Coin Flip Probability Calculator
Enter the number of flips, the number of heads you want to find the probability for, the comparison type (exactly, at least, at most, more than, less than), and the probability of heads per flip. Results update instantly and include the percentage chance, 1-in-X odds, and a full probability distribution chart across all possible outcomes.
Formula
Worked example
Flip a fair coin 10 times. What is the probability of exactly 5 heads? C(10,5) = 252. Then 252 x 0.5^5 x 0.5^5 = 252 / 1024 = 0.2461, or about 24.61%.
How coin flip probability works
When you flip a fair coin, there are two equally likely outcomes: heads or tails, each with a 50% chance. When you flip the same coin multiple times, the results of each flip are independent - one flip does not affect the next. The probability of getting a specific number of heads across multiple flips follows the binomial distribution, named after the two possible outcomes per trial. The binomial formula P(X = k) = C(n, k) x p^k x (1-p)^(n-k) gives the probability of getting exactly k heads in n flips, where p is the probability of heads on any single flip. C(n, k) is the binomial coefficient, which counts how many different arrangements of k heads across n flips are possible. For a fair coin, p = 0.5, so p^k x (1-p)^(n-k) simplifies to (0.5)^n for any k. For cumulative probabilities - such as "at least 5 heads" or "at most 3 heads" - the calculator sums the individual PMF values across the relevant range of k.
What each probability mode means
The five modes let you answer different questions: - Exactly k heads: the chance of getting that precise count, no more, no less. - At least k heads: the chance of k or more (includes k, k+1, ..., n). - At most k heads: the chance of k or fewer (includes 0, 1, ..., k). - More than k heads: strictly greater than k (includes k+1, ..., n). Note this is slightly different from "at least". - Fewer than k heads: strictly less than k (includes 0, 1, ..., k-1). For a fair coin with 10 flips, "exactly 5 heads" gives about 24.6%, while "at least 5 heads" gives about 62.3% because it includes 5, 6, 7, 8, 9, and 10 heads.
Biased and weighted coins
Not all coins are fair. A coin is biased (or weighted) when one outcome is more likely than the other. You can model any coin by adjusting the "probability of heads" input. For example, a coin that lands heads 60% of the time uses p = 0.6. The binomial formula still applies - only the value of p changes. Real-world applications of biased probability include modeling loaded dice in board games, estimating conversion rates in A/B tests (where each visitor either converts or does not), quality control pass/fail inspections, and sports win-probability models. The binomial distribution is the correct tool for any two-outcome repeated independent trial.
Expected value and standard deviation
Two additional outputs help you understand the distribution beyond a single probability. The expected value E[X] = n x p tells you the average number of heads you would expect over many repetitions of the n-flip experiment. For 10 flips of a fair coin, E[X] = 5. The standard deviation SD = sqrt(n x p x (1-p)) measures how spread out the results tend to be. For 10 fair-coin flips, SD = sqrt(10 x 0.5 x 0.5) = 1.58. This means typical results tend to cluster within about 1.58 heads of the expected value of 5, so 3 to 7 heads is the common range. A wider distribution (larger SD) means more variability in outcomes.
Common coin flip probabilities (fair coin, p = 0.5)
| Flips (n) | Heads (k) | Probability | Odds (1 in X) |
|---|---|---|---|
| 1 | 1 | 50.000% | 2.00 |
| 2 | 2 | 25.000% | 4.00 |
| 3 | 3 | 12.500% | 8.00 |
| 4 | 4 | 6.250% | 16.00 |
| 5 | 5 | 3.125% | 32.00 |
| 6 | 6 | 1.563% | 64.00 |
| 10 | 10 | 0.098% | 1,024 |
| 10 | 5 | 24.609% | 4.06 |
| 20 | 10 | 17.620% | 5.68 |
| 100 | 50 | 7.959% | 12.57 |
Exact probability of getting exactly k heads in n flips with a fair coin.
Frequently asked questions
What is the probability of getting heads 10 times in a row?
For a fair coin, the probability of exactly 10 heads in 10 flips is (0.5)^10 = 1/1024, or about 0.098%. This is rare but does happen roughly once every 1,024 experiments of 10 flips. If your coin is biased - say p = 0.6 - the probability rises to 0.6^10, which is about 0.605%.
Does each coin flip affect the next?
No. Coin flips are independent events. The outcome of one flip has absolutely no influence on the next. This is the "memoryless" property. A coin that has just landed heads five times in a row still has exactly a 50% chance of landing heads on the next flip (for a fair coin). Believing otherwise is called the gambler's fallacy.
What is the binomial distribution?
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. A coin flip experiment is the textbook example: n flips, each with probability p of heads, and you count the total heads. The distribution is fully described by two parameters: n (number of trials) and p (probability of success).
How do I find the probability of getting at least 1 head?
The easiest approach is to use the complement rule. The probability of at least 1 head equals 1 minus the probability of 0 heads. P(X = 0) = (1-p)^n. For 5 fair-coin flips, P(X = 0) = 0.5^5 = 0.03125, so P(at least 1 head) = 1 - 0.03125 = 0.96875, or about 96.9%. Select "at least" mode and set k = 1 in this calculator for an instant answer.
Why does the probability of exactly k heads decrease for large n?
As n increases, probability spreads across more possible outcomes (0 through n), so each individual count becomes less likely. With 10 flips, "exactly 5" holds about 24.6% of the probability. With 100 flips, "exactly 50" holds only about 7.96%. However, the probability of landing close to the expected value (within one standard deviation) stays fairly constant - the distribution just becomes wider and flatter in absolute terms.