1. What is the Binomial Probability Formula?

The binomial probability formula calculates the probability of getting exactly k successes in n independent Bernoulli trials (like coin flips) with probability p of success on each trial.

2. How Does the Calculator Work?

The calculator uses the binomial probability formula:

Where:

• \( P(k) \) — Probability of exactly k successes
• \( C(n, k) \) — Binomial coefficient ("n choose k")
• \( n \) — Number of trials (flips)
• \( k \) — Number of successes
• \( p \) — Probability of success on each trial (0.5 for fair coin)

Explanation: The formula accounts for all possible sequences that give exactly k successes and multiplies by the probability of each sequence.

3. Importance of Binomial Probability

Details: Binomial probability is fundamental in statistics for modeling binary outcomes like coin flips, success/failure experiments, and yes/no scenarios.

4. Using the Calculator

Tips: Enter number of flips (n), desired number of heads (k), and probability of heads (p, usually 0.5 for fair coin). All values must be valid (n ≥ k, 0 ≤ p ≤ 1).

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: What if I want at least k successes?
A: You would sum probabilities from k to n. This calculator gives exactly k successes.

Q3: What's a fair coin probability?
A: For a fair coin, p = 0.5. This is the default value in the calculator.

Q4: How accurate is this for large n?
A: Computationally accurate up to about n=1000. For very large n, normal approximation may be better.

Q5: Can I use this for non-coin scenarios?
A: Yes, any binary outcome with constant probability p can use this (e.g., success rates, survey responses).