Drop Rate Math Guide 2025: EV, Union Probability, Binomial Trials & Confidence Intervals

This practical, reproducible guide explains the three pillars of drop-rate math that power our calculators: expected value (EV), union probability for "at least one", and binomial trials for rate estimation, wrapped up with confidence intervals so you can reason about uncertainty. Every formula is paired with a plug-in-your-numbers workflow you can verify in our tools: Diablo 4 Helltide Calculator, WoW Delves Calculator, and OSRS Barrows Calculator.

Expected Value (EV) definition & long-tail loot examples

Expected value is the average reward per attempt if you could repeat the exact same action a huge number of times. EV is neutral to time; when you care about speed you multiply by attempts per hour to get GPH. In loot games you typically compute EV from a table of outcomes × prices × probabilities.

EV formula and gamer-friendly interpretation

EV is the sum of each outcome's value times its probability.

Worked example — EV per action for a map

Suppose a map yields small currency shards with high probability and rare uniques with very low probability. Multiply each item's market price by its drop probability and add them up. That's your EV per action.

Step 1 — Build a price snapshot

Use market APIs or a community price sheet; freeze prices to avoid mid-run bias.

Step 2 — Multiply and add

Compute EV per action; multiply by actions per map to get EV/map; multiply by maps per hour to get GPH.

Union probability for "at least one" rare drop

If each run has success chance p and runs are independent, the chance of getting at least one success across n runs is the classic union-probability result:

Why gamers should care

Players often think in sessions: "What's the chance I see a unique within 50 chests?" The union formula answers that directly and explains why pacing (more attempts per hour) shifts your real-world experience even if EV/action is unchanged.

Worked example — Target farming a 5.8% drop

Plugging p = 0.058 into 1 − (1 − p)ⁿ yields:

• n = 10 → ~45%
• n = 25 → ~77%
• n = 50 → ~94%

You can reproduce these in our Barrows calculator.

Step-by-step

1. Measure or estimate per-run success p.
2. Decide a session size n (maps, chests, bosses).
3. Compute P(any); compare different routes by their n/hour.

Common pitfalls

• Pity mechanics break independence; use the specific pity curve instead (see soft pity, hard pity).
• Mixed-p sessions require weighted products: P(none) = Π(1−pᵢ).

Binomial distribution to estimate drop rate from samples

When you don't know the true drop rate, the binomial model lets you estimate it from data. If you saw k successes in n independent trials, the naïve estimator is p̂ = k/n.

From counts to confidence

The probability of exactly k successes is P(K=k) = C(n,k) pᵏ (1−p)ⁿ⁻ᵏ. We invert that logic to build an interval of plausible p given the data.

Clopper–Pearson (exact) interval

The exact 95% interval is conservative but safe for small samples. Many in-game tests have low counts, so exact methods are preferred.

Step 1 — Record data properly

Log total attempts and wins; avoid cherry-picking only lucky streaks.

Step 2 — Compute CI

Use an online binomial CI calculator, statistics library, or our blog code sample; store the lower/upper bound together with your run notes.

Confidence intervals & how much data you need

CI answers "how precise is my measured rate?". A 95% CI means "if you repeated the full experiment many times, 95% of the CIs would contain the true rate." It is not the probability the true rate lies inside a particular interval (that's a Bayesian statement).

Planning runs with CI width targets

Decide a tolerable CI width (e.g., ±2 percentage points) and plan the number of attempts accordingly. This converts vague "I'll do some runs" into a measurable plan.

Worked example — OSRS unique estimate

You want ±3% around your estimate. Compute the required n with an approximate normal method, or iterate using exact binomial intervals until the width target is met.

Step-by-step

1. Pick target half-width (e.g., 0.03).
2. Start from a prior guess of p (e.g., 0.06) and estimate n.
3. Collect data, recompute CI; stop when the width target is reached.

Measurement errors & bias to watch for

Price drift and mixed markets

EV depends on prices. Use consistent snapshots; for volatile items, prefer medians over means (Glossary: median).

Control: freeze and label prices

Freeze price CSVs with timestamps; note the market (SSF vs Trade) and region if applicable.

Checklists

• Are you mixing league snapshots?
• Are boss-specific drops priced correctly?

Route variance and time accounting

Two routes with the same EV/action can have very different GPH if one has better travel time. Always log actions/hour.

Control: shared timing sheet

Benchmark on your machine; include load times; report medians and 90th percentiles.

Tooling

Our calculators expose these trade-offs explicitly: Helltide · Delves · Barrows.

FAQ — drop rate math, EV/GPH & probability

Is GPH always better than EV?

They answer different questions. Use EV for per-action value, GPH for time efficiency, and P(any) for session risk. "Best" depends on your goal.

How do I combine mixed chances across a session?

Multiply "no-drop" probabilities: P(none) = Π(1 − pᵢ); then P(any) = 1 − P(none).

What if rolls are not independent?

Use the pity curve (soft/hard). Union formula overestimates if odds increase with attempts, and underestimates if odds decrease.