The Shrine

Most games involve math in some way, and gachas in particular often deals with probability. This scenario is something I see all the time. Suppose the chance of getting some rare thing is 1%. The rare thing is something you need more than one of. You pull 50 times. What is your expected number of copies of the rare item you're looking for after 50 pulls? Well, it's actually pretty much just 50 * 0.01 = 0.5. What annoys me is there is always going to be someone who may have taken high school probability telling you this isn't the way because he may have heard about the binomial theorem but don't even know what it's called anymore. Yes, the chance of getting 1 such item with these odds is 30.6%. However, I mentioned that this is something you need more than one of. Therefore, your long term goal is looking for 2 or more of such items. That means when you consider all the possible cases (getting 0, 1, 2, 3, and so on) in the long run they're likely to contribute on average 0.5 copies for your long term goal. I mean, I remember back in high school it's like if you just do the average you don't sound sophisticated enough. If I flipped a fair coin 50 times and I get 27 heads, you don't sound smart if you just say 'well the average number of heads is 25 so that looks about right' (here we assume those sequence of 27 heads itself is ordinary, as opposed to say rolling head 27 times in a row followed by 23 tails) even though you just rolled some perfectly average results. Yes you can use average incorrectly for certain things, like if I did roll head 27 times in a row followed by 23 tails I shouldn't say 'well it averaged to roughly 50%' but if we assume we're dealing with non-rigged situations there's probably indeed nothing eventful if you get some number that's perfectly within the average. The fancy statistical analysis usually comes in if you suspect there's some bias to the data, and it's meaningless to go to them if you're pretty sure that the data is indeed fair and yet people always want to show off that they know all these random fancy terms they learned in high school. You'll see people do some analysis if you crit 3 times in 25 hits in a video game when you've a 5% crit rate and come up with some fancy analysis of perhaps this game is totally rigged. But honestly, do you really think the crit rate you're told in any respectable game is actually rigged? What would they have to gain from even doing that?

Oh, and the opposite spectrum, let's suppose the item you need is something you need exactly 1 of and it's the typical game breaking overpowered stuff that you must get or you instantly become a second class citizen and should quit the game. How many pulls are you expected to do to get it? The answer is however many it takes until your money runs out, or quit the game if your money runs out. Also no statistical analysis needed!