Editing Openai/692e9127-f13c-800a-8bd6-4bf3c4a385c8 (section)

=== Assistant: Your inputs still conflict, so any answer has two branches. ===
Your inputs still conflict, so any answer has two branches.

Problem first:
In the sample you gave:
* Waves: 3732
* Packages: 1934
* Skipped waves: 1336

Then
* Non-skipped waves = 3732 − 1336 = 2396
* Package rate on non-skipped waves = 1934 / 2396 ≈ 80.7%

You said the game’s rate is 78%. Your observed skip rate is about 35.8%, not 19%. So either the in-game percentages are wrong, or your run has modifiers, or the game is not using those numbers in a simple way.

So I will give you two answers:
* One using the stated mechanics: 19% skip, 78% package on non-skipped waves.
* One using the observed rates from your run.

==== Step 1: chance of a package on any given wave ====
* Probability wave is not skipped: 1 − 0.19 = 0.81
* Given not skipped, probability of package: 0.78

Per wave:

P(package per wave)=0.81×0.78=0.6318 or 63.18%P(\text{package per wave}) = 0.81 \times 0.78 = 0.6318 \text{ or } 63.18\%P(package per wave)=0.81×0.78=0.6318 or 63.18%
Step 2: chance of a common module on any given wave

Every package has a 1.2% chance of a common module:

P(module per wave)=0.6318×0.012=0.0075816≈0.758%P(\text{module per wave}) = 0.6318 \times 0.012 = 0.0075816 \approx 0.758\%P(module per wave)=0.6318×0.012=0.0075816≈0.758%
Step 3: expected commons per 4500-wave run

E[modules per run]=4500×0.0075816≈34.1E[\text{modules per run}] = 4500 \times 0.0075816 \approx 34.1E[modules per run]=4500×0.0075816≈34.1
So under the official 19% / 78% numbers:

Expected common modules per 4500-wave run ≈ 34.

Step 4: what is a “minimum” in a normal run

True minimum is 0. You could get insanely unlucky and see no modules.

If you want “how low before it is very unlikely,” use a binomial approximation:
* Mean ≈ 34.1
* Standard deviation ≈ 5.8

Then, roughly:
* About 95% of runs will be above: 34.1 − 1.645 × 5.8 ≈ 25 modules
* About 99% of runs will be above: 34.1 − 2.33 × 5.8 ≈ 21 modules

So, with the stated mechanics, a realistic “minimum”:
* Normal bad run (about 1 in 20): around 25 modules
* Very bad run (about 1 in 100): around 21 modules

==== From your sample: ====
* Packages per wave (including skips): 1934 / 3732 ≈ 0.518 or 51.8%

Use that instead of the theoretical 63.18%.

Step 1: chance of module per wave

P(module per wave)≈0.518×0.012≈0.006216P(\text{module per wave}) \approx 0.518 \times 0.012 \approx 0.006216P(module per wave)≈0.518×0.012≈0.006216
Step 2: expected commons per 4500-wave run

E[modules per run]=4500×0.006216≈28.0E[\text{modules per run}] = 4500 \times 0.006216 \approx 28.0E[modules per run]=4500×0.006216≈28.0
So based on your actual results:

Expected common modules per 4500-wave run ≈ 28.

Step 3: “minimum” here

For this case:
* Mean ≈ 28.0
* Standard deviation ≈ 5.3

Then:
* About 95% of runs will be above: 28.0 − 1.645 × 5.3 ≈ 19 modules
* About 99% of runs will be above: 28.0 − 2.33 × 5.3 ≈ 16 modules

==== If you define “minimum” as: ====
* Hard mathematical minimum: 0 modules, always.
* Expected average with the official 19% skip and 78% package numbers: about 34 common modules per 4500-wave run.
* Expected average using your observed package rate: about 28 modules per run.
* Normal “worst case” with official numbers (95% of runs or better): about 25 modules.
* Normal “worst case” with your observed rates: about 19 modules.

If you want one planning number that reflects your current reality, use:

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