1. What is Binomial Probability?

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 (success/failure) of each trial.

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 each trial
• \( \binom{n}{k} \) — Binomial coefficient ("n choose k")

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

3. When to Use Binomial Probability

Details: Use this when you have a fixed number of trials, independent outcomes, constant probability, and binary outcomes. Examples include coin flips, success/failure experiments, or pass/fail tests.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

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

Q2: What if I want cumulative probability (P(X ≤ k))?
A: You would need to sum probabilities from 0 to k. This calculator gives exact P(X = k).

Q3: What are the assumptions of binomial distribution?
A: Fixed number of trials, independent trials, same success probability, and binary outcomes.

Q4: Can I use this for large n values?
A: Yes, but very large n (e.g., >1000) may cause computational limits. For n>100, normal approximation may be better.

Q5: What if p changes between trials?
A: Then you would need a different distribution (like Poisson binomial). The binomial requires constant p.