How to use (3 steps)
- Choose uniform (equal chances) or weighted (rarities).
- Enter n and optional t/target; paste weights or probabilities if needed.
- Check the results, then run the simulation for intuition or verification.
Assumptions: independent draws with fixed probabilities. Pity/guarantee systems are out of scope.
Inputs
Simulation results are not stored in the URL—only settings and the seed are shared.
Results
Completion curve (CDF)
Simulation (Monte Carlo)
Runs locally in your browser. Use a seed to reproduce the same run.
Examples
Uniform example (n=50)
With 50 equally likely types, the expected draws is 50·H_50 ≈ 224.96. The 90% completion point is much higher than the mean.
Rare item example (1%)
If one type has probability 0.01 and the others share the remaining 0.99, the rare item dominates the completion time. Use weighted mode to see how the expectation jumps.
Interpretation (why it takes so long)
Uniform case: the classic n·Hₙ
If all n types are equally likely, the expected number of draws is E[T] = n·Hₙ where Hₙ is the harmonic number. A useful approximation is:
E[T] ≈ n·(ln n + γ) where γ ≈ 0.57721 (Euler–Mascheroni constant).
For n = 50, this gives about 50·(ln 50 + 0.577) ≈ 224.5, close to the exact value.
Why the “last coupon” dominates
Near the end, most draws are duplicates. The remaining unseen probability becomes small, so the waiting time for the last missing type can be large. This is why high-percentile targets (t90, t99) are often much larger than the mean.
Weighted case: rarities matter
- If probabilities are uneven, the rarest types can dominate the completion time.
- For large n or very skewed weights, simulation is often the practical way to understand typical completion times and variability.
References
How to read expected draws, t90, and weighted rarity
Mean vs percentile targets
E[T] is the average number of draws across many runs, but it is often too optimistic for planning a real collection. Use t90 or t99 when you need a safer answer such as “How many pulls do I need before I am usually done?”
Uniform vs weighted mode
Uniform mode assumes every type has the same chance. Weighted mode is for rarities, where one or a few low-probability items dominate the tail. If your setting looks like packs, loot tables, or gacha rates, weighted mode is usually the right model.
When to trust the exact result
For moderate sizes, the exact or DP-based outputs are the best reference. When the number of types is large or the weights are highly skewed, use the simulation to see how wide the completion-time spread becomes rather than relying on a single mean value.
What this does not model
This page assumes independent draws with fixed probabilities. Pity systems, guaranteed drops, trading, duplicate protection, and changing rates over time need a different model.
FAQ
Why does the last item take so long?
Once most types are collected, each new draw is likely a duplicate. The waiting time for the last unseen type grows like 1/p_min, so the final item often dominates the total time.
Is the uniform formula exact?
Yes. For equal probabilities the expectation is n·H_n and the DP curve gives exact completion probabilities up to the computed t range.
What if probabilities are not uniform?
Use weighted mode with probabilities or weights. Rare items can dominate the completion time, so high-percentile targets and simulation are often more informative than the mean alone.
What is the difference between E[T] and t90?
E[T] is the average completion time across many runs. t90 is the draw count by which 90% of runs have finished, so it is usually better for budgeting or planning.
Does this include pity or guarantees?
No. This tool assumes independent draws with fixed probabilities. Guarantees, pity counters, trading, and duplicate protection need a different model.