Coin Flip Probability Calculator
Calculate the probability of getting a specific number of heads (or tails) in a series of coin flips. Supports biased coins and shows exact, cumulative, and expected values using the binomial distribution. See also our Binomial Distribution Calculator, Probability Calculator, and Combinations Calculator.
How to Use the Coin Flip Probability Calculator
Coin flipping is the simplest example of a Bernoulli trial — an experiment with exactly two outcomes (heads or tails). When you flip a coin multiple times, the number of heads follows a binomial distribution. This calculator uses the binomial probability formula to compute exact probabilities for any number of flips and desired outcomes.
Enter the number of flips (n), the desired number of heads (k), and optionally the coin bias (probability of heads on a single flip, default 0.5 for a fair coin). The calculator shows the probability of getting exactly k heads, at least k heads, and at most k heads. It also displays the expected number of heads and standard deviation.
For a biased coin, change the bias from 0.5. For example, a coin that lands heads 60% of the time has bias = 0.6. This affects all calculations — a biased coin makes certain outcomes more or less likely than a fair coin. The calculator works for any bias between 0 and 1, allowing you to model loaded coins, weighted decisions, or any binary probability experiment.
The binomial coefficient C(n,k) counts the number of different sequences that produce exactly k heads in n flips. For example, with 3 flips and 2 heads, there are C(3,2)=3 sequences: HHT, HTH, THH. Each sequence has probability p²(1-p), and the total probability is C(n,k) × p^k × (1-p)^(n-k). This formula works for any number of flips and any bias value.
Formula
Binomial Probability (exact k heads):
P(X = k) = C(n,k) × p^k × (1-p)^(n-k)
Combination:
C(n,k) = n! / (k! × (n-k)!)
Cumulative (at least k):
P(X ≥ k) = Σ P(X=i) for i = k to n
Cumulative (at most k):
P(X ≤ k) = Σ P(X=i) for i = 0 to k
Expected Value and Std Dev:
E(X) = np, σ = √(np(1-p))
Mode:
Mode = ⌊(n+1)p⌋ or ⌊(n+1)p⌋ - 1
Example Calculation
Flip a fair coin 10 times. What is P(exactly 6 heads)?
n = 10, k = 6, p = 0.5
C(10,6) = 10!/(6!×4!) = 210
P(X=6) = 210 × 0.5^6 × 0.5^4 = 210 × 0.015625 × 0.0625
P(X=6) = 210/1024 = 0.2051 = 20.51%
P(at least 6 heads) = P(6)+P(7)+P(8)+P(9)+P(10)
= 210+120+45+10+1 / 1024 = 386/1024 = 37.70%
Expected heads: 10 × 0.5 = 5
Std dev: √(10×0.5×0.5) = √2.5 = 1.58
Reference Table
| Flips (n) | Heads (k) | Bias (p) | P(exact) | P(at least) |
|---|---|---|---|---|
| 1 | 1 | 0.5 | 50.00% | 50.00% |
| 2 | 2 | 0.5 | 25.00% | 25.00% |
| 5 | 3 | 0.5 | 31.25% | 50.00% |
| 10 | 5 | 0.5 | 24.61% | 62.30% |
| 10 | 7 | 0.5 | 11.72% | 17.19% |
| 10 | 10 | 0.5 | 0.10% | 0.10% |
| 20 | 10 | 0.5 | 17.62% | 58.81% |
| 100 | 50 | 0.5 | 7.96% | 53.98% |
Frequently Asked Questions
What is the probability of getting all heads?
For a fair coin flipped n times, the probability of all heads is (1/2)^n = 1/2^n. For 5 flips: 1/32 = 3.125%. For 10 flips: 1/1024 ≈ 0.098%. The probability halves with each additional flip. For a biased coin with P(heads) = p, the probability of all heads is p^n. This exponential decrease explains why long streaks of heads are extremely rare even though each individual flip has a 50% chance.
Does a coin have memory of previous flips?
No. Each coin flip is independent — previous results have no effect on future flips. This is a common misconception called the gambler's fallacy. After flipping 9 heads in a row, the probability of heads on the 10th flip is still exactly 50% (for a fair coin). The coin has no memory. However, the probability of getting 10 heads in a row before starting is (1/2)^10 ≈ 0.1%, which is different from the conditional probability after 9 heads.
What is a biased coin?
A biased (or unfair) coin has unequal probabilities for heads and tails. If P(heads) = 0.6, then P(tails) = 0.4. Real coins have very slight biases due to weight distribution, but for practical purposes they are considered fair. In probability theory, biased coins model any binary event with unequal probabilities — like a basketball player's free throw percentage (e.g., 80% success = bias of 0.8).
How many flips to get a reliable estimate of the bias?
To estimate a coin's bias within ±0.05 with 95% confidence, you need approximately n = (1.96/0.05)² × 0.25 ≈ 385 flips. For ±0.01 precision: about 9,604 flips. The standard error of the proportion is √(p(1-p)/n), which decreases with more flips. With 100 flips, you can distinguish a fair coin from one with 60% bias, but not from one with 52% bias. More flips give more precision.
What is the law of large numbers for coin flips?
The law of large numbers states that as the number of flips increases, the proportion of heads converges to the true probability (0.5 for a fair coin). After 10 flips, getting 70% heads is common. After 1000 flips, getting 70% heads is virtually impossible. However, the absolute difference between heads and tails tends to grow — you might expect about √n/2 more heads than tails after n flips. The proportion converges, but the count difference diverges.
How is coin flipping used in decision making?
Coin flips are used for fair random selection (who goes first), tie-breaking, and randomized algorithms in computer science. In research, randomized controlled trials use coin-flip-like randomization to assign participants to groups. Interestingly, studies show that people who flip a coin to make a difficult decision (and follow through) report higher satisfaction 6 months later, suggesting that when you're truly undecided, either choice is likely acceptable.