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
- Enter number of trials (n)
- Input probability of success per trial (p)
- Specify number of successful outcomes (k)
- Choose calculation type (exact/cumulative)
- 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.