A small hourglass beside a row of matching generic coins. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
Two run questions use different outputs
“How long, on average, until three consecutive heads?” asks about a waiting-time distribution with no fixed final flip. “What is the chance of three consecutive heads within four flips?” fixes a horizon and asks for one event probability. An expected waiting time is not a deadline at which success becomes certain or even necessarily 50% likely.
Check the fair-coin waiting-time formula
For independent flips with heads probability p strictly between zero and one, the expected number of flips until k consecutive heads is (1−pᵏ)/[(1−p)pᵏ]. For a fair coin and k = 3, it equals 14 flips. The count includes flips in interrupted attempts; failed partial runs consume time. Simply using 1/pᵏ = 8 ignores how overlapping opportunities and resets affect waiting.
Use a finite horizon for a planning probability
Within four fair flips, a three-head run occurs in HHHT, THHH or HHHH, giving 3/16 = 18.75%. The Coin Toss Streak Calculator computes such fixed-horizon probabilities with a finite-state recursion. Increasing the horizon changes the probability, while the expected waiting time remains a property of the same ongoing process. The tool does not claim to report a full waiting-time or longest-run distribution.
Handle limiting cases as separate models
If p = 1, the waiting time for k heads is exactly k. If p = 0 and k is positive, a head run never occurs. The algebraic expression containing divisions by p and 1−p should not be applied blindly at these endpoints. For either-side streaks, the head-only waiting formula is also not the appropriate formula. Read the overlapping-window example before approximating event probabilities by adding windows. Use the Coin Flipper for an observed educational sequence, keeping that random observation separate from an exact model probability.
Check the model behind your result
Use the Coin Toss Streak Calculator for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26