Coin Flip Calculator

Binomial Probability & Bernoulli Trials Formulation

A sequence of n coin tosses is a classic Bernoulli process where each trial is independent with two mutually exclusive outcomes (Heads with probability p, Tails with probability 1-p):

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• Exact Probability Mass Function (PMF)**:

P = nk p^k (1-p)^n-k = (n!) / (k!(n-k)!) p^k (1-p)^n-k

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• Cumulative Probability Functions**:

• At Least k Heads: P(X k) = Σ_i=k^n P

• At Most k Heads: P(X k) = Σ_i=0^k P

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• Statistical Moments**:

• Expected Value (Mean): μ = n · p

• Variance: σ^2 = n · p · (1-p)

• Standard Deviation: σ = √(n · p · (1-p))

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• Consecutive Streak Probability**: The probability of flipping k heads consecutively in k flips is simply p^k.

How to Use This Calculator

Enter your Total Number of Flips (n) (e.g. 10), Target Number of Heads (k) (e.g. 5), Probability of Heads on Single Toss (p) (e.g. 0.5 for fair coin), and Probability Case Benchmark — choose from Custom Toss Parameters, Fair Coin: Exactly 5 Heads in 10 Flips (P = 24.61% / μ = 5.0), Fair Coin: At Least 7 Heads in 10 Flips (P ≥ 7 = 17.19%). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Formula Reference: Coin Flip Calculator

Input variables:

numberOfFlips — Total Number of Flips (n)

targetHeadsCount — Target Number of Heads (k)

coinFairnessProbability — Probability of Heads on Single Toss (p)

Computed outputs:

exactProbabilityPercent — Exact Probability P(X = k)

cumulativeAtLeastPercent — Cumulative Probability P(X ≥ k)

cumulativeAtMostPercent — Cumulative Probability P(X ≤ k)

expectedHeadsMean — Expected Heads (Mean μ)

standardDeviationSigma — Standard Deviation (σ)

Mathematical relationships extracted from calculation logic:

logComb = logFactorial(n) - logFactorial(k) - logFactorial(n - k)

logProb = logComb + (k Math.log(p)) + ((n - k) Math.log(1 - p))

variance = n p (1 - p)

Standard: NIST / ISO mathematical definitions

Worked Example

A researcher flipping a fair coin 10 times calculates a 17.19% chance of getting 7 or more heads (At Least 7), showing that such an outcome occurs roughly once every 5.8 experiments by pure chance.

Understanding Your Result

Evaluate your result in the context of your specific use case and relevant industry standards or reference values. A single calculated result is a starting point — test two to three scenarios to understand how sensitive the outcome is to your key input assumptions. For high-stakes decisions, verify your result against a secondary source or subject-matter expert before acting.