# Coin Flip Streak Calculator

URL: https://speedycalculator.com/calculators/statistics/coin-flip-streak/
Last updated: 2026-10-30

> The coin flip streak calculator gives the probability that the longest run of heads (or tails, or either side) in n tosses is exactly, at least or at most k, as an exact fraction or a decimal, plus the expected longest streak and the full distribution.

## Inputs

- Number of flips (n)
- Streak length (k)
- The longest streak is…
- Streak of
- Chance of heads on one flip (%)
- Show the result as

## Results

- Probability
- Question answered
- Probability as a decimal
- Exact fraction
- Odds (1 in …)
- Expected longest streak
- Distribution of the longest streak

## 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)
```

## Explanation

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 runSequencesProbability0TTT1/81HTT, THT, TTH, HTH4/82HHT, THH2/83HHH1/8

## 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.

## FAQ

**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.


## Sources

- [Wolfram MathWorld – Run (longest run of heads)](https://mathworld.wolfram.com/Run.html)
- [Wolfram MathWorld – Coin tossing](https://mathworld.wolfram.com/CoinTossing.html)
- [Penn State STAT 414 – Introduction to Probability Theory](https://online.stat.psu.edu/stat414/)