# Crunching the numbers: Enhance your game with the power of math

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#### BulbaBot

##### Dreams of electric Bulbasaur
Crunching the numbers: Enhance your game with the power of math

When Danielle Detering was a kid, she never brought what she learned in school into games. After an intense five hours of learning and one hour of homework, the last thing anyone would want to do is think about math. But — holy Arceus! — Pokémon is really just a giant game of numbers. Now a math major, Danielle shares her number-crunching Pokémon tips.

Wow, just, wow. That's a lot of math for my feeble mind to understand. Though it does make some things clear in a way. A 40% chance to knock out two slow PKMN with Blizzard and Rock Slide? I wouldn't have thought that at all. Now, if that could be applied within the set-up time limit during WiFi matches........woah. That'd be awesome. o3o ....here's what I read. "numbers math numbers numbers blizzard math formulas chances math spiritomb moar math"

Either way, wow. D8 That is indeed a lot of math there!

Very good article! I don't think the maths there is too advanced at all as I've done a preliminary statistics course and what I've gained from AS level Maths covers everything here. So it's good to think about pokemon in a different way and to work out the probabilities which I wouldn't have thought to do before.

No. Math is witchcraft, the evil kind. I will not be using this, spellshaper.

I'm actually surprised how much I understood all that.

I found an error in your analysis. You said the probability of neither pokemon being knocked out as being 60.31% reasoning that the probability of no KOs is 1-P(AB) [Where P(A) is probability that Pokemon A is knocked out, P(B) same for Pokemon B, and since both can be assumed to be independent, P(AB) = P(A)P(B)]. This is erroneous, as it disregards the probability that one or the other pokemon is knocked out. In fact, what you calculated was the probability that AT LEAST ONE Pokemon survives, or that the attacks won't knock out BOTH (only 0 or 1, not 2). The actual probability of neither Pokemon being knocked out is actually [1-P(A)]^2 or equivalently [1-P(A)][1-P(B)] regardless of independence (the square of the probability of a particular pokemon surviving). Since the probability of a pokemon being KO'ed is 0.63, the probability of its survival is 1-0.63=0.37. The probability of both pokemon surviving [equivalent, the probability that none are knocked out] is the product of the two events since they're considered reasonably independent. Thus the probability of none being KO'ed is (0.37)(0.37) = 13.69%. As an aside, the probability that only 1 pokemon is knocked out is 46.62%. Add these two values with the probability that both are knocked out (in their decimal forms obviously) and you get 1 as you should.

If you want some even more advanced math when it comes to analyzing Pokémon, think about what it would take to make a good AI, or even a good competitive battling tier system.

Stuff like graph theory and population theory comes into the picture. Especially graph theory. Oh man it would be complicated for Pokémon.

I've always loved the statistics and math within Pokemon -- it's part of the reason I love it (I'm going to be a statistics major next year). Thank you, Danielle Detering, for showing everyone how involved math actually is in the pokemon world.

Overall a good article and quite interesting. However, it might be have been nice to expand from probability into other areas of math too, but I assume probability is your specialty.

I did have a couple of issues, though. You seem to have put the introduction to TCG in the paragraph below where you first referred to it. It did get a little confusing, jumping into something which was explained at a later point. Also, "Spiritombs" bothered me.

If you want some even more advanced math when it comes to analyzing Pokémon, think about what it would take to make a good AI, or even a good competitive battling tier system.

Stuff like graph theory and population theory comes into the picture. Especially graph theory. Oh man it would be complicated for Pokémon.

Where would graph theory come to mind? I assume somewhere in the AI (I'm not well-versed on graph theory or computer science). Anyway not strictly mathematical but the application of mathematical economics (Game Theory comes to mind) would also be very much of interest. One idea that I've been considering is the optimal EV spread for a particular Pokemon, especially given your opponent's likely own strategy in this regard (e.g. is the prevailing wisdom of putting max EVs to an attack stat instead of some split between it and an inferior defensive stat the best option?)

I literally just learned about combinations in math class an hour ago.. That's probably the best part of my stats classes.. Applying it to Pokemon. Fantastic article Where would graph theory come to mind? I assume somewhere in the AI (I'm not well-versed on graph theory or computer science).

The AI would have to examine the possible consequences of its actions, and how the game would play out from there. Each state is a point in the graph.

Anyway not strictly mathematical but the application of mathematical economics (Game Theory comes to mind) would also be very much of interest.

Economics is also biology.

One idea that I've been considering is the optimal EV spread for a particular Pokemon, especially given your opponent's likely own strategy in this regard (e.g. is the prevailing wisdom of putting max EVs to an attack stat instead of some split between it and an inferior defensive stat the best option?)

I'd be hard-pressed to put that into one aspect of game theory.

Dx
But wow, that must have taken a while to write. I applaud you Her last name is "Detering". This has certainly deterred me from maths for a while!

Thanks for the awesome responses everyone. I'm a super huge fan of your constructive criticism too. As you might guess, English is not one of my strong points, so please don't hold back on what I could do better.

The AI would have to examine the possible consequences of its actions, and how the game would play out from there. Each state is a point in the graph.

This actually already exists in game theory. Except, in Game Theory, the graph is called a game tree, and each node is called a ply. I'm not sure how they do the pokemon ai, but when it comes to chess computers, the computer will search so many ply ahead (depending on the power of the computer), and will pick a move that will give it the highest chance of an optimal position further in the game. I hope I'm making sense... If not, I suggest searching for the Horizon effect.

And of course, there are all sorts of other applications involivng logic, math, and pokemon. Videogames are made by computer programmers afterall.

Interesting article, altough I didn't understand half of it. xD My reaction ↑ This is incredible.

Ow, math.
X_x
But that is SO cool!