1. What is the Binomial Probability Formula?

The binomial probability formula calculates the probability of having exactly k successes in n independent Bernoulli trials with success probability p. It's fundamental in probability theory and statistics.

2. How Does the Calculator Work?

The calculator uses the binomial probability formula:

Where:

• \( C(n,k) \) — Combination of n items taken k at a time
• \( n \) — Number of trials
• \( k \) — Number of successes
• \( p \) — Probability of success on a single trial

Explanation: The formula accounts for all possible ways k successes can occur in n trials, weighted by the probability of each specific sequence.

3. Importance of Binomial Probability

Details: Binomial probability is used in quality control, medical testing, risk assessment, and any scenario with binary outcomes and fixed number of trials.

4. Using the Calculator

Tips: Enter positive integer for n, non-negative integer for k (≤n), and probability p between 0 and 1. The calculator computes the exact probability.

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. Normal can approximate binomial when n is large and p isn't extreme.

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

Q3: What are common applications?
A: Clinical trial analysis, quality control (defect rates), genetics (inheritance probabilities), and more.

Q4: When is binomial distribution appropriate?
A: When trials are independent, probability is constant, and there are only two possible outcomes per trial.

Q5: How does this relate to the binomial theorem?
A: The probabilities sum to 1 because (p + (1-p))^n = 1 by the binomial theorem.