1. What is the Binomial Probability Distribution?

The binomial probability distribution describes the number of successes in a fixed number of independent trials, each with the same probability 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:

• \( P(k) \) — Probability of exactly k successes
• \( C(n, k) \) — Binomial coefficient ("n choose k")
• \( n \) — Number of trials
• \( k \) — Number of successes
• \( p \) — Probability of success on a single trial

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

3. Importance of Binomial Probability

Details: The binomial distribution is fundamental in statistics for modeling binary outcomes, quality control, genetics, and many other applications with dichotomous outcomes.

4. Using the Calculator

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

Q2: What are the requirements for binomial distribution?
A: Fixed number of trials, independent trials, two possible outcomes, constant probability of success.

Q3: How is C(n, k) calculated?
A: C(n, k) = n! / (k!(n-k)!), where "!" denotes factorial.

Q4: What if I need cumulative probability?
A: Sum individual probabilities for all relevant k values (P(X ≤ k) = P(0) + P(1) + ... + P(k)).

Q5: When is binomial distribution not appropriate?
A: When trials aren't independent, probability changes, or there are more than two possible outcomes.