The Probability of a Coin Flip

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:

1. Each flip is independent. The coin has no memory.
2. Streaks happen. Long runs of one outcome are expected, not anomalous.
3. 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 coin tends to land on the side that started face-up slightly more than half the time. The effect is small — usually a percent or two — but it is real and has been measured in large-scale studies.
• 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:

1. The underlying random source is uniform — every output is equally likely.
2. 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?

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.

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.

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.