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 commonly used for coin flips, where each flip is independent and has the same probability of heads or tails.

2. How Does the Calculator Work?

The calculator uses the binomial probability formula:

Where:

• \( n \) — Number of trials (coin flips)
• \( k \) — Number of successes (desired heads/tails)
• \( p \) — Probability of success on each trial (0.5 for fair coin)
• \( \binom{n}{k} \) — Binomial coefficient ("n choose k")

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

3. Importance of Binomial Probability

Details: Binomial probability is fundamental in statistics for modeling binary outcomes. It's used in quality control, genetics, finance, and many scientific experiments.

4. Using the Calculator

Tips: Enter the number of trials (n ≥ 1), number of successes (0 ≤ k ≤ n), and probability of success (0 ≤ p ≤ 1). For a fair coin, p = 0.5.

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: How is this different from Bernoulli trials?
A: Bernoulli is a single trial (n=1), while binomial extends this to multiple independent trials.

Q3: What if I want cumulative probability?
A: You would sum probabilities for all values up to k. This calculator gives exact P(k).

Q4: Can I use this for unfair coins?
A: Yes, just adjust p (e.g., p=0.6 for a coin biased towards heads).

Q5: What's the binomial coefficient?
A: It counts the number of ways to choose k successes from n trials: \( \frac{n!}{k!(n-k)!} \).