Polymerization Preparation

# Gacha Odds Calculator

Understanding the numbers is vital if you're going to play gacha games. I find that people often overestimate how good their odds are for getting the character they want (especially if their pull count is low), which inevitably leads to disappointment and complaints about rate-up being a lie when they don't get it.

So, this is a calculator that will tell you how likely you are to get a rate-up 6 star (or any 6 star at all) in whatever number of pulls you have.

Someting important to note is that this calculator is giving you a worst-case scenario because there's no convenient formulaic way to include the pity rates. As a result, you'll most likely see better results than your calculated prediction, but better to be pleasantly surprised than unpleasantly surprised!

A spreadsheet version of this calculator is available as well.

## How this works

The formula this calculator uses is P = 1 - (1 - r)n, where P is our final probability, r is the rate for the 6 star you want, and n is the number of pulls you do. If you're interested in why specifically this works, continue reading!

Boring probability lesson

Gacha draws are akin to dice rolls and coin flips, which we refer to as having "independent probability" because the outcome of each roll has nothing to do with all other rolls you do. You have the same probability of getting heads on your first coin flip as you do on your fifth, or hundredth.

Let's go back and use the coin flip as our "toy" example. The probability of getting heads 3 times in a row is 0.5 * 0.5 * 0.5 = 0.125, which is also the probability of getting tails 3 times in a row because their probabilities are the same.

Things get weirder if you wanted to look at the probability of getting heads twice and tails once in 3 flips, though, because there are multiple ways this could happen. You could get tails first, then heads twice, or heads then tails then heads, or two heads followed by tails. The probability for each of these events happening is still 0.5 * 0.5 * 0.5 = 0.125, but now we have to add the probability of each event together to account for every possible flip combination, which gives us 0.375.

But what if we wanted the probability of getting tails at least once? Then we'd have to add in the probabilities of tails 3 times, tails twice then heads, heads followed by two tails, and tails then heads then tails. Adding the rest of these together gives us 0.875. This wasn't overly terrible for 3 flips, but imagine 100 flips!

Fortunately, there's a much easier way to deal with this. We could also just consider what we call the "complement" of getting tails at least once, or the event where we don't get tails at least once. This means we'd need to get heads on every flip, which brings us back to 0.5 * 0.5 * 0.5 = 0.53 = 0.125. Of course, the total probabilities of any set of events need to add to 1 or 100%, so that means we can find the probability of at least one tails flip by subtracting the probability of its complement from 1, or 1 - 0.125 = 0.875. Which is exactly what we got doing this manually!

We can generalize this approach for more complex situations because all we need is the probability of the thing you want and the number of times you're going to try for it. From the last example, you'll notice our probability for tails is 0.5 and we flipped the coin 3 times, so we get the formula I gave earlier with r = 0.5 and n = 3: P = 1 - (1 - 0.5)3 = 0.875.