Probability

Coin Flip Simulator

Flip a fair or biased coin as many times as you like and watch the running proportion of heads settle toward the true probability.

Flip the coin

Probability of heads

0.5 is a fair coin. Change it to see a biased coin. Changing it does not erase flips already made.

Up to 50,000 flips in total. Flips use your browser's random number source.

Running proportion of heads

Horizontal axis: number of flips (log scale)

Flips

Heads

Tails

Proportion heads

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How to use this tool

Leave the probability of heads at 0.5 for a fair coin, or set your own value for a biased coin.
Press a Flip button to add 1, 10, 100, 1,000, or 10,000 flips. Flips accumulate until you press Reset.
Watch the blue line on the chart. It swings widely at first, then narrows toward the dashed line as the number of flips grows.
Read the summary under the tallies, which compares the result with the standard error you would expect.

Formula Reference

Proportion of heads

p̂ = (number of heads) ÷ n

Standard error of a proportion

SE = √[ p(1 − p) ÷ n ]

Expected heads

n × p

ArticleThe Law of Large Numbers ArticleThe Binomial Distribution ArticleThe Central Limit Theorem CalcBinomial Probability Calculator GlossaryGlossary: Gambler's Fallacy

Frequently Asked Questions

Why does the proportion wobble so much at the start?+

With only a few flips, a single result moves the proportion a long way: one head in two flips is 50%, but a second head makes it 100%. As n grows, each flip matters less, and the proportion settles near the true probability. This is the law of large numbers.

Does a run of heads make tails more likely next?+

No. Each flip is independent, so past results do not change the next one. Runs happen by chance, and in a long series of flips fairly long runs are normal. Believing otherwise is the gambler's fallacy.

Is the flip truly random?+

The simulator uses your browser's built-in random number generator, which is designed for good statistical randomness. It is a simulation, not a physical coin, and it is suitable for learning and demonstrations.

Does the number of heads converge to exactly half?+

The proportion converges, but the raw count does not. The difference between heads and tails can keep growing, in absolute terms, even while the proportion of heads gets closer to 0.5.

How to use this tool

Leave the probability of heads at 0.5 for a fair coin, or set your own value for a biased coin.

Press a Flip button to add 1, 10, 100, 1,000, or 10,000 flips. Flips accumulate until you press Reset.

Watch the blue line on the chart. It swings widely at first, then narrows toward the dashed line as the number of flips grows.

Read the summary under the tallies, which compares the result with the standard error you would expect.