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Binomial Probability Explorer

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About This MicroSim

The binomial distribution is one of the most important distributions in statistics, and this explorer helps you build intuition for how it works. Adjust the parameters and watch the distribution change shape before your eyes!

How to Use

  1. Drag the n slider to change the number of trials (1 to 50)
  2. Drag the p slider to change the probability of success (0.01 to 0.99)
  3. Drag the k slider to select a specific value and see its probability
  4. Hover over any bar to see its exact probability
  5. Use the preset buttons to see common patterns

Preset Buttons

  • Fair Coin: n=20, p=0.5 - perfectly symmetric distribution
  • Skewed Left: n=20, p=0.8 - distribution concentrated on the right
  • Skewed Right: n=20, p=0.2 - distribution concentrated on the left
  • Reset: Return to default settings (n=10, p=0.5)

Key Concepts

The BINS Conditions

Before using the binomial distribution, verify:

  • Binary outcomes (success or failure)
  • Independent trials
  • Number of trials is fixed
  • Same probability for each trial

The Binomial Formula

\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

Where:

  • \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) is the number of ways to choose k successes from n trials
  • \(p^k\) is the probability of k successes
  • \((1-p)^{n-k}\) is the probability of (n-k) failures

Distribution Shape Patterns

  • When p = 0.5: Distribution is symmetric
  • When p < 0.5: Distribution is right-skewed (tail stretches right)
  • When p > 0.5: Distribution is left-skewed (tail stretches left)
  • As n increases: Distribution becomes more bell-shaped (approaches normal)

Learning Objectives

After using this MicroSim, you'll be able to:

  • Calculate binomial probabilities using the formula
  • Predict how changing n and p affects the distribution shape
  • Find the mean and standard deviation of a binomial distribution
  • Identify when a distribution is symmetric vs. skewed
  • Apply the binomial formula with actual values

Lesson Plan

Introduction (5 minutes)

Ask: "If you flip a fair coin 10 times, what's the probability of getting exactly 7 heads?" Use the explorer to find out!

Guided Exploration (15 minutes)

  1. Symmetric case: Set p = 0.5 and vary n from 5 to 50. Watch the distribution become more bell-shaped.

  2. Skewness: With n = 20, slide p from 0.2 to 0.8. Notice how the direction of skewness changes.

  3. Formula verification: For n = 10, p = 0.5, k = 7:

  4. C(10,7) = 120
  5. 0.5^7 = 0.0078125
  6. 0.5^3 = 0.125

  7. P(X = 7) = 120 * 0.0078125 * 0.125 = 0.1172

  8. Mean and SD: Verify that mean = np and SD = sqrt(np(1-p))

Discussion Questions

  1. Why does the distribution become more symmetric as n increases?
  2. What happens to the spread (SD) as p moves closer to 0 or 1?
  3. Why must the BINS conditions be met to use this formula?

Calculator Practice

On TI-83/84:

  • binompdf(n, p, k) gives P(X = k)
  • binomcdf(n, p, k) gives P(X <= k)

Sylvia Says

"The binomial distribution is like a recipe - once you know n and p, you can calculate the probability of any outcome. It's one of the most practical tools in your statistical toolkit!"

Embedding This MicroSim

1
<iframe src="https://dmccreary.github.io/statistics-course/sims/binomial-probability-explorer/main.html" height="602px" scrolling="no"></iframe>

Technical Notes

  • Built with p5.js 1.11.10
  • Uses canvas-based sliders and buttons
  • Binomial coefficient calculated using iterative method for stability
  • Responsive width design
  • Drawing height: 500px, Control height: 100px

References

  • Chapter 13: Random Variables
  • Concepts: Binomial Setting, Binomial Conditions, Binomial Distribution, Binomial Probability, Binomial Formula, Binomial Mean, Binomial Standard Dev