Trying to find math inside everything else

I recently wrote a post about some math I did for Genshin Impact on Reddit, and I figured I’d post it here, more for the math part than the game part. See below (with additional edits for clarification for non-players.)

With the release of Kuki Shinobu and her expedition talent (which Yelan and Shenhe also have, but I didn’t pull them, so I wasn’t thinking about it), I started to wonder about the comparative benefits of these talents vs the quicker expedition talents of Bennett/Fischl/Chongyun/Keqing/Kujo Sara.

Shinobu has the following talent: “Gains 25% more rewards when dispatched on an Inazuma Expedition for 20 hours.”

Sara has the following talent: “When dispatched on an expedition in Inazuma, time consumed is reduced by 25%.”

The first thought is to compare them directly. Shinobu gives 25% extra rewards every 20 hours, and Sara gives regular rewards 25% faster, so every 15 hours. While both say 25%, you are actually collecting rewards with Sara 33% more frequently, and so it is a better talent. (Over the course of 5 days, Sara would get you 40000 Mora, while Shinobu would get you 37500.)

Of course, that requires doing your expeditions immediately upon completion, which will require a shifting schedule and waking up in the middle of the night and such, and is thus fairly unrealistic. So let’s look at a more realistic model.

It would be reasonable to check Shinobu‘s expedition once a day, as 20 hours is close to 24. With Sara’s 15, however, you could do a 2-1 cycle: on the first day, check right when you wake up and right before bed (as most people are awake 16 hours), and then the next day check it in the middle of the day. (For example, 7 AM, 10PM, and then between 1 and 4 PM the next day, so it ready by 7AM the next day. This gives you some wiggle room.)

With this method, Sara is doing 50% more expeditions compared to Shinobu‘s 25% bonus, an even bigger difference than before! Over a 6-day period, Shinobu would bring 37500 Mora, while Sara would get 45000.

However, it’s pretty easy to mess up that 2-1 cycle. Sometimes I would have a class when my expeditions were done, and so couldn’t check, and then wouldn’t remember until after work, which would make my morning expedition late, and then my night one would fall until after I went to sleep. So now my question is, how often can I mess up the cycle and still have it be better than Shinobu?

Consider that same 6-day period. If I mess up on one day, it actually doesn’t change anything. (My 2-1-2-1-2-1 cycle becomes 2-1-1-2-1-2, and the next cycle of 6 days is 1-2-1-2-1-2, so still 9 expeditions per cycle.) However, the second mistake will make it so there’s only 8 expeditions per 6 days, which is equal to Shinobu’s. Similarly, the 3rd mistake is fine but the 4th one will drop you below Shinobu’s bonus rate.

So one way to look at it is if you can keep a rate of 2 double-days every 6 days, Sara and Shinobu are tied. If you can do more frequently, Sara is better. If you can’t, Shinobu is better.

Another way is to think more long term. Over a 30-day period, you would need 11 or more double-days for Sara to beat out Shinobu. So you could mess up on 9 days and still come out ahead (30% error rate). Over a 300-day period, you need 101 or more double-days, so you can mess up 99 times (33% error rate). As you might be able to tell, this limit approaches 1/3, so you can mess up on average (fewer than) 1/3 days and still come out tied or ahead.

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