Default is λ=0 (no bonus triggers). Because bonus triggers add free attempts, actual items needed approaches c·S/(1+λ) — smaller than the default output. Only fill this in if you have measured it.
1 = triggers add one attempt each (min variance).How to measure λ, μK: make several hundred attempts, then enter the items on hand before/after, actual attempts count (and trigger count).
Monte Carlo verification: compares the closed-form 95% against a distribution generated by running the RNG one attempt at a time.
Each attempt consumes c items and yields p points (PK Battle: c=20, p=2).
Bonus triggers fire probabilistically per attempt, so items needed to reach the threshold is a random variable. This tool returns not the average but the number that exceeds the threshold with 95% probability (the 95th percentile). Confidence is fixed at 95%.
Required attempts S = ⌈(threshold − current) ÷ p⌉. By a renewal-process approximation,
mean paid attempts E[X] = S/(1+λ), variance Var[X] = S·(λμ_K − λ²)/(1+λ)³,
items needed = ⌈c·(E[X] + 1.645·√Var[X])⌉ (1.645 = 95th percentile).
With default λ=0, no bonus triggers, so items needed = c·S is deterministic. Add λ and the mean drops, but we round to the 95% side to absorb variance.