Streaks look suspicious to the human eye. When people are asked to write down a “random” sequence of 100 coin flips, they rarely include a run longer than four or five, yet real sequences of 100 flips almost always contain a run of six or more. Knowing how long streaks really are helps you judge sports “hot hands”, trading runs, quality-control alarms and whether a sequence was faked.
The probability of a streak cannot be found from the number of heads alone, because it depends on the order of the flips. The standard approach counts sequences that never contain a run longer than k. For heads-only runs that count follows a generalised Fibonacci recurrence – each valid sequence ends in a tail followed by 0 to k heads – and for runs of either side a similar recurrence applies.
This calculator handles up to 1,000 flips, any coin bias and three kinds of question (exactly, at least, at most). For a fair coin and up to 64 flips it shows the exact fraction, and it always gives the expected longest streak and a chart of how the longest streak is distributed.
How to use the Coin Flip Streak Calculator
- Number of flips (n)
- Streak length (k)
- The longest streak is…
- Streak of
- Chance of heads on one flip
- Show the result as
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The tool runs a dynamic program over the length of the current run: at each flip the run either grows by one (if it stays within k) or resets. Summing the surviving probability gives P(longest run ≤ k); differences and complements give the other two forms. For exact fractions the same recurrence is run with whole-number counts.
Worked example
Ten fair flips, at least 3 heads in a row. Sequences with no run of three heads satisfy the “Tribonacci” recurrence f(n) = f(n − 1) + f(n − 2) + f(n − 3) with f(0) = 1, f(1) = 2, f(2) = 4, which gives f(10) = 504. So P(longest heads run ≤ 2) = 504/1,024 and P(at least 3) = 1 − 504/1,024 = 520/1,024 = 65/128 ≈ 50.78%.
All outcomes of three flips
| Longest heads run | Sequences | Probability |
|---|---|---|
| 0 | TTT | 1/8 |
| 1 | HTT, THT, TTH, HTH | 4/8 |
| 2 | HHT, THH | 2/8 |
| 3 | HHH | 1/8 |
Formula
P(longest run ≤ k) = f(k, n) ÷ 2ⁿ, where f counts sequences without a run of k + 1 (a (k + 1)-step Fibonacci recurrence) · P(≥ k) = 1 − P(≤ k − 1) · P(= k) = P(≤ k) − P(≤ k − 1)
Worked example
Inputs — Number of flips (n): 10; Streak length (k): 3; The longest streak is…: At least k; Streak of: Heads; Chance of heads on one flip: 50 %; Show the result as: Exact fraction (fair coin, up to 64 flips) and decimal.
Results — Probability = 50.7813 %; Question answered = Probability that the longest run of heads in 10 flips is at least 3; Probability as a decimal = 0.5078125; Exact fraction = 520 / 1,024 = 65 / 128; Odds (1 in …) = 1 in 1.97; Expected longest streak = 2.799.
Key terms
- Run (streak)
- A maximal block of identical consecutive outcomes.
- Longest run
- The length of the longest streak in a sequence of trials.
- Recurrence relation
- A formula that builds a count for n from counts for smaller n, like the Fibonacci numbers.
Frequently asked questions
What is the chance of 3 heads in a row in 10 flips?
65/128 ≈ 50.78%. Of the 1,024 sequences, 504 never contain three heads in a row, so 520 do.
What is the chance of 5 heads in a row in 100 flips?
About 81%. A run of at least 5 heads is more likely than not in 100 tosses.
How long is the longest streak in 100 flips?
For runs of either side the expected longest run is about 6.98; for heads only it is about 5.99.
Is a streak of 7 in 100 flips evidence of cheating?
No – a run of 7 or more of the same side happens in about 54% of fair sequences of 100 flips.
Why is "exactly" not symmetric?
Exactly k means the longest run equals k, so very short and very long streaks are both rare and the distribution is skewed to the right.
Does the Fibonacci sequence really appear?
Yes – the number of sequences of n flips with no two heads in a row is the Fibonacci number F(n + 2), e.g. 144 for 10 flips.