Binomial Distribution Formula Calculator

Binomial Probability Formula:

\[ P(k) = C(n,k) \times p^k \times (1-p)^{n-k} \]

Binomial Probability:

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1. What is Binomial Distribution?

The binomial distribution describes the probability of having exactly k successes in n independent Bernoulli trials with the same probability p of success. It's used when there are exactly two mutually exclusive outcomes of a trial (success/failure).

2. How Does the Calculator Work?

The calculator uses the binomial probability formula:

\[ P(k) = C(n,k) \times p^k \times (1-p)^{n-k} \]

Where:

\( n \) — Number of trials
\( k \) — Number of successes
\( p \) — Probability of success on an individual trial
\( C(n,k) \) — Combination of n items taken k at a time

Explanation: The formula calculates the probability of getting exactly k successes in n independent trials, where each trial has success probability p.

3. Applications of Binomial Distribution

Details: Used in quality control, medical testing, risk assessment, and any scenario with fixed number of independent trials with binary outcomes.

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.

5. Frequently Asked Questions (FAQ)

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

Q2: What if k > n?
A: The probability is 0 (can't have more successes than trials).

Q3: What's the expected value of a binomial distribution?
A: E(X) = n × p (mean number of successes).

Q4: What's the variance of binomial distribution?
A: Var(X) = n × p × (1 - p).

Q5: When is binomial distribution not appropriate?
A: When trials aren't independent or probability changes between trials.