Binomial Distribution Calculator

Binomial Distribution:

\[ \text{Mean} = n \cdot p, \quad \text{Variance} = n \cdot p \cdot (1-p) \]

Mean (μ):

Variance (σ²):

Standard Deviation (σ):

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1. What is Binomial Distribution?

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

2. How Does the Calculator Work?

The calculator uses the binomial distribution formulas:

\[ \text{Mean} = n \cdot p \] \[ \text{Variance} = n \cdot p \cdot (1-p) \] \[ \text{Standard Deviation} = \sqrt{n \cdot p \cdot (1-p)} \]

Where:

\( n \) — Number of trials
\( p \) — Probability of success on each trial

Explanation: The binomial distribution models the probability of getting exactly k successes in n independent trials.

3. Importance of Binomial Distribution

Details: Binomial distribution is widely used in statistics, quality control, medical testing, and any scenario involving yes/no outcomes with fixed probabilities.

4. Using the Calculator

Tips: Enter the number of trials (n) as a positive integer and probability of success (p) as a decimal between 0 and 1.

5. Frequently Asked Questions (FAQ)

Q1: What are the assumptions of binomial distribution?
A: Fixed number of trials, independent trials, only two outcomes (success/failure), constant probability of success.

Q2: When should I use binomial distribution?
A: When you have a fixed number of independent trials with constant probability of success, such as coin flips, pass/fail tests, or yes/no surveys.

Q3: What's the difference between binomial and normal distribution?
A: Binomial is discrete (counts successes), normal is continuous. For large n, binomial approximates normal distribution.

Q4: How do I calculate probability for exact number of successes?
A: Use the binomial probability formula: \( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \)

Q5: What if p is greater than 1?
A: Probability values must be between 0 and 1. Values outside this range are invalid for binomial distribution.