Flipping a coin looks like pure luck, but the odds behind it are actually a precise branch of mathematics called the binomial distribution. This coin flip probability calculator turns that math into one simple result: given a number of flips and a target number of heads, it tells you exactly how likely you are to see that outcome — and also how likely you are to see at least that many, or at most that many.
Whether you are settling a bet, modeling a streak, or just curious about what "fifty-fifty" really means after ten tosses, the numbers update the moment you move a slider. The expected number of heads and the standard deviation give you a sense of how far a real run of flips is likely to wander from the average.
How to Use
- Set the Number of Flips — how many times you intend to toss the coin. Ten flips is the default, but you can go up to large numbers to see the distribution smooth out.
- Enter the Heads You Want — the exact count of heads whose probability you want to measure. For ten flips, the most likely single outcome is five heads.
- Adjust the Chance of Heads if your coin is not fair. A normal coin sits at 50%; a weighted or biased coin might be 60%, 70%, or anything between 0% and 100%.
- Read the result card. The highlighted line is the probability of getting exactly the number of heads you asked for. Below it you get the odds of getting at least that many and at most that many, plus the expected value and spread.
The math uses the binomial probability mass function. If you want the chance of a range rather than a single number, use the "at least" and "at most" lines — for example, the odds of getting five heads or more out of ten flips is about 62.3%, not the 24.6% that the exact figure suggests.
The Binomial Formula
Each coin flip is an independent trial with two possible results: heads or tails. When you run n flips and want exactly k heads, the probability is:
P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ
Where:
- n is the number of flips.
- k is the number of heads you want.
- p is the chance of heads on a single flip (0.5 for a fair coin).
- C(n, k) is the combination "n choose k," the number of different ways to arrange
kheads amongnflips.
The expected number of heads is simply n × p, and the typical spread around that average — the standard deviation — is √(n × p × (1 − p)).
Worked example on ten fair flips aiming for five heads: C(10, 5) = 252, p⁵ = 0.03125, and (1 − p)⁵ = 0.03125. Multiplying gives 252 × 0.03125 × 0.03125 = 0.2461, or 24.6%. The expected number of heads is 10 × 0.5 = 5, and the standard deviation is √(10 × 0.5 × 0.5) ≈ 1.58. That means most ten-flip runs land between roughly three and seven heads.
Why "Exactly" Is So Small
A common surprise: the single most likely outcome in ten fair flips — five heads — still happens only about 24.6% of the time. The other 75% of the time you get four, six, or some other count. This is the nature of probability: the peak of the distribution is narrow, and real results scatter around it.
That scatter grows with the number of flips, but proportionally it shrinks. With 1,000 flips you expect 500 heads, and the standard deviation is about 15.8 — so nearly all runs fall between 470 and 530 heads. With 1,000,000 flips you expect 500,000 heads with a spread of about 500, which is a tiny fraction of the total. This is the law of large numbers in action: the more you flip, the closer the actual proportion of heads gets to 50%.
Examples
Example 1 — Ten fair flips, five heads. Set flips = 10, heads = 5, chance = 50%. The exact probability is 24.6%, the odds of five or more heads are 62.3%, and the expected count is 5. If someone bets you "exactly five heads in ten flips," remember that a one-in-four chance is a long shot, not a lock.
Example 2 — A weighted coin. Suppose a trick coin lands heads 60% of the time. Flip it 20 times and ask for 12 heads: the exact probability is about 18.0%, the chance of 12 or more is roughly 59.6%, and the expected number of heads is 12.0. Raising the chance-of-heads slider from 50% to 60% shifts the whole distribution toward heads.
Example 3 — Long-run proportion. Flip a fair coin 1,000,000 times and ask for exactly 500,000 heads. The exact probability is only about 0.08% — an extremely precise single count — yet the expected value is 500,000 with a standard deviation of 500. The exact figure is tiny, but the probability of landing near 500,000 is very high, which is why casinos and insurers rely on large numbers rather than single events.
FAQ
What is the probability of getting heads in a coin flip?
For a fair coin, the probability of heads on a single flip is 50%, and tails is also 50%. This calculator extends that single-flip odds to any number of flips, giving you the chance of seeing an exact number of heads, or a range of heads, across the whole run.
How do I calculate the probability of getting exactly k heads in n flips?
Use the binomial formula P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ, where n is the number of flips, k is the heads you want, and p is the chance of heads per flip. For ten fair flips and five heads, this works out to about 24.6%. Enter your own numbers above to get the exact result instantly.
Is it really 50/50?
A fair, balanced coin is very close to 50% heads and 50% tails, but real coins can be slightly biased, and the way you flip them can also matter. The calculator lets you set any chance of heads between 0% and 100%, so you can model a weighted coin or a coin that favors one side.
What does "expected number of heads" mean?
The expected number of heads is the long-run average: n × p. Over many sets of flips, the average number of heads tends toward this value. It is not a guarantee for any single set — ten fair flips will not always give exactly five heads — but it is the center of the distribution.
Why is the exact probability so low even for the most likely outcome?
Because a single count is narrow. In ten fair flips, five heads is the most likely result, but it still happens only about 24.6% of the time; the rest of the outcomes spread across four, six, and other counts. Use the "at least" and "at most" results to see the probability of a range, which is usually what you actually care about.
What happens as I flip the coin more times?
The expected number of heads grows linearly with the number of flips, and the proportion of heads converges toward the true chance. The standard deviation grows, but more slowly than the total, so the proportion becomes more and more stable. This is the law of large numbers: the more you flip, the closer your results get to the theoretical odds.
Disclaimer
This coin flip probability calculator is provided for informational purposes only and does not constitute gambling, financial, or statistical advice. Probabilities are computed from the binomial formula using the inputs you enter; actual coin outcomes are random and independent. Always use your own judgment for any real-world decision, and remember that past flips have no effect on future ones — the coin has no memory.