bias. A flipped coin tends to land on the side that started face-up slightly more than half the time. In 2007, Persi Diaconis, Susan Holmes and Richard Montgomery showed why. A tossed coin wobbles (precesses) in the air, so it spends a little more time with its starting face up, and their model predicted about a 51% same-side rate. In 2023, a team led by František Bartoš tested this with 350,757 hand flips by 48 people and found 50.8% landed on the starting side. The effect varied between flippers, but no coin showed a heads-versus-tails bias of its own.
- Catch-vs-bounce. If you let the coin bounce on a hard surface, the result depends on the coin's physical properties (weight distribution, edge profile). If you catch it in your palm, the same-side bias gets stronger. Spinning the coin on its edge instead of flipping is much more biased — heavier portraits often dominate.
None of this matters for casual decisions. It does matter if you're running a serious experiment or designing a game where the difference between 50% and 51% has real consequences.
What "fair" means for a digital coin
A digital coin is fair when two conditions are both true:
- The underlying random source is uniform — every output is equally likely.
- The mapping from random number to "heads or tails" doesn't introduce a skew (for example, by reading two bits and discarding one of the four outcomes asymmetrically).
This site uses crypto.getRandomValues, which is a cryptographically secure random source built into your browser, and a simple even/odd check on a 32-bit unsigned integer — both halves of the integer space are equal in size, so the mapping introduces no bias. Why that matters in detail is on how online coin flippers work.
Worked example: the eight-heads-in-a-row problem
Suppose you flip a fair coin and get eight heads in a row. What's the probability that the ninth flip is heads?
Answer: 0.5. Same as every other flip. The coin doesn't know what just happened.
Now a different question: before you started, what was the probability of getting eight heads in a row? That's (1/2)^8 = 1/256, about 0.39%. Both answers are correct because they're answers to different questions. Confusing them is the gambler's fallacy: the belief that a "due" outcome becomes more likely after a streak. It doesn't.
The flip side is the hot-hand belief: the idea that a streak makes more of the same outcome likely. For a fair coin, that's also wrong. Each flip resets to 50/50.
How long are streaks supposed to be?
If you flip a fair coin 100 times, you should not be surprised to see a run of six or seven heads (or tails) in a row somewhere in that sequence. By 1,000 flips, expected longest runs are around nine or ten. By a million flips, expected longest runs are around twenty.
This is one of the most counter-intuitive facts about randomness, and it's why people accuse random number generators of being "broken" when they see clusters. Truly random sequences look clumpy. Sequences that look perfectly alternating — H, T, H, T, H, T — are the suspicious ones.
Can a coin land on its edge?
Yes, a real one can, but very rarely. In a 1993 paper in Physical Review E, Daniel Murray and Scott Teare modelled a tossed coin and estimated that a US nickel lands on its edge roughly once in 6,000 tosses onto a hard surface. Thicker coins, relative to their diameter, land on edge more often. A cylinder nearly as thick as it is wide (more like a short stack of coins) lands on its side about a third of the time. On a soft surface such as grass or a carpet, edge landings are more likely still.
For everyday purposes the edge is ignored and the flip is re-done. The simulator has exactly two outcomes, so the question never arises there.
Odds of streaks: a quick reference
Because flips are independent, the probability of a specific run is the per-flip probability multiplied by itself: (1/2)n. If you only care that all flips match, heads or tails, double it, because there are two ways to get a uniform run.
| Run length | All heads | All the same side |
|---|---|---|
| 2 in a row | 1 in 4 (25%) | 1 in 2 (50%) |
| 3 in a row | 1 in 8 (12.5%) | 1 in 4 (25%) |
| 5 in a row | 1 in 32 (3.1%) | 1 in 16 (6.25%) |
| 6 in a row | 1 in 64 (1.6%) | 1 in 32 (3.1%) |
| 10 in a row | 1 in 1,024 (0.098%) | 1 in 512 (0.20%) |
| 20 in a row | 1 in 1,048,576 | 1 in 524,288 |
These are the odds for a run starting now. The chance that a run of that length shows up somewhere in a long sequence is much higher. That's why streaks feel common when you flip a lot (see below).
How likely is exactly 50/50?
