1. What is Binomial Distribution?

The binomial distribution describes the probability of having exactly k successes in n independent Bernoulli trials with success probability p. 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
• \( C(n,k) \) — Combination of n items taken k at a time

Explanation: The formula calculates the probability of getting exactly k successes in n independent trials, each with success probability p.

3. Importance of Binomial Probability

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

4. Using the Calculator

Tips: Enter number of trials (n ≥ 1), number of successes (0 ≤ k ≤ n), and probability of success (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 discrete (counts successes), while normal is continuous. For large n, binomial approximates normal.

Q2: When is binomial distribution appropriate?
A: When trials are independent, probability is constant, and there are only two possible outcomes per trial.

Q3: What if k > n?
A: Impossible - number of successes can't exceed number of trials. Calculator will show error.

Q4: What's the expected value of binomial distribution?
A: The mean is n × p. The variance is n × p × (1-p).

Q5: Can I calculate cumulative probabilities?
A: This calculator gives exact P(k). For P(X ≤ k), you'd sum probabilities from 0 to k.