1. What is Binomial Probability?

The binomial probability describes the probability of having exactly k successes in n independent Bernoulli trials with success probability p. It's fundamental in statistics for modeling yes/no outcomes.

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 essential in quality control, medical testing, survey analysis, and any scenario with binary outcomes. It helps determine likelihoods for decision-making.

4. Using the Calculator

Tips: Enter number of trials (n ≥ 1), number of successes (0 ≤ k ≤ n), and probability (0 ≤ p ≤ 1). All values must be valid for calculation.

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 k > n?
A: The calculator prevents this as it's impossible to have more successes than trials.

Q3: What are common applications?
A: Clinical trial analysis, manufacturing defect rates, A/B testing, and genetics (dominant/recessive traits).

Q4: How accurate is the calculation?
A: Mathematically exact for given inputs, though real-world scenarios may have dependencies between trials.

Q5: Can I calculate cumulative probabilities?
A: This calculator gives exact P(k). For P(X≤k), you'd sum probabilities from 0 to k.