"At 1/20 shiny rate, you will need 60 raids for ~95% chance at shiny Rayquaza, and 90 raids for ~99% (with coins cost analysis)"

#PokemonGO: Seeing as shiny Rayquaza is bound to be popular, I thought a post to educate people on the number of raid passes they might end up spending on to get this beautiful black dragon would be helpful.So, past shiny legendaries like Groudon, Kyogre, Latios, Latias, Cresselia have been at a shiny rate of 1/20. That does not mean you are expected to get 1 shiny after 20 raids. Here's how the math works out.This calculation shows you how to do the mathThe chance of failure (no shinies) after 1 raid is 20/20 - 1/20 = 19/20, which is 0.95 = 95%The chance of failure (no shinies) after 2 raids is 0.95 × 0.95 = 0.952 = 0.9025 ≈ 90%The chance of success (at least 1 shiny) after 2 raids is 1 - 0.9025 = 0.0975 ≈ 10%This calculation is for people who expect a shiny after 20 raids (credit u/BenPliskin for the idea)The chance of failure (no shinies) after 20 raids is 0.9520 = 0.36 ≈ 36%The chance of success (at least 1 shiny) after 20 raids is 1 - 0.36 = 0.64 ≈ 64%This calculation is for people who only use free raid passes, but do so every day of the event, from July 31 to September 2, not counting extra passes for Suicune (credit u/Durian881 and u/Stormrycon for the idea)The chance of failure (no shinies) after 33 raids is 0.9533 = 0.184 ≈ 18%The chance of success (at least 1 shiny) after 33 raids is 1 - 0.184 = 0.816 ≈ 82%How many tries do I need for 95% chance of success?The chance of failure (no shinies) after 60 raids is 0.9560 = 0.046 ≈ 05%The chance of success (at least 1 shiny) after 60 raids is 1 - 0.046 = 0.954 ≈ 95%How many tries do I need for 99% chance of success?The chance of failure (no shinies) after 90 raids is 0.9590 = 0.0099 ≈ 01%The chance of success (at least 1 shiny) after 90 raids is 1 - 0.0099 = 0.9901 ≈ 99%Therefore, from the calculations, you will need 60 raids for ~95% chance at shiny Rayquaza, and 90 raids for ~99%. Which means that the unluckiest ~5% will not have a shiny in 60 raids, and the unluckiest ~1% will not have a shiny in 90 raids.Coins cost analysisYou'll get around 33 free raid passes for this raid boss. So to reach 60 raid passes, you'll need an extra 27 premium passes. To reach 90 raid passes, you'll need an extra 57. The current Ultra Box provides 14 raid passes per 1480 coins. You'll need 2 Ultra Boxes for 27 raid passes, so that's 2960 coins for a ~95% chance. You'll need 4 Ultra Boxes to hit a total of 89 raid passes, so that's 5920 coins for a ~99% chance.However, this Ultra Box might be replaced. So what's the best case scenario based on history? According to Gamepress, historically, the Kanto event box of 2018 had the best ever value for raid passes. That box provided 24 premium raid passes for 1480 coins. With 1 such box, you'll have 57 total passes. With 2 such boxes, you'll have 81 total passes (2960 coins).Credit to u/Durian881 for pointing out that I should include free raid passes in this calculation!Additional analysis 1: for the 1/10 raid events that only last 3 hours (Zapdos, Moltres, Raikou, Entei), ~95% of success requires 30 passes, and ~99% of success requires 66 passes.Additional analysis 2: A simple rule of thumb to get an estimate is to multiply the shiny rate denominator by 3 to get the ~95% chance of success. The denominator is X, when the rate is a fraction 1/X. So, for example, if the shiny rate is 1/450, then 450 × 3 = 1350. So, 1350 encounters are needed for ~95% chance of a 1/450 shiny.Extra note: although I won't factor this into calculations, yes, (1) if you live in a city/country in a late time zone, say, Phoenix, the United States is UTC−7, you can get 4 extra raid passes on August 16 once Suicune is active on New Zealand's August 17 raid hours (or even earlier, Kiritimati) and so on. (2) if you live in a city/country in an early time zone, say, Brisbane, Australia is UTC+10, then you can get 4 extra raid passes on August 18, up to when Suicune is active in American Samoa's (Pago Pago) August 17 raid hours. Credit to u/mrragequit456 and u/Durian881 via /r/TheSilphRoad https://ift.tt/3109QpB