A common surprise: although heads and tails are equally likely, getting exactly half heads gets less likely the more you flip. There are simply more near-miss results to land on. The probability of exactly n/2 heads in n flips is C(n, n/2) / 2n:
| Flips | Chance of an exact 50/50 split |
|---|---|
| 2 | 50% |
| 4 | 37.5% |
| 10 | 24.6% |
| 20 | 17.6% |
| 50 | 11.2% |
| 100 | 8.0% |
| 1,000 | 2.5% |
| 10,000 | 0.8% |
So if you flipped 62 or 162 times and didn't get a perfect split, that's completely normal. You had roughly a 90% and 94% chance respectively of not hitting it exactly. What does settle toward 50% is the percentage of heads, which is the law of large numbers.
Worked example: expected number of flips
How many flips, on average, until the first heads? Two. Each flip has a 1/2 chance of ending the wait, and the expected waiting time for an event with probability p is 1/p.
A fair coin is tossed until a head appears or five tails occur. What's the expected number of tosses? The game reaches toss k (for k = 1 to 5) only if all earlier tosses were tails, which happens with probability (1/2)k−1. Adding those up gives the expected count:
1 + 1/2 + 1/4 + 1/8 + 1/16 = 31/16 ≈ 1.94 tosses.
How many flips, on average, until two heads in a row? Six. Waiting for a head followed by a tail (HT) takes only four on average. That asymmetry sits behind Penney's game, a coin game where the second player can win most of the time.
Common mistakes
- Treating short runs as evidence. Ten flips that go 7-3 is well within normal variation for a fair coin. You need hundreds, often thousands, of flips before a small bias becomes visible against the noise.
- Assuming the next flip "owes" you the other outcome. This is the gambler's fallacy. Lottery players, roulette players, and people watching coin flips fall into it constantly.
- Conflating "improbable in advance" with "improbable now". Any specific sequence of 20 flips has the same probability as any other:
(1/2)^20. The sequence is only striking after you single it out. - Spinning instead of flipping a real coin. Spinning a coin on its edge is a much more biased process than flipping it through the air. If you want close-to-fair physical randomness, flip and let it land flat on a soft surface.
Frequently asked questions
What is the probability of each side of a coin?
For a fair coin, heads and tails each have a probability of 1/2 (0.5, or 50%). The two probabilities always add up to 1, because a flip must land on one side or the other. The rare edge landing is the only exception for a physical coin.
Is a coin flip really 51/49?
For a real coin flipped by hand, roughly. The 2023 Bartoš et al. experiment found coins landed on the side they started on 50.8% of the time across 350,757 flips. That makes the starting side about a 51/49 favourite. If you don't know which side started face-up, the bias averages out, and heads and tails are still 50/50 overall.
Is heads heavier, so tails comes up more often?
Not in a way that matters for a flipped coin. The raised design on each face shifts the centre of mass by a tiny amount. That has a negligible effect on a coin spinning through the air, and the same-side effect is much larger. Weight differences only matter when a coin is spun on its edge on a table, which is a different, much more biased process.
What does it mean that coin flips are independent?
Independent means the result of one flip gives no information about the next. The probability of heads on flip 10 is 0.5 whether the previous nine were all heads, all tails or a mix. Mathematically, P(A and B) = P(A) × P(B) for any two flips, which is why the chance of two heads in a row is 0.5 × 0.5 = 0.25.
Is the theoretical probability of heads always 0.5, even for a biased coin?
No. 0.5 is the theoretical probability for a fair coin. A biased coin has a different true probability, say 0.55, and with enough flips its observed frequency will converge to that number, not to 0.5. That's the law of large numbers.
What are the chances of flipping the same side 6 times in a row?
The chance of six heads in a row is (1/2)6 = 1/64, about 1.6%. The chance of six of the same side, either heads or tails, is 2/64 = 1/32, about 3.1%. Rare in six flips, but in a long session of flipping a run of six becomes almost certain.
Try it on the simulator
The simplest way to feel these ideas is to use them. Open the coin flip tool, flip 50 times, and watch the heads-percentage stat. It'll wander above and below 50%. Reset, flip 200 times, and watch it settle. That settling is the law of large numbers in action. The streaks you saw along the way are also expected. Both can be true at once.
Read next
Sources
- Diaconis, P., Holmes, S. & Montgomery, R. (2007). Dynamical bias in the coin toss. SIAM Review, 49(2).
- Bartoš, F. et al. (2023). Fair coins tend to land on the same side they started: evidence from 350,757 flips. arXiv:2310.04153.
- Murray, D. B. & Teare, S. W. (1993). Probability of a tossed coin landing on edge. Physical Review E, 48(4).