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Coin Flip Probability Calculator
What are the odds of flipping 7 heads in a row? How likely is it to get exactly 60 heads in 100 flips? The mathematics of coin flips — the binomial distribution — underlies probability theory, statistics, gambling, finance, and genetics. Calculate any coin flip probability instantly.
A fair coin has a 50% chance of heads on every flip. But over many flips, seemingly improbable sequences are actually expected — and seemingly likely outcomes can be surprisingly rare.
Coin Flip Probability
10 flips • 50% heads probability
Probability: 0%
Full Probability Summary
Probability Distribution (10 flips)
| Heads | Exact Prob. | Cumul. (≤) | Cumul. (≥) |
|---|
Streak Probability Reference
| Streak Length | Prob. in 10 flips | 1 in X | Flips needed for 50% chance |
|---|
"After 10 heads in a row, the next flip is still 50/50. The coin has no memory. This is the hardest thing for the human brain to accept about probability."
— Probability and Cognitive Bias
The binomial distribution explained
The coin flip is the simplest possible random experiment: two outcomes, fixed probability, independent trials. The mathematics that describes it — the binomial distribution — is one of the most fundamental in all of statistics. It answers: given n independent trials each with probability p of success, what is the probability of exactly k successes?
The formula: P(X = k) = C(n,k) × pk × (1−p)n−k. C(n,k) is the number of ways to choose k items from n (combinations). For 10 flips and 3 heads: C(10,3) × 0.53 × 0.57 = 120 × 0.125 × 0.0078125 = 11.72%.
The expected number of heads in n flips is n × p. The standard deviation is √(n × p × (1−p)). For 100 fair coin flips: expected 50 heads, standard deviation 5. About 68% of the time you’ll get between 45 and 55 heads — not always exactly 50.
lightbulb Surprising Coin Flip Facts
| Scenario | Probability |
|---|---|
| Exactly 50 heads in 100 flips | 7.96% |
| At least 60 heads in 100 flips | 2.84% |
| 10 heads in a row (in 10 flips) | 0.098% |
| At least one streak of 5 in 100 flips | 97.1% |
| Getting any streak of 6 in 100 flips | 80.6% |
| All heads in 20 flips | 0.0001% |
| All heads in 30 flips | 1 in 1,073,741,824 |
Coin Flip FAQs
Does a coin “remember” previous flips?
No — this is the gambler’s fallacy. Each flip is completely independent of every previous flip. After 10 consecutive heads, the probability of heads on the 11th flip is still exactly 50%. The coin has no memory. What “should” happen to balance out the sequence exists only in our intuition, not in the physics of the coin.
Is a real coin actually fair?
Close, but not perfectly. Physical coin flips have been studied carefully. The Stanford coin-flipping study (Diaconis et al.) found that coins landed on the same side they started approximately 51% of the time due to precession in the spin, not a 50/50 split. However, for most practical purposes, a coin flip is close enough to 50/50 that the difference is immaterial unless you’re making thousands of high-stakes bets on coin flips.
What is the law of large numbers?
The law of large numbers states that as the number of trials increases, the observed proportion of heads will converge toward the true probability (50%). After 10 flips you might see 7 heads (70%). After 1,000 flips, you’ll very likely be within 2–3% of 50%. After 1,000,000 flips, you’ll be within 0.1%. The law does not mean individual streaks get “corrected” — they get diluted by the enormous volume of subsequent flips.
Terminology
Binomial Distribution
The probability distribution for the number of successes in n independent trials with probability p per trial. Describes coin flips, yes/no surveys, quality control sampling, and many other scenarios. Mean = np, variance = np(1−p).
Combinations C(n,k)
The number of ways to choose k items from n items without regard to order. Also written as “n choose k” or nCk. The number of distinct sequences of n coin flips with exactly k heads. C(10,3) = 120 — there are 120 ways to get exactly 3 heads in 10 flips.
Expected Value
The mean outcome over many trials. For coin flips, expected heads = n × p. The expected value is not necessarily the most likely single outcome — it’s the average over infinite repetitions.
Standard Deviation
A measure of spread around the expected value. For binomial: σ = √(np(1−p)). For 100 fair flips: σ = 5. About 68% of results fall within one standard deviation of the mean (45–55 heads), 95% within two standard deviations (40–60 heads).
Gambler’s Fallacy
The mistaken belief that past random outcomes influence future ones. Believing a coin is “due” for tails after a streak of heads. Each flip is independent; history is irrelevant to the next outcome. The fallacy underlies many costly gambling decisions.
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