1. What is Binomial Probability?

Binomial probability calculates the probability of getting exactly k successes in n independent Bernoulli trials, where each trial has the same probability p of success.

2. How Does the Calculator Work?

The calculator uses the binomial probability formula:

Where:

• \( C(n,k) \) — 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 exactly k successes in n independent trials with constant success probability p.

3. Importance of Binomial Probability

Details: Binomial probability is fundamental in statistics and probability theory, used in quality control, medical testing, survey analysis, and many other fields where binary outcomes are analyzed.

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 within their respective ranges.

5. Frequently Asked Questions (FAQ)

Q1: What are the assumptions of binomial distribution?
A: Fixed number of trials, independent trials, only two possible outcomes per trial, and constant probability of success.

Q2: What if I need cumulative probability?
A: This calculator gives the probability of exactly k successes. For cumulative probabilities (≤k or ≥k), you would need to sum multiple binomial probabilities.

Q3: When is binomial distribution appropriate?
A: When you have a fixed number of independent trials with constant success probability and binary outcomes.

Q4: What is the binomial coefficient C(n,k)?
A: It represents the number of ways to choose k successes from n trials, calculated as n!/(k!(n-k)!).

Q5: How does this relate to normal approximation?
A: For large n, the binomial distribution can be approximated by a normal distribution with mean np and variance np(1-p).