Coin Flip Probability Chart: Odds for 1 to 10 Flips
For n independent fair coin flips, the probability of getting all heads is (1/2)^n: 50% for 1 flip, 25% for 2, 12.5% for 3, down to about 0.098% for 10 flips. Meanwhile the number of distinct possible flip sequences doubles with every extra flip, reaching 1,024 possible sequences at 10 flips.
A worked example with numbers
| Flips (n) | Possible sequences (2^n) | P(all heads) | P(exactly half heads) |
|---|---|---|---|
| 1 | 2 | 50% | — |
| 2 | 4 | 25% | 50% |
| 3 | 8 | 12.5% | — |
| 4 | 16 | 6.25% | 37.5% |
| 5 | 32 | 3.125% | — |
| 6 | 64 | 1.5625% | 31.25% |
| 8 | 256 | 0.390625% | 27.34% |
| 10 | 1,024 | 0.09765625% | 24.61% |
How to set the rule before the result
Use this kind of table to judge whether an observed run is worth a second look: if a stated all-heads run’s row shows 3.125% or higher, it is uncommon but unremarkable across many people flipping coins; a row far smaller than that, combined with a plausible reason to suspect the coin or the method, is worth checking with the streak simulator across a much larger, preplanned sample rather than judging from the single observed run. The same table is also useful in reverse, as a planning tool: a party game that wants a roughly 1-in-8 chance of a bonus round can look up the 3-flip row and require three heads in a row as the trigger, rather than guessing at a probability and checking it later.
Common mistakes that change the odds or the process
Reading this table backward is the most common error: after seeing four heads in a row, the 6.25% figure describes the chance of that specific four-flip block before any flips happened — it does not mean the fifth flip is somehow “due” for tails, or that the fifth flip’s probability changes at all. Each flip keeps its own independent 50% chance regardless of what came before. A second mistake is applying the table to a streak found by scanning a long session for the best run rather than to a pre-named block of flips; searching many overlapping windows for the longest streak is a different, more likely event than naming one block in advance. A third mistake is comparing two people’s streaks from sessions of different lengths as if length did not matter — someone who flips 500 times has many more opportunities to produce a notable run than someone who flips 20 times, so a longer streak from the larger session is not evidence of a “luckier” or more biased coin without accounting for how many chances each session actually gave the streak to appear.
Where this method stops being appropriate
This table assumes truly independent flips at exactly 50% each. It does not describe a hand-flipped coin’s small measured same-side bias — about 50.8% rather than 50%, discussed in a related guide — nor does it apply to a coin that has already been shown to be damaged or weighted. For casual use these distinctions rarely matter, but a table built on a slightly wrong probability compounds the error with every added flip. The table also only describes flips of the same coin under the same conditions throughout; mixing results from different coins, different flippers, or a coin swapped mid-session into one combined sequence breaks the single consistent probability the calculation assumes.
How the random source fits into the rule
NIST’s binomial-distribution reference defines the binomial-probability formula used to calculate the “exactly half heads” column from independent trials each with probability 0.5. The same reference’s combinatorial formulas extend directly to any other independent two-outcome repeated trial with a stated success probability, including a weighted coin, a pass/fail test, or a dice-pool success count with a threshold other than 0.5.
Why the odds halve with every extra flip
Each additional flip is an independent event with its own 1/2 chance, and the all-heads probability for n flips is the product of n separate one-half factors. Multiplying by one-half is the same operation as halving, so every extra flip exactly halves the previous row’s probability — 50%, then 25%, then 12.5%, and so on — while the count of possible sequences doubles for the same reason, since each new flip independently doubles the number of ways the whole sequence can turn out.
The same doubling rule explains why a 20-flip all-heads run feels almost unimaginable next to a 10-flip one: ten more halvings after the 10-flip row’s 0.09765625% brings the probability down by a further factor of 1,024, landing near 0.0000954%. Small per-flip odds compound quickly, which is the same underlying arithmetic behind why a long winning streak in any independent 50/50 game becomes vanishingly rare well before it becomes impossible.
