Binomial Probability Distribution Calculator
A binomial probability distribution calculator helps determine the probability of getting exactly k successes in n independent trials with a fixed probability (p). It's essential for statistics, quality control, and risk assessment, enabling users to calculate success/failure probabilities in experiments, surveys, and manufacturing processes.
Calculator
Formula
\[ P(X = k) = C(n, k) \times p^k \times (1-p)^{n-k} \]
Where \( C(n, k) = \frac{n!}{k!(n-k)!} \)
How to Use
Enter the total number of trials (n), desired successes (k), and probability of success (p). Click Calculate to get the probability. The calculator computes combinations, then applies the binomial formula. Ensure 0 ≤ p ≤ 1 and k ≤ n. Use Clear to reset inputs.
Derivation Process
The binomial formula combines combinatorics and probability: (1) Calculate combinations of n trials taken k at a time, (2) Multiply by success probability raised to k, (3) Multiply by failure probability (1-p) raised to (n-k). This accounts for all ways k successes can occur in n independent trials with constant probability p.
FAQs
1. What is binomial probability?
Binomial probability calculates the chance of exactly k successes in n independent trials with binary outcomes and constant success probability p. It's used in quality control, surveys, and experiments.
2. Can I use decimal probabilities?
Yes, enter p as decimal between 0 and 1. For example, use 0.25 for 25% probability. The calculator handles decimal inputs automatically.
3. What's the maximum n value allowed?
While technically unlimited, extremely large n values (above 1000) may cause calculation delays due to factorial computations in combinations.
4. How accurate are the results?
Results are precise to 10 decimal places. However, very small probabilities might show in scientific notation for readability.
5. Can I calculate cumulative probabilities?
This calculator shows exact probabilities. For cumulative results (≤k or ≥k successes), consider additional implementation of cumulative distribution functions.
6. Why is my result 0?
If n and k are valid but result is 0, the probability might be extremely small. Try increasing decimal places in display or check if p=0.
7. What if k > n?
The calculator automatically validates inputs. If k exceeds n, you'll receive an error message to correct your inputs.
8. Is this calculator mobile-friendly?
Yes, responsive design ensures proper functionality on both desktop and mobile devices. Inputs and buttons adjust automatically.
9. How are combinations calculated?
Combinations C(n,k) use factorial mathematics optimized for computational efficiency, avoiding direct factorial calculations for large numbers.
10. Can I use this for hypothesis testing?
While useful for individual probabilities, hypothesis testing requires additional calculations like p-values and significance levels beyond this basic calculator's scope.