Hey everyone, i’m trying to understand probability.
If we say a character has a 1% pull chance. After 200 pulls is there any increased chance of getting said character.
If i work it out in terms of “chance of not having said character after 200 pulls” its a smaller chance the more pulls i work it out after. But the base chance per pull is still 1%
So how does probability actually work in this context.
Am i any more likely to recieve said character after x amount of pulls?
In: 3
*Drawing without replacement*: Suppose you have 300 Scrabble tiles, mark 3 of them as “winners”, mix them up and put them in a bag. You draw tiles one at a time and set them aside.
*Drawing with replacement*: Suppose you have 300 Scrabble tiles, mark 3 of them as “winners”, mix them up and put them in a bag. You draw tiles one at a time, then you put the tile back in the bag and mix them up before drawing again.
These both have “1% odds” but they are very different. For drawing without replacement, you get a rising chance to win. For example, after drawing 200 tiles with no winner, there are still 3 winners among the 100 tiles left in the bag, so you have a 3% chance to win. After drawing 297 tiles with no winner, you *know* the next tile will win — the only logical possibility is that the three tiles left are the three winning tiles!
For drawing with replacement, your chance is always 1%. Earlier draws don’t affect the current draw since you always start with all 300 tiles mixed up. For events that don’t affect each other (*independent* events), you can multiply probabilities to get the probability of both events happening.
We can use this to calculate the probability of failing on 3 draws, it’s 0.99 x 0.99 x 0.99, which is 0.970299. This only applies to independent draws, i.e. draws with replacement.
In general, the probability that you draw (with replacement) n times and have them all fail is 0.99^n. This means you’re never *guaranteed* to win, there’s always some nonzero chance of failure no matter how many you draw. 0.99^200 is 0.133980 meaning there’s about a 13% chance you might not see that 1% winner in 200 pulls.
Some intuition for how this can happen: On average you’ll have seen 2 winners in 200 draws, but you can get lucky and sometimes see more than 2 winners. Mathematically this possibility of getting lucky has to be balanced by a possibility of getting unlucky and sometimes having less than 2 winners in 200 draws.
Now you’re asking about gacha. The answer is, it’s completely up to what the programmers do. Programmers can simulate drawing with replacement, or drawing without replacement, or they can do something else entirely. For example they could decide your prize odds based on how much money you’ve spent, your login patterns, what timezone your phone’s clock is set to, your age / gender, and so on, trying to basically look into your eyes and figure out whether (a) you’re frustrated you haven’t gotten a prize in a while and you’re about to quit, so they should give you a prize to make you keep playing, or (b) you’re ready to spend money on a mountain of loot boxes, so they should give you a long dry spell to extract as much cash as possible before you get the prize.
Some games (notably many cryptocurrency online casinos) have provably fair odds, meaning that basically they not only publish the odds of their games, they also use some clever math that proves the games are as described and they aren’t cheating. But AFAIK this kind of transparency is rare to nonexistent in the gacha industry. Even though some jurisdictions are starting to require publishing of loot box odds, AFAIK nobody’s actually checking what the game’s code is actually doing. They could be doing “Look-into-the-eyes AI model except we randomly distribute extra prizes / dry spells if the prizes actually handed out start to deviate from the published odds so we don’t get in trouble with regulators.”
To add onto this surely probability is relative to number of tries. Eg there’s a much higher chance that you get something if you roll 10,000 times compared to only 10 times. As such how does the math work out then?
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