How many attempts until the rare item? Drop rates and pity systems

Why it makes a good exploration

Many games publish drop rates; few players understand what they imply. The geometric distribution answers “how many tries?” and pity systems give a modified distribution worth analysing — with clear consumer implications.

The mathematics you'll need

• Geometric distribution: mean, variance, percentiles
• Cumulative probability and logarithms
• Truncated distributions with a pity rule
• Expected cost

Where the data comes from

Use a game's published drop rates; optionally record your own attempts.

• Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

A possible outline

1. Explain the geometric model and independence.
2. Compute mean, median and 90th percentile.
3. Add a pity rule and recompute.
4. Compare expected cost with and without pity.
5. Reflect on regulation and player psychology.

Showing personal engagement

• Use a game you play.
• Test the published rate with your own records.
• Discuss whether loot boxes are gambling.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

Taking it further

Model a soft pity system where the probability rises after each failure.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL games & puzzles exploration adds.