1. What is Binomial Distribution?

The binomial distribution describes the probability of having exactly k successes in n independent Bernoulli trials with the same probability p of success. It's used when there are exactly two mutually exclusive outcomes of a trial (success/failure).

2. How Does the Calculator Work?

The calculator uses the binomial probability formula:

Where:

• \( n \) — Number of trials
• \( k \) — Number of successes
• \( p \) — Probability of success on a single trial
• \( C(n,k) \) — Combinations of n items taken k at a time

Explanation: The formula calculates the probability of getting exactly k successes in n trials, accounting for all possible ways those successes can occur.

3. Importance of Binomial Probability

Details: Binomial distribution is fundamental in statistics for modeling binary outcomes. It's used in quality control, medical testing, risk assessment, and many other fields.

4. Using the Calculator

Tips: Enter number of trials (n ≥ 1), number of successes (0 ≤ k ≤ n), and probability (0 ≤ p ≤ 1). All values must be valid.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between binomial and normal distribution?
A: Binomial is for discrete counts of successes, while normal is for continuous data. For large n, binomial approximates normal.

Q2: What if my probability is greater than 1?
A: Probabilities must be between 0 and 1. Values outside this range are invalid.

Q3: How is C(n,k) calculated?
A: It's the combination formula: n! / (k! × (n-k)!), representing the number of ways to choose k successes from n trials.

Q4: What if k > n?
A: This is impossible - you can't have more successes than trials. The calculator will not compute invalid inputs.

Q5: Can I calculate cumulative probabilities?
A: This calculator gives exact probabilities. For cumulative probabilities (≤k or ≥k), you'd sum multiple binomial probabilities.