1. What is Binomial Distribution?

The binomial distribution describes the probability of having exactly k successes in n independent Bernoulli trials with success probability p. It's a fundamental discrete probability distribution with applications in statistics, quality control, and risk assessment.

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
• \( C(n,k) \) — Combinations of n items taken k at a time

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 crucial for understanding processes with binary outcomes, such as quality control (defective/non-defective), medical testing (positive/negative), and many other yes/no scenarios.

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.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between binomial and normal distribution?
A: Binomial is discrete (counts successes in fixed trials), 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: What does C(n,k) represent?
A: The combination formula, calculated as n!/(k!(n-k)!), counts ways to choose k successes from n trials.

Q4: What if p is 0 or 1?
A: At p=0, P(k=0)=1; at p=1, P(k=n)=1. All other k values give P(k)=0 in these cases.

Q5: How accurate is this calculation?
A: Mathematically exact for given inputs, though real-world applications may have additional complexities.