Run experiments, watch empirical probabilities converge to theoretical values, and explore the Law of Large Numbers in real time.
Coin Flip Experiment
Explore the binomial distribution — each flip has a 50/50 chance. Watch how results converge to 50% as trials increase.
Total Flips
0
Heads
0
Tails
0
Heads %
—
Expected %
50%
Law of Large Numbers: With a fair coin, each flip is independent and has a 50% chance of heads. While short runs can be wildly uneven, the empirical proportion approaches 50% as the number of trials increases. Try changing the bias to see skewed distributions!
Dice Roll Experiment
Roll dice and watch the uniform distribution emerge. Add multiple dice to see the Central Limit Theorem — sums approach a normal distribution.
Total Rolls
0
Mean
—
Expected Mean
3.5
Std Dev
—
Last Roll
—
Central Limit Theorem: A single die produces a uniform distribution (every face equally likely). But when you add multiple dice, the sum's distribution becomes bell-shaped. This is the Central Limit Theorem in action — the sum of many independent random variables approaches a normal distribution.
Card Draw Experiment
Draw from a standard 52-card deck with or without replacement. See how the probabilities change when the deck isn't reshuffled.
Total Draws
0
Cards Left
52
Last Card
—
Most Common
—
Sampling Without Replacement: When you don't reshuffle, each draw changes the probabilities for future draws. If you draw 3 Hearts in a row, the chance of the next card being Hearts drops from 25% to about 21%. This is fundamental to understanding dependent vs independent events in probability.
Birthday Paradox
With just 23 people in a room, there's a 50% chance two share a birthday. With 70 people, it's over 99%. Run simulations to see this counterintuitive result.
Simulations
0
Matches Found
0
Empirical %
—
Theoretical %
50.7%
Difference
—
Why it works: With N people, there are N(N-1)/2 pairs to compare. For 23 people that's 253 pairs — much more than intuition suggests. Each pair has a 1/365 chance of matching, so the probability of no match is (364/365)^253 ≈ 49.3%, meaning at least one match is 50.7%.
Normal Distribution Sampler
Generate samples from a normal distribution. Watch the bell curve emerge from randomness and explore how mean and standard deviation shape the distribution.
Samples
0
Sample Mean
—
Target Mean
0
Sample Std Dev
—
Target Std Dev
1
The Normal Distribution: The bell curve appears everywhere in nature — heights, test scores, measurement errors, particle velocities. It's defined by just two parameters: the mean (μ) and standard deviation (σ). About 68% of values fall within 1σ, 95% within 2σ, and 99.7% within 3σ of the mean.