Binomial Probability Calculator
A binomial probability calculator helps determine the likelihood of a specific number of successes in fixed trials with two possible outcomes. It's essential for statistics, quality control, and risk assessment. Researchers use it to predict experiment outcomes, while businesses apply it for decision-making and probability analysis.
Calculator
Formula
P(X = k) = C(n,k) * pk * (1-p)n-k
Where C(n,k) = n!/(k!(n-k)!)
How to Use
1. Enter total number of trials (n)
2. Input desired success count (k)
3. Provide success probability (p) between 0-1
4. Click Calculate
5. Results show probability and interpretation
Derivation Process
The binomial formula combines:
1. Combination calculation for success arrangements
2. Probability of successes (p^k)
3. Probability of failures ((1-p)^(n-k))
Developed through probability theory principles, it assumes independent trials with constant probability. Jacob Bernoulli's work established the foundation for this fundamental probability distribution.
1. What is binomial probability?
Binomial probability calculates the chance of exactly k successes in n independent trials with binary outcomes. It's used when outcomes are success/failure, probability remains constant, and trials are independent. Common applications include quality control and survey analysis.
2. When should I use this calculator?
Use when analyzing scenarios with fixed trials, two possible outcomes, and constant probability. Examples include coin flips, product defect probabilities, or election prediction models. Not suitable for dependent events or changing probabilities.
3. What's the difference between binomial and normal distribution?
Binomial is discrete (exact counts), normal is continuous. Binomial handles yes/no outcomes, normal covers range-based probabilities. Use binomial for small samples and exact counts, normal for large samples and approximations.
4. Can I use decimal probability values?
Yes, enter probability as decimal between 0-1. For 25% success rate, input 0.25. The calculator automatically handles decimal conversions and validates input range to prevent errors.
5. What does C(n,k) mean?
C(n,k) represents combinations - number of ways to choose k successes from n trials. Calculated using factorial formula n!/(k!(n-k)!). This accounts for different success arrangements without considering order.
6. How accurate are the results?
Results are precise up to 8 decimal places. Accuracy depends on correct input values. The calculator uses JavaScript's precision math but may show rounding differences in extremely small probabilities.
7. Can I calculate cumulative probabilities?
This calculator shows exact probability (P(X=k)). For cumulative probabilities (P(X≤k) or P(X≥k)), you need to sum multiple binomial probabilities. Future versions may include cumulative options.
8. Why am I getting zero probability?
Zero probability indicates extremely unlikely events, often when p is 0/1 or k exceeds possible range. Check input values - ensure k ≤ n and p is between 0-1. Very small probabilities may display in scientific notation.
9. What's considered a valid probability?
Probability (p) must be between 0 (impossible) and 1 (certain). Values outside this range trigger alerts. For percentages, convert to decimal (25% → 0.25). The calculator validates inputs before calculation.
10. Can I use this for medical trials?
Yes, but consult a statistician for critical applications. While accurate for binomial models, real-world medical trials often involve complex factors beyond basic binomial assumptions like changing probabilities or dependent outcomes.