Openai/692e9127-f13c-800a-8bd6-4bf3c4a385c8

This conversation was exported from ChatGPT

gpt-5-1

User: I need help doing some calculations for a game.

I need help doing some calculations for a game.

Every wave in what the game calls a "run", I have a 78% chance of getting a "Recovery Package"

However, every wave has a 19% chance of skipping a wave. Each skipped wave means that I don't get a recovery package.

It seems like my overall "Recovery package" percentage is about 50-60% when factoring in every wave.

For example, im on wave 3690 and have only received 1911 recovery packages.

Every recovery package has a 1.2% chance of dropping a common module.

So here's my question... if I run an average of 4500 waves per run. What is the minimum amount of common modules I would get per run?

Assistant: This has several flaws you need to fix before you can trust any number.

This has several flaws you need to fix before you can trust any number.

1. Your stated rates contradict your observed data. • Expected recovery package chance per wave equals 0.78 times the probability the wave is not skipped. • You gave a 19 percent skip rate, so the expected package rate becomes 0.78 times 0.81 equals 0.6318 or 63.18 percent. • Your observed 1911 packages out of 3690 waves equals 51.8 percent. That does not match 63.18 percent. • Either the skip rate or the package rate is not constant or the game uses a non-independent system.
2. You ask for the minimum number of common modules per run but do not define minimum. • Math minimum tends to zero because probabilities do not guarantee drops. • You likely want the expected value, not the minimum. Say that if correct.

Let me clarify the assumptions so the math actually works. I need:

1. Should I treat 78 percent as the chance only on non-skipped waves or as the chance before the skip check meaning the game rolls package first then skip overrides it?
2. Do you want expected average common modules per 4500-wave run or the lowest number you would expect in a normal outcome for example the 1st percentile?
3. Should I trust your observed 51.8 percent package rate instead of your stated 63.18 percent theoretical rate?

gpt-5-1-thinking

User: 1.

1. The game rolls the "wave skip" calculation before it rolls the recovery package chance
2. Can I get both?
3. At the time of the calculation I should have included the number of waves skipped to help. For example: same run: wave = 3732. Packages = 1934. Waves skipped. 1336

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:

>

User: Does probability change in any way if I do longer runs?

Does probability change in any way if I do longer runs? For example, if I go to an easier level, I can hit up to 5500 waves. If I go to an even easier level, 9000

Assistant: You are mixing two things that need to stay separate.

You are mixing two things that need to stay separate. The probability per wave does not change with longer runs. Only the spread of results and the total drops change.

Let me clarify.

No.

If the game rules stay the same and each wave is independent:

• Skip chance stays 19 percent.
• Recovery package chance on a non skipped wave stays 78 percent.
• Common module chance per package stays 1.2 percent.

So the chance on any single wave is the same whether you do 500, 4500, 5500, or 9000 waves.

What does change:

• The expected total number of packages and modules goes up in direct proportion to waves.
• The percent you see in practice (like 51 to 53 percent packages) wobbles less when you have more waves.

Short run: luck can push you high or low. Long run: your observed percent gets pulled toward the true rate.

I will use your observed rate, since that reflects reality better than the stated 19 percent and 78 percent.

From your data:

• Packages per wave ≈ 1934 / 3732 ≈ 0.518 or 51.8 percent.
• Common module chance per package = 1.2 percent.
• So common modules per wave ≈ 0.518 × 0.012 ≈ 0.006216.

Now scale by total waves.

- 4500 waves: 4500 × 0.006216 ≈ 28 modules per run.

• 5500 waves: 5500 × 0.006216 ≈ 34 modules per run.
• 9000 waves: 9000 × 0.006216 ≈ 56 modules per run.

So if the game keeps behaving like your current run, longer runs just mean more modules in a straight line way. About 28 → 34 → 56 as you add more waves.

True minimum never changes. It is always 0 modules. You can always get absurdly unlucky.

What changes is how unusual that kind of bad run is:

• With more waves, the distribution tightens around the average.
• So your “normal worst case” gets closer to the mean in percent, even though the raw gap in module count is a bit bigger.

Rough idea using your observed behavior:

• At 4500 waves: Mean ≈ 28 modules. A really bad but not insane run might land in the high teens.
• At 9000 waves: Mean ≈ 56 modules. A really bad run might be in the low 40s.

So:

• Per wave probability does not improve with longer runs.
• Longer runs give you more modules on average and make extreme bad luck less common.

