Coin flip odds: how fair is heads or tails?

Is a coin flip really 50/50?

In theory, a fair coin has two equally likely outcomes, so heads and tails each have a probability of one half. A good random flip also has no memory. The result of the last flip has no effect on the next one.

Real coins are very slightly different. A study by statisticians Persi Diaconis, Susan Holmes and Richard Montgomery found that a coin tossed by hand lands on the same side it started on about 51% of the time. A digital coin flip, driven by a random number generator, has no such bias.

Why streaks are normal

People expect random sequences to alternate, but real random sequences contain runs. Five heads in a row has a probability of 1 in 32 (about 3%) for any particular five flips.

Over a longer series, runs are almost certain. In 20 flips, there is about a 46% chance of a run of at least five of the same side. In 100 flips, the longest run is around 7 on average, and a run of six or more occurs about 81% of the time.

The gambler’s fallacy

After five heads in a row, many people feel that tails is “due”. It is not. The coin has no memory, so the next flip is still 50/50. The belief that a streak must be corrected is called the gambler’s fallacy.

The opposite mistake is also common: assuming a coin is biased because of a streak. A streak of five heads in a row is no evidence of a rigged coin.

What happens over many flips

The share of heads gets closer to 50% as the number of flips grows, but the gap in the raw counts does not shrink. After 1,000 flips you would expect between 469 and 531 heads about 95% of the time, which is a swing of 31 either way from 500.

This is the law of large numbers at work. It is about proportions settling down, and it never means that the totals must even out.

Using a coin flip to decide fairly

1. Agree what each side means before you flip, so nobody can change their mind afterwards.
2. Decide whether you are using a single flip or best of three. Best of three gives each side the same chance and only takes longer.
3. Flip once and accept the result.

More than two choices

A coin only decides between two options. For three or more, use a dice roller, the random number generator or the spin the wheel tool. For a list of names, a list randomizer can shuffle them or pick winners.

A useful trick: decide by how you feel

Some people use a coin flip as a mirror for their own preference. Flip the coin and notice whether you feel relieved or disappointed by the result. That reaction can tell you what you actually wanted. You can always ignore the coin and choose what you prefer.

Try it

The coin flip tool flips one coin or up to 1,000 at once and keeps a running tally, so you can see streaks and the law of large numbers for yourself. Results are not stored.