Binomial Probability Calculator
Calculate exact and cumulative probabilities for binary outcome experiments using the binomial formula. Get P(X = k), P(X ≤ k), and key summary statistics for a binomial distribution.
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.
How to Interpret the Results
- P(X = k) is the probability of getting exactly k successes in n trials.
- P(X ≤ k) is the probability of getting at most k successes, meaning 0 through k.
If you need a right tail probability, you can use P(X ≥ k) = 1 - P(X ≤ k - 1). This is common in risk analysis and hypothesis testing.
Worked Example
Example: You flip a fair coin 10 times (n = 10, p = 0.5). What is the probability of getting exactly 7 heads? Enter n = 10, p = 0.5, k = 7 to compute P(X = 7). If you want the chance of 7 or fewer heads, use the same inputs and read P(X ≤ 7).
Assumptions and Limitations
The binomial model is a good fit when:
- The number of trials n is fixed in advance.
- Each trial has two outcomes (success or failure).
- The success probability p stays constant across trials.
- Trials are independent.
If independence or constant probability does not hold, you may need a different model such as a hypergeometric, negative binomial, or beta-binomial distribution.