How this table differs from a streak found after the fact
The probabilities above describe one specific, pre-named block of flips considered before any of them happen. Searching a much longer session for the longest run anywhere within it is a different calculation, because there are many overlapping possible starting positions for a streak to begin — a long enough session makes even an unlikely-looking run something to expect eventually. Treating a found streak the same as a predicted one is the error behind most “this looks rigged” reactions to an otherwise ordinary sequence.
A rough way to feel the difference: in a session of 1,000 flips, there are 991 overlapping possible starting points for a ten-flip window, each with its own small chance of being all heads. Even though any one specific window has only a 0.0977% chance, having roughly a thousand chances for one of them to succeed makes finding at least one such run across the whole session far more likely than the single-window figure suggests.
This is the same reasoning that explains why, across millions of people flipping coins worldwide every day, some individual will experience an eight- or nine-flip all-heads run purely by chance without anything being unusual about their coin. A rare event considered across one trial and the same event considered across a very large number of independent trials are different questions, even though the underlying per-trial probability never changes.
The complete table, all ten rows
| Flips (n) | Possible sequences | P(all heads) | P(exactly half heads) |
|---|---|---|---|
| 1 | 2 | 50% | — |
| 2 | 4 | 25% | 50% |
| 3 | 8 | 12.5% | — |
| 4 | 16 | 6.25% | 37.5% |
| 5 | 32 | 3.125% | — |
| 6 | 64 | 1.5625% | 31.25% |
| 7 | 128 | 0.78125% | — |
| 8 | 256 | 0.390625% | 27.34% |
| 9 | 512 | 0.195312% | — |
| 10 | 1,024 | 0.097656% | 24.61% |
Flips 7 and 9 have no “exactly half” entry because an odd number of flips can never split evenly — half of 7 is 3.5, which no head count can equal. The odd rows only ever offer an all-heads probability and its mirror, all-tails, each exactly halving the row above it, matching the doubling of possible sequences shown in the middle column.
The same table for the measured 50.8% same-side bias
| Flips (n) | P(all heads) at 50% | P(all heads) at 50.8% | Difference |
|---|---|---|---|
| 1 | 50.0000% | 50.8000% | 0.8000 |
| 2 | 25.0000% | 25.8064% | 0.8064 |
| 4 | 6.2500% | 6.6597% | 0.4097 |
| 6 | 1.5625% | 1.7186% | 0.1561 |
| 8 | 0.3906% | 0.4435% | 0.0529 |
| 10 | 0.0977% | 0.1145% | 0.0168 |
This is the version of the chart that applies only to a caught, hand-flipped coin where the starting face is visible and called every time, as discussed in this site’s guide to whether a coin flip can be predicted. The difference column shrinks after n = 2: the absolute gap between the fair and biased predictions peaks early and then narrows, because both probabilities are being raised to higher powers and shrinking toward zero together. A digital flip has no starting face and no bias to apply this second table to at all — the plain 50% chart is the correct one for it.
The “it must happen about a quarter of the time” mistake
The 24.61% figure for exactly half heads in 10 flips describes an expected long-run share, not a promise about any fixed batch of sessions. Run 100 separate 10-flip sessions, and the expected number landing on exactly 5 heads is 100 × 24.61% ≈ 24.61 — but the actual count varies session-batch to session-batch with a standard deviation of about 4.31, so anywhere from roughly 16 to 33 such sessions is entirely ordinary. Treating 24.61 as a quota that must be hit, and treating a batch of 100 sessions that produced only 18 “exactly half” results as evidence something is wrong, misreads a statistical expectation as a guarantee.
How many sessions it takes to pin the 24.61% figure down
Confirming the exactly-half-heads rate for 10 flips to within ±1 percentage point at 95% confidence takes roughly n = 1.96² × 0.2461 × 0.7539 ÷ 0.01² ≈ 7,128 independent 10-flip sessions — 71,280 individual flips in total. That scale is why this table is built from the binomial formula rather than from an experiment: no realistic manual test could gather enough sessions to verify the figure directly, while the formula, given only n and p = 0.5, produces an exact answer instantly.