1. What is Binomial Probability?

The binomial probability calculates the chance of getting exactly k successes in n independent trials, each with success probability p. For coin flips, this models the probability of getting exactly k heads in n flips.

2. How Does the Calculator Work?

The calculator uses the binomial probability formula:

Where:

• \( P(k) \) — Probability of exactly k successes (dimensionless)
• \( C(n, k) \) — Binomial coefficient (dimensionless)
• \( n \) — Number of trials/flips (dimensionless)
• \( k \) — Number of successes (dimensionless)
• \( p \) — Probability of success on each trial (dimensionless, usually 0.5 for fair coins)

Explanation: The formula combines the number of possible success combinations with the probability of each specific outcome.

3. Importance of Probability Calculation

Details: Understanding binomial probability is essential for statistics, risk assessment, and predicting outcomes in repeated independent events like coin flips, medical trials, or quality control testing.

4. Using the Calculator

Tips: Enter number of flips (n), desired number of successes (k), and probability of success (p, default 0.5 for fair coins). All values must be valid (n > 0, 0 ≤ k ≤ n, 0 ≤ p ≤ 1).

5. Frequently Asked Questions (FAQ)

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

Q2: How is this implemented in Python?
A: Python's math.comb() calculates C(n,k), and the ** operator handles exponents. Similar to our PHP implementation.

Q3: What if I want at least k successes?
A: Sum probabilities from k to n. This calculator gives exactly k successes.

Q4: Why is p usually 0.5 for coins?
A: For fair coins, head/tail probability is 50%. Adjust p for biased coins.

Q5: What are practical applications beyond coins?
A: Used in genetics (allele inheritance), medicine (treatment success rates), and business (conversion rates).