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Fair Coin Heads Probability Calculator — result sheet
Calculate the probability of an exact or at-least number of heads in repeated fair-coin flips, plus the expected count and spread.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: For a fair coin, P(X=k) = C(n,k)/2ⁿ; P(X≥k) = Σⱼ₌ₖⁿ C(n,j)/2ⁿ; E[X]=n/2 and SD(X)=√n/2.
A sequence of independent fair flips produces a binomial count of heads. The calculator distinguishes the probability of exactly the selected count from the probability of reaching or exceeding it, because those are different questions. It also reports the long-run expected number of heads and the standard deviation of the count; neither value predicts the result of one short sequence.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Number of flips — minimum 1; maximum 200
- Heads to evaluate — The exact and at-least probabilities are evaluated for this count.; minimum 0; maximum 200
Worked example
| Input | Value |
|---|---|
| Number of flips | 10 |
| Heads to evaluate | 5 |
Exactly five heads has probability 0.24609375; at least five heads has probability 0.623046875; the expected count is 5.
Assumptions and limits
- Each flip has two outcomes, heads and tails, with probability one half each.
- Flips are treated as independent and the selected count is an integer from zero through the number of flips.
- The exact probability counts the combinations of positions that contain heads, not a single ordered sequence.
- The at-least probability includes the selected count and every larger feasible count.
- Expected value and standard deviation describe the distribution over repeated experiments, not a guarantee for one experiment.
- Biased coins, dependence, physical tossing conditions, and observation errors are outside this model.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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