My unfortunate foray into the world of probability

So I was wondering today, when trying to skill enhance a card that's already at max maturity, is it better to use ten cards at once, or one at a time. So began a dark journey.

Little timmy wants to enhance his awesome new SR Brunhilde, using rares to enhance from skill lvl 1 to 2.

using 10 rares at once he get's 80%, one at a time gives timmy ten shots at 8%. Timmy does the math, 10x8=80%. Hurray! and he'll be saving cards in the process!

WRONG!

8% = 0.08 = 2/25 so the odds of getting a single "hit" is 2 out of 25 times. To calculate the odds of something happening "at least once" across a series of events, you need to first multiply the odds of it not happening by itself n amount of times.
Because MATH!

The odds of not getting a hit = 23/25. So across 2 events the chances of not getting a hit are 23/25 x 23/25, or 23^2/25^2 = 529/625. across n events 23^n/25^n

23^10/25^10 = More numbers than Timmy ever knew existed.

After subtracting the first number (odds of no hit), from the second number (all possibilities), you get the odds of getting a hit, which, when converted back to percentage comes out to a measly 56.4%

Not quite as good as 80%, but since he'll be getting "hits" on the first attempt as often as on the tenth, timmy'll be saving 50% of his cards!
...except for the times where he'll get hits on the 11th card. or the 12th.

So, how many cards does he need to spend to get the overall odds back up to the original 10 card 80%? Because of diminishing returns, the answer is 19 (79.8%)

19 cards, used one at a time, saving 50%, averaging out to 9.5 consumed for the same return as using 10 at once, or a 5% saving in cards spent.

However, diminishing returns also means that the more cards you need to spend for the same chance at a level up, the less you'll end up saving. For example, if we needed 20 cards to get 80%, we'll need to spend 39 one at a time for the same 80%, saving 50% across a large enough sample, we'll end up using 19.5 cards, instead of the original 20. Only a 2.5% saving, and the scaling up is exponential, so at higher skill levels you quickly drop to 0.00x% savings.

After spending countless eons enhancing his beloved Brunhilde finally from skill lvl 9 to 10 one card at a time, Timmy takes a moment to admire his newly grown zz-top beard, and contemplates the value of human life and the transendental meaning of time.

...

Math is stupid, somebody get me a beer.

Just so you know, I wanted to kill myself by the 6th paragraph. Let's put it in simple terms. 80% = way better than 8%. Done. Go get a beer.

hahaha i quit reading at 6th too! thanks anyway for this!

I stopped reading after the kids name was timmy, no offense if your name is 0.0