1. What is Binomial Probability?

The binomial probability describes the probability of having exactly k successes in n independent Bernoulli trials (experiments with two possible outcomes) with success probability p. It's fundamental in statistics for binary outcome scenarios.

2. How Does the Calculator Work?

The calculator uses the binomial probability formula:

Where:

• \( C(n,k) \) — Number of combinations (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 trials, each with success probability p.

3. Importance of Binomial Probability

Details: Binomial probability is essential in quality control, medical testing, risk assessment, and any scenario involving repeated independent trials with binary outcomes.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

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

Q2: When is binomial distribution appropriate?
A: When trials are independent, have same success probability, and fixed number of trials.

Q3: How is combinations (n choose k) calculated?
A: \( C(n,k) = \frac{n!}{k!(n-k)!} \) where ! denotes factorial.

Q4: What if I need cumulative probability?
A: This calculates exact probability for k successes. For P(X ≤ k), you'd sum probabilities from 0 to k.

Q5: What are some real-world applications?
A: Drug efficacy studies, quality control (defect rates), genetics (inheritance probabilities), and more.