Coin Flip Probability
The math behind every coin toss: streak calculators, the law of large numbers, and why the gambler's fallacy will cost you money.
Basic Coin Flip Probability
The probability of a coin flip for a fair coin is one-half for heads and one-half for tails: each side has P = 0.5. This coin flip probability stays the same on every toss because flips are independent. In short, coin flip probability is 50% heads and 50% tails each time.
A fair coin has exactly two outcomes -- heads and tails -- each with a probability of 0.5 (50%). This makes the coin flip a Bernoulli trial: a random experiment with exactly two possible results. The mathematical model was formalized by Jacob Bernoulli in Ars Conjectandi (1713) and remains a foundational concept in probability theory.
The chance of a specific outcome on a single flip is always:
Each outcome is independent -- the result of one flip has zero influence on the next. This makes coin flips a textbook example of independent events in probability theory. A coin that landed heads ten times in a row still has a 50% chance of heads on the eleventh flip. The coin has no memory.
Streak Probability Calculator
For streaks, coin flip probability is (1/2)n for n consecutive same-side results. Use the calculator below to find the odds of any streak:
| Streak | Probability | Odds (1 in X) | Expected flips |
|---|---|---|---|
| 2 | 25% | 1 in 4 | 6 |
| 3 | 12.5% | 1 in 8 | 14 |
| 5 | 3.125% | 1 in 32 | 62 |
| 7 | 0.781% | 1 in 128 | 254 |
| 10 | 0.098% | 1 in 1,024 | 2,046 |
| 15 | 0.003% | 1 in 32,768 | 65,534 |
| 20 | 0.0001% | 1 in 1,048,576 | ~2.1M |
Law of Large Numbers -- Live Demo
The law of large numbers states that as the number of trials increases, the observed frequency of an outcome approaches its long-run expected value. For coin flips, the ratio of heads to tails converges on 50/50.
After 10 flips, seeing a 70/30 split is common. After 1,000 flips, the ratio typically falls within 48-52%. After 10,000 flips, results consistently land within 49-51%. The convergence is not because the coin "corrects" itself -- each flip is independent -- but because the accumulated total overwhelms any short-term deviations.
The Gambler's Fallacy
The gambler's fallacy is the incorrect belief that past random events affect future probabilities. After seeing five consecutive heads, a person committing this fallacy believes tails is "due" or "overdue." In reality, the sixth flip is still exactly 50/50.
This fallacy cost casinos' patrons millions. On August 18, 1913, at the Monte Carlo Casino, the roulette ball landed on black 26 times in a row. Gamblers lost enormous sums betting on red, convinced the streak had to end. The odds of 26 consecutive blacks are roughly 1 in 67 million -- extremely rare, but each individual spin remained independent.
The antidote to the gambler's fallacy is the coin flipper itself: flip 100 times, watch the statistics panel, and observe that runs of 4-6 in a row appear regularly despite a perfectly fair 50/50 mechanism.
Multiple Coin Flips
When flipping multiple coins simultaneously, the chance of a specific combination follows the binomial distribution. For n coins, the chance of getting exactly k heads is:
For quick reference in plain text: with 2 coins, the ordered outcomes are HH, HT, TH, and TT (four cases, each at 1/4). With 3 coins, the ordered outcomes are HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT (eight cases, each at 1/8).
Where C(n,k) is the binomial coefficient "n choose k." For two coins, the outcomes are: HH (25%), HT (25%), TH (25%), TT (25%) -- with "one head, one tail" appearing 50% of the time because there are two ways to achieve it. This multi-toss coin flip probability view matches the binomial formula above. Test this yourself with the coin flipper's multi-coin mode. For binary decisions where the math doesn't matter and you just need an answer, try the yes or no decision flipper or the classic heads or tails game.