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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?

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If you already have the opportunities 'saved up', then combining them is best, as $$ 1 - .98^{10} = 18.29\% < 20\% $$

However, if you have not earned the opportunities yet, we also have to consider the expected time to level up as part of this.

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 $50$.

So in the end, it does not seem to matter!

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