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

== 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.
# 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.
# 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:
# 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?
# 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?
# Should I trust your observed 51.8 percent package rate instead of your stated 63.18 percent theoretical rate?