If you tell me which level you usually farm and roughly how long a 4500, 5500, or 9000 wave run takes, I can help you choose the sweet spot in modules per hour, not just per run.

User: Alright - let's get deeper.

Alright - let's get deeper.

Each module can be one of 4 types: cannon, armor, generator, and core. Each model that drops can provide me with 18 "shards"

These shards are what I use to level up my higher end modules.

Each module at this point requires up to 3800 shards to level up by 1.

How many waves/runs will I need to do just to level up my higher end modules

Assistant: This is missing one key piece: how many high end modules and how many levels you actually want.

This is missing one key piece: how many high end modules and how many levels you actually want. So I will give you the math per +1 level on a single high end module, based on your real observed drops. You can scale it for any number of levels or modules.

I will also treat “module” here as your common modules only, since those are the ones that drop at 1.2% from recovery packages and give 18 shards each.

From your run:

• Waves: 3732
• Packages: 1934

Packages per wave (including skips):

ppkg per wave=1934/3732≈0.518p_{\text{pkg per wave}} = 1934 / 3732 \approx 0.518ppkg per wave​=1934/3732≈0.518 Package → common module: 1.2%:

pmodule per wave≈0.518×0.012≈0.00622p_{\text{module per wave}} \approx 0.518 \times 0.012 \approx 0.00622pmodule per wave​≈0.518×0.012≈0.00622 So on average:

Each module gives 18 shards.

shards per wave=0.00622×18≈0.112 shards per wave\text{shards per wave} = 0.00622 \times 18 \approx 0.112 \text{ shards per wave}shards per wave=0.00622×18≈0.112 shards per wave So:

You said each level costs up to 3800 shards.

waves per level=3800/0.112≈33,950 waves\text{waves per level} = 3800 / 0.112 \approx 33{,}950 \text{ waves}waves per level=3800/0.112≈33,950 waves Call it:

Use your three run sizes.

- Shards per run: 4500×0.112≈504 shards4500 \times 0.112 \approx 504 \text{ shards}4500×0.112≈504 shards

• Runs per level: 3800/504≈7.53800 / 504 \approx 7.53800/504≈7.5

So:

- Shards per run: 5500×0.112≈616 shards5500 \times 0.112 \approx 616 \text{ shards}5500×0.112≈616 shards

• Runs per level: 3800/616≈6.23800 / 616 \approx 6.23800/616≈6.2

So:

- Shards per run: 9000×0.112≈1007 shards9000 \times 0.112 \approx 1007 \text{ shards}9000×0.112≈1007 shards

• Runs per level: 3800/1007≈3.83800 / 1007 \approx 3.83800/1007≈3.8

So:

Let:

• LLL = number of levels you want on one high end module
• MMM = number of high end modules you want to push

Total shard cost:

Total shards≈3800×L×M\text{Total shards} \approx 3800 \times L \times MTotal shards≈3800×L×M Given the per run shard rates above, you can do:

runs needed≈3800×L×Mshards per run\text{runs needed} \approx \frac{3800 \times L \times M}{\text{shards per run}}runs needed≈shards per run3800×L×M​ For example, if you wanted +3 levels on 4 modules, at 4500 wave runs:

• Shard cost = 3800 × 3 × 4 = 45,600
• Runs ≈ 45,600 / 504 ≈ 90 runs of 4500 waves.

If you tell me how many high end modules you care about and how many levels you want on each, I can plug in the numbers and give you a concrete “you are looking at about X runs at 4500 vs Y at 9000.”

User: If I upgraded my recovery package so it increases the chance of dropping a common module from 1.2% to 1.6%.

If I upgraded my recovery package so it increases the chance of dropping a common module from 1.2% to 1.6%. What changes would I see?

Can you calculate the numbers from all the previous requests with this new variable

Assistant: This change only affects one thing.

This change only affects one thing. You increased the “package → common module” chance. So every number that depends on common modules or shards improves by the same factor.

Old per package: 1.2% New per package: 1.6%

Factor increase: 1.6 / 1.2 = 4 / 3 ≈ 1.33

So: • Common modules per wave go up by about 33%. • Shards per wave go up by about 33%. • Waves and runs needed per level go down to about 75% of what they were.

