Binomial Probability Calculator

Calculate exact and cumulative probabilities for binary outcome experiments using the binomial formula. Analyze success/failure probabilities with confidence intervals and distribution visualization.

Calculation Steps

  1. Enter number of trials (n)
  2. Input probability of success per trial (p)
  3. Specify number of successful outcomes (k)
  4. Choose calculation type (exact/cumulative)
  5. Review probability density and mass function

Understanding Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials:

Key Properties:

  • Fixed number of trials (n)
  • Two possible outcomes (success/failure)
  • Constant probability (p) for each trial
  • Independent trials

Formula: P(X = k) = C(n,k) × p^k × (1-p)^(n-k)

Real-World Applications

Binomial distribution is used in many fields:

  • Quality Control: Defect rates in manufacturing
  • Medicine: Success rates of treatments
  • Marketing: Customer conversion rates
  • Genetics: Inheritance patterns

It helps predict outcomes in situations with binary results and multiple trials.

Statistical Measures

Mean (μ): n × p

Variance (σ²): n × p × (1-p)

Standard Deviation (σ): √(n × p × (1-p))

Skewness: (1-2p) / √(n × p × (1-p))

Where n = number of trials, p = probability of success

Normal Approximation

When n is large and p is not extreme:

  • Use normal distribution as approximation
  • Generally valid when n×p >= 5 and n×(1-p) >= 5
  • Apply continuity correction for better accuracy
  • Useful for large-scale calculations

This approximation simplifies calculations while maintaining acceptable accuracy.