Binomial Distribution Calculator

Number of Events (n):

Probability of Success (p):

Calculate:

Number of Successes (r):

From (r0):

To (r1):

1. What is the Binomial Distribution?

Definition: The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

Purpose: Used to analyze outcomes of repeated experiments, such as coin flips or dice rolls.

2. How Does the Calculator Work?

The calculator uses the formula: \( P(X=r) = \binom{n}{r} \times p^r \times (1-p)^{n-r} \), where:

\( \binom{n}{r} \): Number of combinations ("n choose r").
\( p \): Probability of success per trial.
\( n \): Total number of trials.
\( r \): Number of successes.

Steps:

Enter the number of events (n) and probability of success (p).
Select the calculation type and input the relevant success numbers.
Click "Calculate" to get the probability and statistical measures.

3. Why Use the Binomial Distribution?

Useful for:

Quality Control: Testing defect rates.
Surveys: Estimating response proportions.
Gambling: Analyzing win probabilities.

4. Using the Calculator

Example:

n = 5, p = 0.667, Exactly r = 3: \( P(X = 3) \approx 0.329 \).
n = 4, p = 0.5, At most r = 2: \( P(X \leq 2) \approx 0.6875 \).
n = 5, p = 0.5, Between r0 = 1 and r1 = 3: \( P(1 \leq X \leq 3) \approx 0.8125 \).
Measures: Mean = 2.5, Variance = 1.25, Std Dev ≈ 1.118.

5. Frequently Asked Questions (FAQ)

Q: What if p = 0 or 1?
A: The distribution becomes deterministic (all failures or all successes).

Q: Can n be zero?
A: No, n must be at least 1.

Q: Applications?
A: Used in statistics, biology, and engineering for binary outcomes.