Coin Flip Probability Calculator
Covers three different coin questions: the chance of one exact sequence of flips, the chance of a specific number of heads in any order, and the chance of a run of consecutive heads or tails.
Choose a scenario
The one distinction that resolves most coin flip confusion
"What's the probability of 3 heads in 5 flips" and "what's the probability of exactly HHTHT" sound similar but ask completely different questions. The first accepts any arrangement with 3 heads among 5 flips, HHTTT, HTHTH, THHHT, and every other qualifying order. The second asks about one single, specific sequence. The first probability is always larger than the second, because it's counting many outcomes instead of one.
The three formulas
A worked example: sequence versus count
Flip a fair coin 4 times. The exact sequence HHTT has probability 0.54 = 0.0625, or 6.25%. That's true for any specific 4-flip sequence, since a fair coin makes all 16 possible sequences equally likely.
But "exactly 2 heads in 4 flips, any order" is a different question. There are C(4,2) = 6 sequences with exactly 2 heads: HHTT, HTHT, HTTH, THHT, THTH, TTHH. Each has the same 6.25% probability, so the total is 6 × 0.0625 = 0.375, or 37.5%, six times higher than any single sequence, because six different orderings all qualify.
A worked example: how surprising is a 5-flip streak?
The probability of flipping heads 5 times in a row, starting from flip one, is 0.55 = 0.03125, about 3.1%. That's genuinely uncommon for a fair coin, though far from impossible; it would be expected to happen roughly once in every 32 sets of 5 flips.
Worth separating from a different, much easier question: "what's the probability of seeing a streak of 5 somewhere within a much longer sequence of flips," which is far more likely than 3.1%, since there are many more chances for a streak to start somewhere in a long sequence than in a fixed 5-flip window.
Mistakes people make with coin flip probability
The gambler's fallacy. After a run of heads, people often expect tails is "due." A fair coin has no memory. The probability of heads on the next flip is still exactly p, regardless of what came before.
Treating a count question as a sequence question, or vice versa. As shown above, these give very different numbers. Always check whether the question cares about a specific order or just a total count.
Assuming a fair coin when the problem specifies a biased one. The formulas above work for any probability p, not just 0.5. Read carefully whether a problem describes a weighted coin.
Questions people actually ask about this
Is every sequence of coin flips equally likely?
Yes, for a fair coin, every specific sequence of a given length is exactly as likely as any other. HHHHH is just as probable as HTHTH, both 0.5 raised to the power of 5. What differs is how many sequences share a given property, like "contains 3 heads," which is why some totals feel more or less common.
Why does a streak feel more surprising than a mixed sequence?
Purely a matter of counting, not probability. There's only one way to get HHHHH, but many ways to get "3 heads and 2 tails in some order," so mixed outcomes are individually just as likely as a streak but collectively far more common.
How does a weighted or biased coin change these formulas?
Replace 0.5 with the actual probability of heads, p, everywhere it appears. The formulas themselves don't change, only the input value does.
What's the probability of getting at least one head in several flips?
Easier to calculate the complement: the probability of getting all tails, (1−p) raised to the number of flips, then subtract that from 1.