Calculate exact, at-least or at-most heads probabilities for independent fixed-probability coin flips using the binomial model.
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
P(X=k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ. Tails are sums of binomial probabilities.
Assumptions and limits
Independent flips with constant heads probability. Head count ignores ordering; a specified sequence and a streak are separate events. Up to 5,000 flips.
Worked example
For three fair flips, exactly two heads has probability 3/8 = 37.5%. At least two heads includes two or three and has probability 1/2. At most two heads excludes only three heads and has probability 7/8 = 87.5%. These are head-count events; the heads do not have to be consecutive. Enter probability 50 for a fair coin, not 0.5.
Worked reading and practical questions
- Exactly, At Least and At Most: Translate a Coin-Count Question
- A Head Count Is Not a Fixed Coin Sequence
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Frequently asked questions
What does this calculator assume?
Independent flips with constant heads probability. Head count ignores ordering; a specified sequence and a streak are separate events. Up to 5,000 flips.
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.