Guide · The maths of the toss
Coin Flip Probability
Updated: June 2026
A single coin flip is the simplest random event there is, yet the probability questions it raises trip up almost everyone: the odds of a streak, the chance of "exactly k heads," and the stubborn myth that a result can be overdue. This page lays out the formulas plainly and lets you test them by flipping coins by the hundred.
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One flip: the building block
A fair coin has two equally likely results, so the probability of heads is one half and the probability of tails is one half. Everything else in coin probability is built from this one fact by combining flips. The key rule is independence: the result of one flip tells you nothing about the next. That lets you find the chance of a sequence by simply multiplying the chance of each step.
Because the two outcomes are symmetric, anything you can say about heads applies equally to tails. That symmetry is what makes the coin the textbook example of a uniform random variable.
Several flips in a row
To get the same result repeatedly, multiply. Two heads in a row is 0.5 × 0.5 = 0.25; three in a row is 0.5³ = 0.125; and in general, n of the same side in a row is one half raised to the power n. The numbers shrink fast, which is why a long unbroken streak feels remarkable — it genuinely is rare.
| Streak | Probability | Roughly |
|---|---|---|
| 2 in a row | 1/4 | 25% |
| 3 in a row | 1/8 | 12.5% |
| 5 in a row | 1/32 | 3.1% |
| 10 in a row | 1/1024 | 0.098% |
Exactly k heads in n flips
The richer question is how many heads you get across several flips, in any order. This follows the binomial distribution. The number of arrangements with k heads among n flips is the binomial coefficient "n choose k," and each specific arrangement has probability one half to the power n. Multiply the two together:
For 3 flips, the chance of exactly 2 heads is C(3,2) × (1/2)³ = 3 × 1/8 = 3/8, or 37.5%. For 10 flips, the most likely single outcome is 5 heads, but its probability is only about 24.6% — the spread around the middle is wide, which is why you rarely land on a perfect half in a short run. Flip a few hundred coins in the tool and the tally creeps toward 50% as the law of large numbers takes over.
The gambler's fallacy
After a run of heads, it is tempting to feel that tails is "due." It is not. The coin has no memory, so the next flip is exactly 50/50 no matter what came before. The streak that already happened was unlikely before it began, but standing at the end of it changes nothing about the next toss. Confusing "this long sequence was rare to predict in advance" with "the next flip is now skewed" is the single most expensive mistake in games of chance.
Frequently asked questions
What is the probability of heads twice in a row?
0.5 × 0.5 = 0.25, or 25%. Flips are independent, so you multiply.
After five heads, is tails more likely?
No. Each flip stays 50/50. Thinking otherwise is the gambler's fallacy.
What is the chance of exactly 2 heads in 3 flips?
C(3,2) × (1/2)³ = 3/8 = 37.5%.
Why doesn't a short run land on exactly half?
The outcomes spread around the middle. Only over many flips does the proportion settle close to 50%.