Binomial Distribution Calculator

Binomial Distribution is evaluated from Number of Trials, Number of Successes and Probability of Success. The calculation reports P - Exact, P - Cumulative and P - Complement.

Results

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About the Binomial Distribution Calculator

Binomial Distribution is treated here as a quantitative relation between Number of Trials, Number of Successes and Probability of Success and P - Exact, P - Cumulative, P - Complement and Mean.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
P(X=k) = C(n,k) x pᵏ x (1 - p)^(n - k)
mu = np
sigma = sqrt(np(1 - p))

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

P(X=k) = C(n,k) x pᵏ x (1 - p)^(n - k)
mu = np
sigma = sqrt(np(1 - p))

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Coin Flips — Exactly 3 Heads in 10

Inputs

n_val: 10 k_val: 3 p_val: 0.5
P - Exact: 0.1171875. P - Cumulative: 0.171875. P - Complement: 0.9453125. Mean: 5. Std Dev: 1.5811

With Number of Trials = 10, Number of Successes = 3 and Probability of Success = 0.5 as the stated inputs, the result is P - Exact = 0.1171875, P - Cumulative = 0.171875 and P - Complement = 0.9453125. Each value corresponds to the declared output fields.

Example 2: Free Throw Shooting — At Least 7 of 10

Inputs

n_val: 10 k_val: 7 p_val: 0.75
P - Exact: 0.25028229. P - Cumulative: 0.4744072. P - Complement: 0.77587509. Mean: 7.5. Std Dev: 1.3693

With Number of Trials = 10, Number of Successes = 7 and Probability of Success = 0.75 as the stated inputs, the result is P - Exact = 0.25028229, P - Cumulative = 0.4744072 and P - Complement = 0.77587509. Each value corresponds to the declared output fields.

Example 3: Quality Control — Defective Parts

Inputs

n_val: 20 k_val: 2 p_val: 0.05
P - Exact: 0.1886768. P - Cumulative: 0.92451633. P - Complement: 0.26416048. Mean: 1. Std Dev: 0.9747

With Number of Trials = 20, Number of Successes = 2 and Probability of Success = 0.05 as the stated inputs, the result is P - Exact = 0.1886768, P - Cumulative = 0.92451633 and P - Complement = 0.26416048. Each value corresponds to the declared output fields.

Example 4: Survey — Response Rate Prediction

Inputs

n_val: 50 k_val: 35 p_val: 0.7
P - Exact: 0.12234686. P - Cumulative: 0.55316843. P - Complement: 0.56917844. Mean: 35. Std Dev: 3.2404

With Number of Trials = 50, Number of Successes = 35 and Probability of Success = 0.7 as the stated inputs, the result is P - Exact = 0.12234686, P - Cumulative = 0.55316843 and P - Complement = 0.56917844. Each value corresponds to the declared output fields.

Common Use Cases

  • Probability of exactly k successes in n coin flips
  • Chance of at least 3 heads in 10 flips
  • Calculate defect probability in quality control
  • Model success/fail scenarios in surveys