Coin Flip Probability Calculator
Exact probability of getting a given number of heads in n flips — for a fair or biased coin — plus the odds of a streak of k in a row. Full distribution chart and show/hide step-by-step working.
Streak probability
Probability of getting a run of k consecutive heads (or tails) somewhere within n flips.
What is coin flip probability?
A coin flip is the simplest possible random experiment: two outcomes (heads or tails), equally likely on a fair coin, and every flip independent of every other flip. That independence is what makes it the textbook example for the binomial distribution — the exact probability model for the number of "successes" in a fixed number of independent yes/no trials with a constant success probability.
Coin-flip probability, piece by piece
Exact probability of k heads
Coin flips are independent Bernoulli trials with a fixed probability p of heads. The number of heads in n flips follows a Binomial(n, p) distribution, whose PMF combines the binomial coefficient with the probabilities of the required heads and tails.
At least / at most k heads
Real questions usually ask 'k or more' or 'k or fewer' heads, which sums point probabilities across a range. The bar chart to the right shows every P(X = i) for n = 10 flips — adding up the bars from k to n gives P(X ≥ k), and from 0 to k gives P(X ≤ k).
Mean and variance
The expected number of heads is simply n·p, and the variance is n·p·(1 − p). For a fair coin this simplifies to n/2 and n/4 — so the standard deviation grows only as √n, which is why long runs of flips still cluster tightly around 50% heads.
Probability of a streak
The chance of seeing a run of k or more consecutive heads (or tails) somewhere inside n flips is not just k · pᵏ — the possible starting positions overlap. This calculator uses the exact recurrence, so short runs in long sequences come out visibly more common than most people guess.
Why streaks are more common than people expect
A single 5-heads-in-a-row block has probability (1/2)5 = 1/32 ≈ 3.13%. So people intuitively expect a 5-heads streak to be rare — and in a specific 5-flip window it is. But if you flip a fair coin 20 times, you get 16 overlapping windows where a length-5 streak could start (flips 1–5, 2–6, 3–7, …, 16–20). Those windows overlap, so we can't just multiply 16 × 3.13% — but the actual probability that at least one streak of 5 heads appears somewhere in those 20 flips is about 25%, roughly one in four.
That's the gambler's-fallacy trap in reverse: people expect streaks to be rare because a single specific streak is rare, but in a long enough sequence some streak is almost inevitable. The coin is not "on a hot streak" and it is not "due" for tails — it simply has no memory. Independence means each flip has probability p regardless of history, and streaks fall out as a mathematical consequence.
Common mistakes
- The gambler's fallacy. A run of tails does not make heads more likely on the next flip. The coin has no memory; each flip is independent.
- Confusing P(exactly k) with P(at least k). They are very different once k moves away from n/2. Always match the question to the right one.
- Assuming the "average" is the most likely single value. The mean of 10 fair flips is 5 heads, and 5 is the most likely value here — but its probability is still only ~24.6%. In most flips you'll see something other than the mean.
- Multiplying overlapping streak windows. The chance of at least one length-5 streak in 20 flips is NOT 16 × (1/32). Overlaps make the true probability harder — use the streak tool above (or the DP behind it).
Features of this calculator
- Exact probability of exactly, at least, and at most k heads in n flips.
- Fair coin (p = 0.5) by default, with a toggle for any biased probability between 0 and 1.
- Streak sub-tool: probability of a run of k consecutive heads (or tails) somewhere within n flips — computed exactly, not simulated.
- Full binomial distribution bar chart from 0 to n heads with your target highlighted.
- Mean μ = n·p, variance σ² = n·p·(1 − p) and standard deviation printed alongside.
- Show/hide step-by-step working — including the binomial coefficient and the formula substituted.
- Copy the summary as text or download the result panel as a PNG.
Frequently asked questions
+What is the probability of getting 3 heads in 5 flips?
C(5, 3) / 25 = 10 / 32 = 5/16 ≈ 31.25%.
+What is the probability of 10 heads in a row?
(1/2)10 = 1/1024 ≈ 0.098%, or roughly one in a thousand for that exact 10-flip sequence.
+How does this differ from a coin flip simulator?
A simulator generates random flips — you'll see sampling variation. This calculator returns the true probability computed from the binomial formula (or the run-length DP for streaks). No sampling error.
+Can I model an unfair coin?
Yes — untick the fair-coin toggle and enter any p between 0 and 1. Every probability, mean and streak result adjusts automatically.
+What if I want probability of tails instead?
For the binomial tool, use k = n − (target tails) and p = 0.5 (or 1 − p for a biased coin). For the streak tool, switch the toggle to tails.
+How large an n does the calculator handle?
Up to n = 2000. The formula uses log-gamma so probabilities stay numerically stable even for large n and extreme k.