1. What is 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:

• \( n \) — 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 Bernoulli trials.

3. Importance of Binomial Probability

Details: Binomial distribution is fundamental in statistics for modeling binary outcomes. It's used in quality control, medical testing, genetics, and many other fields where you need to calculate probabilities of specific counts of successes.

4. Using the Calculator

Tips: Enter number of trials (positive integer), number of successes (non-negative integer ≤ n), and probability of success (between 0 and 1). All values must be valid (n ≥ k, 0 ≤ p ≤ 1).

5. Frequently Asked Questions (FAQ)

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

Q2: How is this different from Poisson distribution?
A: Poisson is for rare events with many trials, while binomial is for exact counts with known probability.

Q3: What if n is very large?
A: For large n, normal approximation may be used, but this calculator handles exact calculations.

Q4: Can I calculate cumulative probabilities?
A: This calculator gives exact P(X=k). For P(X≤k) you'd need to sum probabilities from 0 to k.

Q5: What's the maximum n this calculator can handle?
A: It depends on server resources, but factorial calculations become impractical around n=170.