Below are the updated numbers for both cases:

1. Using your observed drop behavior (more realistic)
2. Using the “official” 19% skip and 78% package numbers

1. Using your observed data (the realistic branch)

From your run:

• Waves: 3732
• Packages: 1934
• So packages per wave ≈ 1934 / 3732 ≈ 0.5182

1.1. Per wave probabilities

Old module chance per wave (1.2% per package):

• p(module per wave) = 0.5182 × 0.012 ≈ 0.00622

New module chance per wave (1.6% per package):

• p(module per wave) = 0.5182 × 0.016 ≈ 0.00829

So:

• Old: about 0.622% of waves give a common module.
• New: about 0.829% of waves give a common module.

1.2. Expected common modules per run

4500-wave run

• Old: ≈ 27.98 modules
• New: ≈ 37.31 modules

Call it: • Old: about 28 modules • New: about 37 modules

5500-wave run

• Old: ≈ 34.20 modules
• New: ≈ 45.60 modules

So: • Old: about 34 modules • New: about 46 modules

9000-wave run

• Old: ≈ 55.97 modules
• New: ≈ 74.62 modules

So: • Old: about 56 modules • New: about 75 modules

1.3. Shards per wave and per run

Each module gives 18 shards.

Old shards per wave:

• 0.00622 × 18 ≈ 0.112 shards per wave

New shards per wave:

• 0.00829 × 18 ≈ 0.149 shards per wave

So:

4500-wave run

• Old: 0.112 × 4500 ≈ 504 shards
• New: 0.149 × 4500 ≈ 672 shards

5500-wave run

• Old: ≈ 616 shards
• New: ≈ 821 shards

9000-wave run

• Old: ≈ 1007 shards
• New: ≈ 1343 shards

1.4. Waves and runs needed for +1 level (3800 shards)

Cost per level: 3800 shards.

Old waves per level:

• 3800 / 0.112 ≈ 33,948 waves

New waves per level:

• 3800 / 0.149 ≈ 25,461 waves

So the upgrade cuts the wave requirement to about 75% of the old value.

Convert to runs:

At 4500 waves per run

• Old shards per run: ≈ 504 • Runs per level: 3800 / 504 ≈ 7.54 runs
• New shards per run: ≈ 672 • Runs per level: 3800 / 672 ≈ 5.66 runs

So 7–8 runs drop to about 5–6 runs per level.

At 5500 waves per run

• Old: 3800 / 616 ≈ 6.17 runs
• New: 3800 / 821 ≈ 4.63 runs

So about 6 runs drop to about 5 runs.

At 9000 waves per run

• Old: 3800 / 1007 ≈ 3.77 runs
• New: 3800 / 1343 ≈ 2.83 runs

So just under 4 runs drop to just under 3 runs.

Summary for the realistic branch:

• Modules per run increase by one third.
• Shards per wave and per run increase by one third.
• Runs needed per level drop by about one quarter.

2. Using the “official” mechanics (19% skip, 78% package)

Here the game behaves exactly as described:

• Skip chance: 19%
• Not skipped: 81%
• Package on a non skipped wave: 78%
• So packages per wave: 0.81 × 0.78 = 0.6318

2.1. Per wave module chance

Old:

• p(module per wave) = 0.6318 × 0.012 ≈ 0.00758

New:

• p(module per wave) = 0.6318 × 0.016 ≈ 0.01011

2.2. Modules per run

4500-wave run

• Old: 4500 × 0.00758 ≈ 34.12 modules
• New: 4500 × 0.01011 ≈ 45.49 modules

5500-wave run

• Old: ≈ 41.70 modules
• New: ≈ 55.60 modules

9000-wave run

• Old: ≈ 68.23 modules
• New: ≈ 90.98 modules

2.3. Shards and waves per level

Shards per wave:

• Old: 0.00758 × 18 ≈ 0.1365 shards
• New: 0.01011 × 18 ≈ 0.1820 shards

Waves per level:

• Old: 3800 / 0.1365 ≈ 27,845 waves
• New: 3800 / 0.1820 ≈ 20,884 waves

Again the same pattern: about one third more shards per wave, about one quarter fewer waves per level.

What this means in plain terms

Using your real observed rates:

• Old: about 7–8 runs of 4500 waves per level.
• New: about 5–6 runs of 4500 waves per level.

Or:

• Old: about 3.8 runs of 9000 waves per level.
• New: about 2.8 runs of 9000 waves per level.

If you tell me how many high end modules and how many levels you want on each, I can translate this into a total “grind budget” with the upgrade vs without it.