How to Play

🪙 First Flip: See Fairness Take Shape

Begin with 10 coins and a short run. Press Play to watch the coins tumble and settle. The game instantly counts how many show heads and raises the matching bar in the histogram. Pause after a few flips and take a moment to look at where the bars are starting to gather — usually near the middle bins.

Early on, results may look unpredictable. A streak of heads can push the bars unevenly, or a few tails-heavy flips may make the distribution lean. This is randomness up close: surprising but always informative.

🔧 Adjust Your Experiment

Use the Coins per flip slider (1–40) to change how large each batch of coins is. A tiny batch of 3–5 coins spreads results widely across the chart, while a giant batch of 30–40 coins keeps outcomes close to the center. Then set your Flip count — short runs show noisy behavior, while long runs settle into steady patterns.

Play, pause, change settings, and play again. Use Reset whenever you want to clear the chart and try a new configuration. Before each run, pause to predict where the most common heads totals will appear — then test your guess by letting the experiment unfold.

📈 Compare Real Data to the Binomial Curve

Turn on the theoretical curve to see the expected peak and spread for your chosen number of coins. As your flip count rises, the live bars begin to match that curve more and more closely. This comparison builds intuition about fairness, probability, and why large samples reveal stable patterns.

🔁 Observe, Predict, Reflect

Try flipping just 2 coins, then 10, then 40. Notice how the center bars become more dominant as the number of coins grows. Experiment with short bursts of 10 flips and long runs of 500. Repeat experiments to see how randomness behaves differently from trial to trial, even when the underlying probabilities are the same.

Study Notes

⚖️ Fairness, Made Visible

Every coin is a 50/50 chance experiment. The histogram visualizes how often each possible number of heads appears. Instead of seeing fairness as an abstract idea, you watch it materialize in real time as the center bins rise and the outer bins lag behind.

🌊 Spread, Clustering, and Sample Size

Small coin batches produce wide, wobbly spreads because each individual coin has a strong impact on the total. Large batches cluster tightly around the center — flipping 40 coins rarely yields only 5 or 35 heads. Short runs are jumpy and irregular; long runs smooth out beautifully as the law of large numbers takes effect.

🧠 Reflection Starters

• “Which heads total do you expect to appear most often? Why?”
• “Is a streak of heads strange, or just randomness doing its job?”
• “How do the bars change when you flip more coins at once?”
• “Does your histogram come close to the theoretical curve?”

These prompts help players connect the visual patterns to ideas in probability — showing that even simple coin flips contain rich mathematical structure waiting to be explored.