Single post in CSY's Journal: Maths and Musings

Bad shiny luck, bad albino luck, and good melan luck So, I was recently talking to some people and I made the claim that having bad shiny/albino luck was good for your melan chances. As Canta has been annoying me like crazy to substantiate this with a decent explanation I feel like this is an interesting topic to talk about, I'll be talking a bit about the maths, etc. behind it here. This is pretty basic probability, but for the purposes of cohesiveness, and so that this post is self-referential, I might mention some simple concepts. I know most people will know this stuff but like please bear with me haha~ PFQ is based on probability - HM, shiny charms, Ubercharms don't guarantee special pokemon, but increases your chances of getting something. Based on empirical evidence, many agree that agglomerate shiny luck with max boosts is about 1 in 25, so lets use this as the basis. Lets also assume albinos and melans don't exist YET. For simplicity Probability case 1: Investigating shiny chances of at least 1 shiny IMPORTANT NOTE! When I mention "you hatch x eggs", x assumes you've already started the 40 egg chain One egg You hatch one egg. What's the chance of it being a shiny? 1 out of 25. 2 eggs You hatch 2 eggs. What's the chance of getting at least 1 shiny? You can either get a shiny on your first egg, or get a shiny on your second egg, or get 2 shinies. P(Shiny on first egg, nothing on second egg) = 1/25*24/25 = 3.84% P(Shiny on second egg nothing on first egg) = 24/25*1/25 = 3.84% P(Shiny on first egg, shiny on second egg) = 1/25*1/25 = 0.16% P(At least 1 shiny in 2 eggs) = add all the above probabilities up = 7.84% 3 eggs Okay this is gonna get complicated really quickly so I'll be stopping after 3 eggs. You can either (Egg 1, Egg 2, Egg 3) - Case A: Shiny, no, no - Case B: No, Shiny, No - Case C: No, No, Shiny - Case D: Shiny, Shiny, No - Case E: No, Shiny, Shiny - Case F: Shiny, Shiny, Shiny There are 6 permutations that are favorable. Lets do the math now! P (A) = 1/25*24/25*24/25 = 3.69% P (B) = 3.69% P (C) = 3.69% P (D) = 1/25*1/25*24/25 = 0.154% P (E) = 0.154% P (F) = 0.00640% P(at least one shiny in 3 eggs) = 11.38% In fact, if we look at 25 eggs, the probability of getting at least 1 shiny = 64.0% [Additional note: If you want to read more on this sorta calculations, search up Binomial Probability] For reference (% chance of getting at least 1 shiny) 40 eggs - 80.5% 50 eggs - 87.0% 75 eggs - 95.3% 100 eggs - 98.3% Anyways, as is evident, as you hatch more, the chances of getting at least 1 shiny increases. Conclusion 1 - The more you hatch, the more likely you are going to get at least one shiny Probability case 2 - Investigating the chances of getting your "first" shiny upon xxx hatched Okay, so for this one, we're investigating the probability of getting a shiny given that all previous rolls were null (i.e. all previous rolls were regular). P(S=x), where x is an integer, and the number that the shiny finally hatches at. One egg P(S=1) = 1/25 = 4% Two eggs P(S=2) = 24/25*1/25 = 3.84% Three eggs P(S=3) = 24/25*24/25*1/25 = 3.69% More eggs lol P(S=4) = 24/25*24/25*24/25*1/25 = 3.54% P(S=5) = 24/25*24/25*24/25*24/25*1/25 = 3.40% P (S=25) = (24/25)^24 * 1/25 = 1.50% P (S=100) = (24/25)^99 * 1/25 = 0.0703% (Additional note: P(S=x) = (24/25)^(x-1) * 1/25) So, the trend here is as x is increasing, the probability is decreasing, i.e. conclusion 2: The more eggs you hatch, the less likely it is for the shiny to first hatch on that egg. Combining the 2... What does this mean? It might seem anti-intuitive at first, but it actually makes sense when you look at the larger numbers. Conclusion 1: The more eggs you hatch, the more likely you will have at least 1 shiny. Conclusion 2: The more eggs you hatch, the less likely it is for the shiny to first hatch on that egg. This can be interpreted as: The more eggs you hatch, the less likely it is for a shiny to have not hatched yet So the more eggs you hatch, given that you haven't hatched a shiny yet, the more likely the next egg will be a shiny. Now I try to convince you if you haven't hatched a shiny yet, you're more likely to get an albino Albino chances function the same way as shiny chances but at less chance. The more eggs you hatch given that you havent hatched an albino yet, the more likely the next egg will be an albino. It's not proposterous to infer: The more eggs you hatch given that you haven't hatched a shiny or albino yet, the more likely the next egg will hatch both shiny and albino. Which is a melan. Hey I WANTED A MATHEMATICAL PROOF! PROVE THIS TO ME MATHEMATICALLY!!! If you're not satisfied with the above claim, here's more math. Lets assume in this world, there are only shinies or melans. For simplicity. Assume shiny chance for an individual using max boosts = 1 in 25. Assume melan chances for an individual using max boosts = 1 in 3500 (Based on this post. Not exactly equal to the number gotten, but again for simplicity) We are interested in seeing how your shiny luck affects your melan luck. So we will be looking at how long your chain is without getting a shiny and your associated chance of getting a melan. So lets consider, lets say, 4 eggs and 1 melan in that. What are the possible permutations? (Egg 1, Egg 2, Egg 3, Egg 4; S = Shiny, N = Nothing, M = Melan) No shinies: - M N N N - N M N N - N N M N - N N N M 1 shiny: - M S N N - M N S N - M N N S - S M N N - N M S N - N M N S - S N M N - N S M N - N N M S - S N N M - N S N M - N N S M okay its getting p. long so lets just list out 1 permutation of 2 shinies: - M S S N since all permutations of 2 shinies have the same probability And one permutation of 3 shinies: - M S S S Now for the maths. 1 melan, 3 nothings P (M N N N) = 1/3500*24/25*24/25*24/25 = 0.0253% Since you can ONLY have ONE single permutation - i.e. the order matters - you dont get the conglomerate chances for the same combination! - we dont need to add stuff together here. (I.e. your chain is either M N N N OR N M N N OR N N M N OR N N N M if you have 1 melan and 3 regulars. Order matters) 1 melan, 1 shiny, 2 nothings P (M S N N) = 1/3500*1/25*24/25*24/25 = 0.00105% 1 melan, 2 shinies, 1 nothing P ( M S S N) = 1/3500*1/25*1/25*24/25 = 0.00004% 1 melan, 3 shinies P (M S S S) = 1/3500*1/25*1/25*1/25 = 0.00000183% OKAY. You can see increasing no. of shinies in your permutation (i.e. order) will cause the chances to drop drastically. I use a super simplified case with just 4 eggs here, but it applies to larger numbers at a greater scale. Conclusion: Less shinies, higher melan chance. Again, you can infer this to albinos. Less albinos, higher melan chance. Less shinies --> Higher melan chance Less albinos --> Higher melan chance Less shinies & albinos --> Higher melan chance. Conclusion If you're having a shiny drought, thank Sally, because it means you're more likely to get a melan. DISCLAIMER: This is all mathematical probabilities, it doesn't guarantee anything. A bad shiny ratio doesn't mean you WILL get a melan, it means you're more likely to get one. A good shiny ratio doesn't mean you WONT get a melan, it means you're less likely to get one. And remember, a lot depends on your accumulated chances.