Loot Box Expected Cost Per Item

Compute bounded unequal-probability coupon-collection expectation by stable subset waiting-time recurrence for all remaining caller target items, accounting for owned items and non-target mass; one exclusive award per independent box, no pity/trading/refunds.

Description

Collecting all missing items takes longer when draws repeat owned items.

Collecting all missing items takes longer when draws repeat owned items. A subset expectation accounts for unequal item probabilities.^[1]

Enter aligned nonempty lists through 12 items: positive probabilities≤1 summing≤1 and binary owned flags, plus nonnegative draw cost. Evaluate expectations over unowned subsets; all owned gives zero. Return expected draws and cost.

When to use Loot Box Expected Cost Per Item

Use Loot Box Expected Cost Per Item when you need to evaluate the supplied records under the calculation and evidence rules below. At most one listed item is awarded per independent draw; no pity, changing rates or bundles are modeled.

Prepare the supplied game state, observations and explicit variant settings. Follow the card, board, coordinate and state conventions in the calculation rules.

How to use Loot Box Expected Cost Per Item and understand its results

Prepare the supplied game state, observations and explicit variant settings. Follow the card, board, coordinate and state conventions in the calculation rules.
Enter targetItemProbabilities, alreadyOwnedFlags, callerCostPerBox using the field table and calculation rules, then run the tool.
Read unownedTargetItemCount, expectedBoxesToCompleteTargets, expectedAdditionalCollectionCost. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
targetItemProbabilities	Target item probabilities	list of number	Required
alreadyOwnedFlags	Already owned flags	list of number	Required
callerCostPerBox	Caller cost per box	number	Required

E(S)=(1+sum(p_iE(S without i)))/sum(p_i)

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
unownedTargetItemCount	Unowned target item count	number
expectedBoxesToCompleteTargets	Expected boxes to complete targets	number
expectedAdditionalCollectionCost	Expected additional collection cost	number

json

{
  "input": {
    "targetItemProbabilities": [
      0.5,
      0.5
    ],
    "alreadyOwnedFlags": [
      0,
      0
    ],
    "callerCostPerBox": 2
  },
  "output": {
    "unownedTargetItemCount": 2,
    "expectedBoxesToCompleteTargets": 3,
    "expectedAdditionalCollectionCost": 6
  }
}

Limitations and assumptions

At most one listed item is awarded per independent draw; no pity, changing rates or bundles are modeled.

Numeric inputs, where present, must be finite. Follow the calculation-specific rules for overflow, underflow and exceptional cases. Numeric lists are limited to 10,000 entries; string lists to 128 strings of at most 128 characters each. The rules above may impose smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Gacha Pity Counter Expected Cost: A conditional pity schedule gives success probability after each failure count. Certainty at the final step bounds the remaining wait. For a complementary task, use Drop Rate Geometric Attempts To Success: A geometric waiting model assumes an independent fixed success probability on every attempt. A confidence target asks for a minimum whole draw count.

References

Loot Box Expected Cost Per Item — tool.io

Similar or alternative tools

Gacha Pity Counter Expected Cost
A conditional pity schedule gives success probability after each failure count. Certainty at the final step bounds the remaining wait.
Drop Rate Geometric Attempts To Success
A geometric waiting model assumes an independent fixed success probability on every attempt. A confidence target asks for a minimum whole draw count.