# Several small chances or many minuscule

I'm playing a game that has a Gacha element to it... Every so often I can try and level up my character, I get a few chances every day, and I want to get my character to Level 8 as soon as possible. Each "chance" is a 2% chance of increasing their level by 1. Alternatively I can combine my chances into a single roll... At most 10 chances (which gives a 20% chance of leveling). Assuming the math is fair am I better off saving for the combined 20% chances until I've leveled 7 times, or taking as many individual 2% chances as I can?

If you already have the opportunities 'saved up', then combining them is best, as $$1 - .98^{10} = 18.29\% < 20\%$$
If we use opportunities as soon as we get them, there is a $$2\%$$ chance it will occur at the first opportunity, and a $$98\%$$ chance it will take more tries. In this second case, there is a $$2\%$$ chance it will occur at the second opportunity, and so forth. That is, expected time is $$0.02\cdot 1 + 0.98(0.02 \cdot 2 + 0.98(0.02 \cdot 3 + 0.98(\dots)))\\ = \sum_{t=1}^\infty 0.02 \cdot t \cdot 0.98^{t-1}$$ and you can check that this converges to $$50$$.
Our alternate strategy has an expected time $$0.2\cdot 10 + 0.8(0.2 \cdot 20 + 0.8(0.2 \cdot 30 + 0.8(\dots)))\\ = \sum_{t=1}^\infty 0.2 \cdot 10t \cdot 0.8^{t-1}$$ which also converges to $$50$$.