Visualizing Probability

Statistical concepts often feel abstract until they are anchored in physical reality. We’ve built a suite of interactive components—Dice, Coins, Urns, and Plots—to bring these concepts to life. Here is how they work with real-world examples.

1. The Gambler’s Fallacy (Bernoulli Trials)

In a fair coin toss, every flip is independent. The coin has no memory. Yet, humans often believe that after a streak of “Heads”, “Tails” is due.

Real World Data: In 1913, at the Monte Carlo Casino, roulette black came up 26 times in a row. Gamblers lost millions betting on red, convinced the streak had to break.

2. The Sample Space (Dice Rolls)

A single die defines a uniform distribution $S=\{1,2,3,4,5,6\}$. Two dice create a triangular distribution of sums, peaking at 7.

Craps Odds: The most common roll is 7 (6 ways), while 2 (Snake Eyes) and 12 (Boxcars) are the rarest (1 way each).

3. Sampling Without Replacement (The Urn)

Imagine a lottery or raffle. Once a winning ball is drawn, it isn’t put back. This changes the odds for the remaining balls. This is the Hypergeometric Distribution.

Real World Application (Quality Control): A factory produces 1000 widgets, 10 are defective. If an inspector samples 20 random widgets, what is the chance they find at least one defect? We model this exactly like drawing ‘defective’ balls from an urn.

4. The Normal Approximation

When we flip a coin enough times (say, $n=100$), the discrete Binomial distribution starts to look remarkably smooth. This is the Central Limit Theorem in action.

Insight: The discrete bars of the Binomial distribution (left) are perfectly approximated by the continuous Normal bell curve (right) when sample sizes are large. This allows statisticians to use Calculus to solve discrete counting problems!