MCPCalc

Binomial Probability Calculator

Compute P(X=k), cumulative binomial probabilities, and distribution statistics.

Scratchpad (not saved)

What This Calculator Does

This calculator evaluates a binomial model from three inputs: number of trials, success probability, and a success count of interest. It reports point and cumulative probabilities together with the standard distribution moments used to interpret the result.

It combines Number of Trials (n), Success Probability (p), Number of Successes (k) to estimate P(X = k), P(X <= k), P(X >= k).

Formula & Method

Core equations: For X\sim\mathrm{Bin}(n,p), point probability is P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}. Cumulative probabilities are P(X\le k)=\sum_{i=0}^{k}\binom{n}{i}p^i(1-p)^{n-i} and P(X\ge k)=\sum_{i=k}^{n}\binom{n}{i}p^i(1-p)^{n-i}. Moments are \mu=np, \sigma^2=np(1-p), and \sigma=\sqrt{np(1-p)}.

Notation used in the formulas: R = P(X = k); x_{1} = Number of Trials (n); x_{2} = Success Probability (p); x_{3} = Number of Successes (k).

Method summary: inputs are normalized to consistent units, core equations are evaluated, then secondary values are derived and rounded for display.

Use it for repeated yes/no experiments with a constant success probability, such as hit rates, defect counts, quiz guessing, or A/B testing style success tallies.

Inputs Used

  • Number of Trials (n): Used directly in the calculation.
  • Success Probability (p): Used directly in the calculation.
  • Number of Successes (k): Used directly in the calculation.

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