Independent means the conditional probability does not change
Let Xᵢ describe success on attempt i. Attempts are independent for the model when learning earlier results does not change the probability assigned to the next eligible attempt. If every attempt has the same success probability p, the chance of no success in n attempts is(1 − p)ⁿ. The chance of at least one is its complement, 1 − (1 − p)ⁿ.
The formula does not make a player “due.” After a failure, the next independent attempt still has probabilityp. It also does not prove that a game uses independent rolls; independence is an input assumption that needs a rule or a deliberately labeled what-if scenario.
Different known rates can still use a product
Independence does not require identical rates. If the applicable per-attempt probabilities arep₁, p₂, …, pₙ and the events are independent, the probability of at least one success is1 − ∏(1 − pᵢ). The source must establish every rate and the state that chooses it. A table copied from an old patch or a community chart without version provenance is not a current schedule.
A hard guarantee is a state transition
Suppose a rule guarantees a qualifying result by counter H. The counter changes after each eligible failure and resets when the rule says a qualifying result occurs. The probability at the guarantee boundary is not the same fixed p; it is determined by the guarantee. Applying 1 − (1 − p)ⁿ beyond that boundary contradicts the rule. A deterministic progress calculation can still report H − cwishes remaining from current count c without claiming the intervening probability curve.
Selected targets add another conditional stage
“A top-rarity result appears” and “the selected target appears” are different events. A featured guarantee, path point, or designated-item rule is conditional on the top-rarity result and its prior state. Multiplying a per-wish rarity rate by a featured percentage is valid only when both conditional meanings, denominators, and dependencies are represented. A percentage conditional on obtaining a 5-star cannot be relabeled as the probability that the next wish is the featured character.
Genshin provides a concrete conditional-disclosure example
HoYoverse's updated Capturing Radiance Q&A publishes a base trigger probability and a consolidated promotional-character probability when winning a 5-star character in a Character Event Wish. It also describes a guarantee based on a sequence of prior promotional-character outcomes. Those statements define a conditional selection mechanic; they do not publish a complete next-wish 5-star schedule.
The Character Event Wish-2 rules separately document that two Character Event Wish surfaces share a guarantee count while other Wish types have independent counts. This is another reason not to pool every record bearing the word “Wish.” State ownership is part of the probability model.
Version changes invalidate timeless selectors
HoYoverse's Version 5.0 update added Capturing Radiance and reduced the Epitomized Path maximum from two Fate Points to one. A model written for the earlier system can remain historically accurate yet be wrong for a current plan. Store version, verification date, and exact Details text beside every stateful input.
A practical verification hierarchy
- Define the exact event: rarity result, featured selection, or named target.
- Define the eligible counter family and reset event.
- Capture the current rule, version, source URL, and in-game Details.
- Separate deterministic guarantees from disclosed probabilities.
- Preserve every conditional denominator: per wish, per top-rarity result, or per historical sequence.
- Use fixed-rate complement math only within a state where the rate is established and unchanged.
- When the per-attempt schedule is unavailable, report progress and ceilings instead of fabricating a curve.
What an observation log can and cannot show
A real log should preserve version, account/server context, counter before the attempt, applicable state, outcome, counter after the attempt, and rule source. It can identify data-entry errors and describe observed frequencies. It cannot prove a complex hidden schedule from a small convenience sample, and it must never be populated with illustrative rows presented as collected pulls.
Use the state-aware workflow
The Genshin Wish Guarantee & Resource Planner deliberately calculates a ceiling without a soft-pity probability. The Genshin Wish Guarantee Evidence Guide provides the capture and logging workflow. For truly fixed-rate events, use the formulas above only after documenting why independence and the rate apply.