Blank sequence cards with copper coins arranged beside unordered coin piles. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
One sequence is only one arrangement
For five independent fair flips, the specified sequence HHTHT has probability 1/32. Exactly three heads is a larger event: it contains every arrangement with three heads and two tails. There are C(5,3) = 10 such arrangements, so the head-count probability is 10/32 = 31.25%. The specified sequence has probability 3.125%.
Understand the combination multiplier
With a constant heads probability p, any particular five-flip sequence containing three heads has probability p³(1−p)². The binomial formula multiplies that sequence probability by the number of distinct placements of the three heads. Order is ignored only when defining the event; it is still the reason multiple equally weighted sequences contribute to the count probability.
Consecutive heads define another event
Exactly three heads need not include a run of three. HHTHT has three heads but no three-head streak. HHHHH has a three-head streak but does not contain exactly three heads. Neither event contains the other in general. A statement such as “three heads somewhere” therefore needs clarification: does it mean a total count or an adjacent run?
Match the question to the appropriate tool
The Coin Flip Probability Calculator handles head counts. The Combination Calculator can check the number of placements, and the Coin Toss Streak Calculator handles overlapping adjacent runs. For a fixed sequence under independent flips, multiply the relevant per-position probabilities directly; the count multiplier is not part of that event. If probabilities vary by position, different sequences with the same number of heads may no longer have equal weights, so the ordinary fixed-p binomial formula is not applicable. The count-event guide explains the separate distinction between exact counts and cumulative tails.
Check the model behind your result
Use the Coin Flip Probability 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