"Mathematics on seeing X shinies after N raids. ELI15 guide in calculating luckiness"

#PokemonGO: When it comes to shiny odds, we often hear about that trainer who found 3 shinies after only 10 raids, while you raided Ho-oh 40 times and hadn’t found squat! So A) just how lucky was that trainer and B) how unlucky were you?For the answer to B), there are already very good posts that show you how to calculate the odds or probability of seeing NO shinies after N encounters (link in footer below). So we won't repeat that.What about A), the probability of seeing X shinies after N encounters? This is the general case and I haven’t seen a math tutorial for this, so bear with me at my attempt to explain this or just skip to the tldr to get an online tool to do all the calculations for you.ExplanationWe start by looking at the encounter history of our lucky trainer: Dull, Dull, Shiny, Dull, Shiny, Dull, Shiny, Dull, Dull, Dull.Let’s say the odds of a shiny raid boss, p, is 1/20 (it’s probably wrong in real life). The odds of non-shiny are then (1-p) or 19/20. We calculate the probability of other trainers having the same encounter history by multiplying the probabilities of those encounters together because they are independent: (1/20)3 * (19/20)7. We get some insanely small number like 0.009%.But wait, isn’t that very unlikely? Yes, that’s because those are the odds for anybody experiencing the same outcomes in that order. That’s obviously not what we want - we don’t care about the ordering, only that 3 shinies occurred.From our example, there are 120 combinations (mathematically known as 10 choose 3) 3 shinies can occur in a sequence of 10 encounters. In other words, our result of 0.009% only considered 1 of those 120 combinations. To get our final result, we need to sum the probabilities of all 120 combinations (we add them because those combinations can never happen at the same time for the same trainer). Thankfully, each combination has the same probability, so we can just multiply 0.009% by 120 to get our final answer of around 1%. Meaning in a park of 100 trainers, only 1 is expected to be as lucky.To summarize, our final formula (10 choose 3 = 120) * (1/20)3 * (19/20)7 finds the odds of seeing exactly 3 shinies in 10 trials, ordering be damned. If we keep the same number of trials and repeat our formula from 0 shinies to 10, we arrive at a probability distribution where the sum of all probabilities is always 100%. We have the whole picture now.Num. ShinyProbability060%132%27%31%etc.~0%10~0%We can see from this distribution that in our park of 100 trainers each doing 10 raids, the majority will go home without a shiny, a third will have 1, and the trainer with 10 probably works at Niantic.tldr: In math, they call this the Binomial Distribution which has these 3 parameters:n, the number of trials (in our case, encounters = 10)p, the probability of success (our fake shiny rate for raids = 1/20 or 0.05)x, the number of successes (number of shinies = 3)Because of how common this distribution is, you can simply plug in n, p, x into an online calculator to find the probability of seeing x shinies after n encounters. Wolfram Alpha has a good one. For example, an input of this:binomial distribution n=10 p=0.05 x=3Produces this, confirming our answer to A), which is ~1%:https://ift.tt/2krhrds larger n, it's better to explore a range of success values. So with n = 1000, instead on considering only the odds of exactly x = 50 successes, maybe we want to know the probability for the range of 40 < x < 60, or even x >= 50. You can enter these inequalities in Wolfram or manually add probabilities for each bin.Finally a reminder that no matter how lucky/unlucky you are, the shiny rate of your next encounter will always be 1/20. Past encounters do not affect the present. The Binomial Distribution just calculates the likelihood of historical, mutually exclusive outcomes.How to calculate odds of finally getting 1 shiny after n encounters:https://ift.tt/2siNeAH via /r/TheSilphRoad https://ift.tt/2ksJFVd