Coin Toss Streak Calculator
Find the likelihood of getting a run of consecutive heads when flipping a coin. Adjust the number of tosses, streak length, and coin bias.
Probability of At Least One Streak
78.70%
Run of 3 heads
Probability of No Streak
21.30%
No run of 3 heads
Approx. Expected Streaks
1.125
Estimated occurrences
How to Use This Calculator
- Enter how many coin tosses you will perform and the streak length of heads you care about.
- Set the probability of heads (0.5 for a fair coin, other values for biased coins).
- Read the probability of witnessing at least one streak and the complementary chance of no streak.
- Use the expected streak estimate as a rough guide for average streak occurrences.
Formula
The calculator uses a dynamic programming approach to track the probability of having exactly i consecutive heads without yet reaching the target streak.
Let Si(n) be the probability that after n tosses you have i consecutive heads (i < streak) and have not seen a run of length streak.
S0(n + 1) = (1 − p) · ∑i=0streak−1 Si(n)
Si+1(n + 1) = p · Si(n), for 0 ≤ i < streak − 1
The probability of observing the streak is 1 minus the remaining probability mass over the states Si after the final toss.
Full Description
Streaks fascinate gamblers and statisticians alike. Even with independent coin tosses, long runs appear surprisingly often. This calculator applies dynamic programming to enumerate all possible states — how many heads in a row you currently have — while conditioning on never having reached the target streak yet.
The method is exact for the given parameters and works for fair or biased coins. Use it to estimate the likelihood of hot streaks, the risk of unusual patterns in random processes, or to calibrate expectations in games of chance.
Frequently Asked Questions
Why does the probability approach 1 for large toss counts?
Given enough trials, it becomes almost certain that any finite streak will eventually appear. The DP calculation shows how quickly that convergence happens for your parameters.
Can the calculator handle streaks of tails?
Yes. Simply set the probability of heads to 1 − p, or reinterpret the result — the math is symmetric between heads and tails.
Why is the expected streak count only approximate?
Counting exact streak occurrences requires more elaborate state tracking. The provided formula is a quick approximation useful for intuition, especially when p is near 0.5.
What happens if the streak length exceeds the number of tosses?
The probability drops to zero because you cannot observe a longer run than the number of available tosses. The calculator highlights this situation automatically.