Roll for Deception - The Biased Dices

Is My d6 Die Fair? A Statistical Analysis After 2000 Rolls

Have you ever wondered if a small tweak in design could affect something as “random” as a dice roll? As a maker and board game enthusiast, I decided to find out.

🛠️ The Maker Twist: A Custom Weighted Die

I designed and 3D-printed my own 6-sided die (d6), but with a small internal modification — I added extra weight directly beneath face “1.”

Why?

⚖️ The Physics Behind It

In a fair die, the mass is evenly distributed, and each face should have an equal chance (1 in 6) of landing face-up. But if one side is heavier, it naturally wants to settle on the bottom when thrown.

So if face 1 is heavier, then its opposite — face 6 — should come up more often.

This is a basic principle of center of gravity. When the mass is unbalanced, the heavier face resists rotation and settles downward more frequently.

🎯 The Hypothesis

If I increase the weight on face 1, face 6 will appear more often than the others.

To put this theory to the test, I rolled the die 2,000 times and recorded every single result.

📊 Results: Observed vs. Expected Frequencies

In a perfectly fair die, each face should appear about 333.33 times out of 2,000 rolls.

Here’s how my die performed:

🔍 Interpretation:

• Face 6 appeared 544 times — 63% more than expected!
• Face 1, the weighted side, appeared just 180 times, which is 46% less than expected.

That’s a clear, dramatic pattern. But is it statistically valid?

🧪 Chi-Squared Test: Is the Die Fair?

To test fairness mathematically, I ran a Chi-Squared Goodness-of-Fit Test. This compares the observed results to the expected results, and tells us if any differences are likely due to chance or to actual bias.

Test Results:

• Chi-squared statistic: 188.61
• p-value: 7.75 × 10⁻³⁹

In statistics, a p-value < 0.05 means the results are significant.
Mine? Basically zero.

📈 Stats Summary

• 🎯 Most frequent face: 6
• 📉 Least frequent face: 1
• 🧮 Standard deviation of counts: high (strong spread from expected)
• 📊 Deviation pattern: consistent with intentional weight shift

Even with 2,000 rolls — a robust sample — the trend holds strong.

🧠 Why This Matters

This experiment shows just how sensitive dice are to weight and balance. A tiny bit of extra infill on one face can cause large deviations over time.

It’s a fun project, but it also highlights something serious:

If your game relies on randomness, unfair dice can ruin the experience — or rig it completely.