Chances of 2 shiny pikachu in viridian forest?

Question

So I was wondering, in Pokemon FR/LG, what are the chances of encountering 2 shiny pikachu within 10 tiles of grass of each other? (no this didnt happen to me)

Comments

answer can be for either game

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The chance of finding a shiny Pokemon on any given encounter in FRLG is 1/8192. The chance of encountering Pikachu in Viridian Forest is 1/20. Therefore, to find the chance of finding two shiny Pikachu, solve the following:

(1/8192*1/20)^2

This comes out to 3.7252903e-11, or about 0.00000000373%.

I don't know how to find the odds of this happening in 10 tiles.

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when you say within 10 tiles, do you mean they passed through 10 tiles? Cause then it would just be 10 potential encounters, so the probability would be multiplied by 10.
EDIT: nvm, the encounter probability is based on when you encounter it
So the chance of encounter being 320/2879 or ~11%, then the probability would be instead around 6.713867188e-7 for 1 shiny pika, and doing by using binomial probability/plug into my calc, the probability would be 2.07103246e-11, so
0.00000000207103246%.

So, could you clarify if thats what 10 tiles means? Otherwise, there's your answer

https://bulbapedia.bulbagarden.net/wiki/Wild_Pok%C3%A9mon#Generation_III

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10 tiles means like, you travel at most ten tiles of grass between the two pika encounters. also, its assuming encounters with other pokemon dont matter, all that matters are the shiny pikas

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So, for the sake of the question, you're assuming Pikachu is the only spawn and it must be encountered within 10 tiles and ONLY 10 tiles?

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yes, but the spawn rates are still the same for pikachu.

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between the two? or they travel 10 tiles, because that's what I calculated. Does it satisfy the question?
Also @JATP, wdym by assuming pikachu is the only spawn?

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What I mean is that we're assuming they're the only Pokémon that spawn in Viridian Forest.

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what i mean is i encounter a shiny pikachu, then at most ten tiles later, i encounter another.

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Why are you specifying ten tiles, or rather, why are you specifying an amount of tiles at all?

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its me wondering something

Answer 1

It depends on a few factors, but assuming you're not using any encounter modifiers like White Flute or Illuminate, and aren't riding the bike, you have an 11% chance of getting an encounter per step in tall grass. And there's a 5% chance that you'll get a Pikachu for any given encounter.

Using binomial distribution, we see that we have a 21.43% chance of getting exactly 2 encounters within 10 steps, with an additional 8.84% change of getting 3 or more encounters, meaning a total of 30.28% chance of >= 2 encounters (though we'll have to treat the >2 scenarios a bit differently).

Pikachu has a 5% encounter rate, so in the exactly-2-encounter scenario, it's a 5% * 5% = 0.25% chance of them both being Pikachu. Accounting for the fact that these odds are only applicable in the 21.43% chance that we get exactly 2 encounters, we'll multiply 21.43% * 0.25% to get 0.0536%, or about 1 in 1867.

However! We still need to take into account the chance that we run into more than 2 Pokémon within our 10 steps. I went and repeated the process above for each of the 3-10 encounter scenarios (using binomial distribution instead of multiplication as necessary) to get a cumulative total of 0.0786% from all of these. Since these are separate scenarios from the exactly 2 encounters we calculated above, we can add the two rather than multiplying them: 0.0536% + 0.0786% = 0.1322%, or about 1 in 757

But that's just the odds of getting two Pikachu within 10 steps, we still need to account for shininess. Technically, there's a whole bunch of additional math we could do to account for running into 3+ Pikachu, and only needing two of them to be shiny, but the odds of that are so unbelievably small and would take so much effort to get that I'm excluding that here; I can guarantee you that it wouldn't noticeably change our answer even if we did find it. So that just leaves us with the simple version:

1/8192 * 1/8192 * our answer above, 1/756.56 for a final answer of 1 in 50,771,882,148

Source for encounter rates per step
Viridian Forest encounter table

Comments

maths is satisfying