1. What is Binomial Probability?

The binomial probability formula calculates the probability of getting exactly k successes in n independent Bernoulli trials (like coin flips), each 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:

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

3. Python Implementation

Here's the Python code that implements this calculation:

from math import comb

def binomial_probability(n, k, p):
    """Calculate binomial probability P(k) in n trials with success probability p"""
    return comb(n, k) * (p ** k) * ((1 - p) ** (n - k))

# Example usage:
n_flips = 10  # Number of coin flips
k_heads = 5   # Number of desired heads
p_head = 0.5  # Probability of heads (fair coin)

probability = binomial_probability(n_flips, k_heads, p_head)
print(f"Probability: {probability:.4f} ({probability*100:.2f}%)")
                

4. Using the Calculator

Tips: Enter number of flips (n), desired number of successes (k), and probability of success (p, 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 outcomes (exact counts), while normal is continuous. For large n, binomial approximates normal.

Q2: How is C(n, k) calculated?
A: It's the combination formula: n! / (k! × (n-k)!), representing "n choose k" ways to get k successes.

Q3: What if I want at least k successes?
A: You'd sum probabilities from k to n. For "at least 3 heads in 10 flips", calculate P(3)+P(4)+...+P(10).

Q4: Why is p usually 0.5 for coins?
A: For fair coins, heads and tails are equally likely. For biased coins, adjust p accordingly.

Q5: Can this be used for non-coin scenarios?
A: Yes! Any binary outcome with constant probability (success/failure, yes/no) can use binomial probability.