Free tool

Flip a Coin Online

Heads or tails, no coin required. Flip once for a decision, or flip a coin 100 times in one press and watch the split settle toward an even 50/50.

Is a digital coin flip actually fair?

More even than a real one, as it happens. This flip draws from your browser's random number generator and maps the result to heads below the midpoint and tails above it. There is no memory between flips, no drift toward the side you picked last, and no thumb on the scale. Each toss is an independent event with a probability of exactly one half.

Physical coins are slightly less honest. A well-known study of coin tossing by Persi Diaconis and colleagues at Stanford found that a flipped coin has a small bias toward landing on the same face it started on, because the coin precesses about an axis rather than spinning cleanly. The effect is around 51 to 49, small enough to ignore for choosing who buys lunch and large enough to be measurable over tens of thousands of tosses. Coins spun on a table are worse still, since a slightly heavier face tends to end up down.

One caveat worth stating plainly. A browser's random number generator is pseudo-random: given the same internal state it would produce the same sequence. That is fine for settling an argument and unsuitable for anything requiring certified randomness, such as gambling or a prize draw.

Why the tally rarely looks like 50/50

Flip ten times and getting seven heads is unremarkable. It happens roughly one time in six. People read a run like that as evidence the coin is loaded, because we expect randomness to look tidy at small numbers. It does not. Streaks of four or five identical results are common, and a fair sequence that alternated neatly would in fact be the suspicious one.

The share only converges on half as the count grows, and it converges slowly, at a rate proportional to the square root of the number of flips. At a hundred flips a result anywhere between 40 and 60 heads is ordinary. At ten thousand, the band tightens to roughly 49 to 51 per cent. The counter above is a cheap way to watch this happen: press the button thirty times and watch the heads share wander, then keep going and watch it stop wandering.

The companion error is the gambler's fallacy, the belief that a run of tails makes heads overdue. The coin has no memory. After six tails the chance of heads is still one half, and it will be one half after sixty.

Flip a coin 100 times in one press

The three batch buttons run 10, 100 or 1,000 independent tosses and report the result immediately, which is the useful thing a heads or tails simulator does that a real coin cannot. Flipping a hundred coins by hand takes about four minutes and a flat surface. Here it takes one press, and the totals join the same tally as your single flips.

Batches are the quickest way to see why probability is unintuitive. Run 10 flips a few times and the heads share swings between 20 and 80 per cent. Run 100 and it usually lands between 40 and 60. Run 1,000 and it sits close to 47 to 53. Nothing about the coin changed between those runs; only the sample size did.

Anyone asked to flip a coin 100 times for a statistics exercise is usually being shown exactly this, along with the way real sequences contain long streaks. In a hundred tosses, a run of six or seven identical faces is more likely than not. A handwritten data set with no streak longer than four tends to have been invented rather than tossed.

When flipping a coin is the right method

A coin is the correct tool in a narrower set of cases than people assume, and in those cases it is genuinely better than thinking harder.

The first is when the options are close to equal. If you have weighed two choices and cannot separate them, that is information: the difference is smaller than your ability to measure it, and further deliberation will not produce a better answer, only a later one. The second is when the cost of deciding exceeds the cost of deciding wrongly, which covers most restaurant menus. The third is fairness, where what matters is not which outcome occurs but that neither party chose it.

There is also a diagnostic use. Flip the coin, look at the answer, and notice your reaction before you act on it. Relief and disappointment both tell you what you actually wanted, which is often the thing the deliberation was obscuring. In that case, ignore the coin and follow the reaction. Steven Levitt ran a large experiment along these lines, asking people facing real decisions to let a coin flip break the tie; those nudged toward change reported being happier months later, which says less about coins than about how strongly people default to the status quo.

And the cases where a coin has no business being involved: anything with an asymmetric downside, anything irreversible, anything affecting someone else's health, money or safety. Randomness is a shortcut for choices that do not matter. It is not a decision procedure.

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