Calculate the probability of at least one head-only or either-side streak in a fixed number of flips, accounting for overlapping runs.
How to use this tool
- Choose the event or model and review the assumptions below.
- Enter the labeled inputs. Probability inputs are percentages from 0 to 100, not decimal fractions.
- Select Calculate and read the labeled outputs.
- Change an input and calculate again; Reset restores the example inputs.
Calculation method
A finite-state recursion tracks current run length while accumulating the probability of first reaching k.
Assumptions and limits
Independent flips with constant heads probability. Computes at least one overlapping run anywhere in n flips, not the number of runs or the longest-run distribution. Work is capped for browser responsiveness.
Worked example
For a run of three heads anywhere in four fair flips, HHHT, THHH and HHHH qualify. Their total probability is 3/16 = 18.75%. The two possible length-three windows overlap, so their separate probabilities cannot simply be added. Selecting either-side streaks also includes the three analogous tail-run sequences, giving 6/16 = 37.5%. This output is not an expected waiting time.
Worked reading and practical questions
- Why Overlapping Streak Windows Cannot Be Added
- Waiting Time for a Head Run vs Probability Within a Fixed Horizon
Browse Statistics calculators or explore other tools on the GreenCalc homepage.
Frequently asked questions
What does this calculator assume?
Independent flips with constant heads probability. Computes at least one overlapping run anywhere in n flips, not the number of runs or the longest-run distribution. Work is capped for browser responsiveness.
How do I enter and update the inputs?
Use the labeled fields, then select Calculate. Percent fields use 0 to 100. After editing, calculate again; Reset restores the example inputs.