Binomial Pdf Calculator

Here’s the comprehensive documentation for your Binomial PDF Calculator in Markdown format:

Binomial PDF Calculator Overview

What is It?

A specialized calculator that computes:

• Exact probability P(X=k) - chance of exactly k successes
• Cumulative probability P(X≤k) - chance of k or fewer successes

Used in quality control, medical trials, risk analysis, and scientific research.

The Formula

Exact Probability (PDF):

P(X=k) = C(n,k) times p^k times (1-p)^{n-k}

Cumulative Probability (CDF):

P(X leq k) = sum_{i=0}^k C(n,i) times p^i times (1-p)^{n-i}

Where:

• ( n ) = Number of trials (positive integer)
• ( k ) = Successes (integer, 0 ≤ k ≤ n)
• ( p ) = Success probability per trial (0 ≤ p ≤ 1)
• ( C(n,k) ) = Combinations = (\frac{n!}{k!(n-k)!})

Formula derived from Bernoulli trial principles in probability theory.

How to Use

1. Enter Parameters:

   • Success probability (p) as decimal (e.g., 0.3)
   • Number of trials (n) as integer (e.g., 20)
   • Successes (k) as integer (e.g., 7)

2. Select Mode:

   • “Exact” for P(X=k)
   • “Cumulative” for P(X≤k)

3. View Results:

   • Probability value (scientific notation available)
   • Combination count
   • Interactive distribution chart

Key Features

✔ Works for n ≤ 500 with precision
✔ Visualizes probability distribution
✔ Explains calculation steps
✔ Mobile-responsive design
✔ Sample datasets for quick testing

Common Questions (FAQs)

Q: When should I use binomial distribution?
A: When events are independent, have fixed trials, and constant success probability.

Q: Why does my probability show as 0?
A: Either extremely low probability (<1e-10) or invalid parameters (check k ≤ n).

Q: Can I calculate P(X≥k)?
A: Yes! Use 1 - P(X≤k-1).

Q: How accurate are the results?
A: Precision to 12 decimal places for n ≤ 120.

Terminology

PDF (Probability Density Function)
Probability of exactly k successes.

CDF (Cumulative Distribution Function)
Probability of up to k successes.

Bernoulli Trial
Single experiment with success/failure outcome.

Expected Value (μ)
Average successes: μ = n×p

Technical Notes

Limitations:

• Maximum n = 500 (for performance)
• Probability p must be 0-1
• k must be integer

Calculation Method:

• Uses logarithmic gamma function for large n
• Dynamic precision adjustment

Pro Tips

1. Bookmark common configurations
2. Screenshot graphs for reports
3. Compare multiple probabilities using browser tabs
4. For n >100, prefer cumulative mode