Coin Flip Streak Calculator
How likely are you to flip five heads in a row during 20 coin flips? This calculator computes the probability of getting at least k consecutive successes (heads, wins, or any binary outcome) in n independent trials with a fixed per-trial success probability. It uses a dynamic programming recurrence that is exact for any n and k. This is useful for analyzing win streaks in games, validating game balance, and teaching probability concepts around the "hot hand" fallacy.
Streak probability formula
Let Q(i) = P(no streak of k in first i trials)
Q(0) = 1, Q(i) = sum over j=0..min(i,k)-1 of p^j*(1-p)*Q(i-j-1), plus p^k*0
P(at least k in a row in n trials) = 1 - Q(n)
This recurrence computes the exact probability. Q(i) is built up iteratively for i from 1 to n. The calculator handles this exactly for any reasonable n and k.
Streaks and the hot hand fallacy
- The "hot hand fallacy" is the mistaken belief that a player is more likely to succeed after recent successes. For independent events (like fair coin flips), past outcomes do not affect future ones.
- In 100 fair coin flips, there is about a 96% chance of seeing a streak of at least 6 consecutive same-side outcomes, which feels surprising but is mathematically expected.
- Streaks are natural features of random sequences. Game designers can use streak probability to validate that their shuffle or RNG algorithms are not suppressing expected runs.
- Anti-streak RNG algorithms (pity systems) are intentionally non-random in order to smooth player experience at the cost of statistical independence.
Coin flip streak: frequently asked questions
How is the probability of a streak calculated?
The probability of getting a streak of at least k consecutive successes in n independent Bernoulli trials uses a recurrence relation. P(n, k) = 1 minus Q(n, k), where Q(n, k) is the probability of no streak of length k in n trials, computed iteratively.
Why is a streak more likely than people expect?
People systematically underestimate streak probabilities. In 100 fair coin flips, there is about a 96% chance of getting at least 6 consecutive heads or tails. This is because there are many possible positions where a streak could start.
Does this apply to win/loss streaks in games?
Yes, for any sequence of independent outcomes with fixed probability. A player winning 60% of matches has a significant probability of a long winning streak over many games, which can mislead analysis of true skill if sample sizes are small.
What is the expected length of the longest streak in n flips?
The expected longest run in n fair coin flips is approximately log2(n) for a biased coin adjust by log(1/p). This calculator shows probability of at least a specific streak length, not the expected maximum streak.
Can I use this for biased coins or unfair game outcomes?
Yes. The success probability can be set to any value between 0% and 100%. Use 50% for a fair coin. Set higher values for events more likely to succeed, such as a player winning 70% of matches.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: itl.nist.gov/div898/handbook/.
- National Council of Teachers of Mathematics: nctm.org.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.