This is the first article in a series of articles about probabilities in the Pokémon TCG. In this particular article, we’ll discuss the probabilities behind the opening of the game (a fitting place to start article #1).
Don’t worry if you are not a “mathematically inclined” TCG player, as I have done my absolute best to make the content of this article a “user friendly” experience that will help every player (not just the math geeks).
Because this is going to be a long article, I have broken it up into sections and subparts for easier reading. I also realize that some of you may want to return to certain parts of the article when analyzing certain aspects of the game (such as a deck list); thus the organization of the article is crucial.
Here is how we are going to organize the article: Each section will be numbered, as well as the subpart for that section (if any). Example: “3.2” means section 3 subpart 2. The equations and data tables that we will use in this article will be numbered to match the section they will be contained in.
For example, “3.1; Formula 1” refers to the single equation in section 3 subpart 1, which is the first formula in the article (hence the name “Formula 1”). As an example for tables; “3.1; Table 1” refers to the first table in section 3 subpart 1.
There will be only one equation per subpart in section 3, which makes up all of the equations we will use in this article. I hope you learn a lot through reading this!
Contents
- Table of Contents
- SECTION 1.0: Probability
- SECTION 2.0: Probabilities in the Pokémon TCG
- SECTION 3.0: Data, data, data!
- SECTION 4.0: A Bit of Application
- Conclusion
Table of Contents
- 1.0: Probability
- 2.0: Probabilities in the Pokémon TCG
- 3.0: Data, data, data!
- 4.0: A Bit of Application
- Conclusion
SECTION 1.0: Probability
(Please skip this section if you already understand probability well.)
Before I cover this “prerequisite,” I want you all to know that it is not necessary that you fully understand this content about probability. But if you don’t know as much about probability, please read this and do your best to understand as much as you can, it will help later.
To begin our analysis, and just so everyone is up to speed, I’m going to define the word “probability”:
Probability – The chance that an event will occur.
As a really simple example of this really simple idea, we’ll talk about flipping coins. You have probably heard that there is a “50-50” chance for getting heads when you flip a coin. This refers to the percent chance that heads is flipped. To get the “percent chance” that you flip heads, we must first find the “numeric probability” that you flip heads (“numeric probability” is just the number that represents the chance).
Because it is equally likely that we flip heads or tails, we can take the number of ways to get the ideal outcome and divide it by the number of total possible outcomes to get the “numeric probability.”
So the probability that we flip heads is the number of ways to flip heads, divided by the total number of possible outcomes. We see that this results in: 1/2, or (0.5). So the number 0.5 represents the probability that I will flip heads.
To simplify this, I’m going to convert this to a percentage by multiplying the probability by 100 (mostly because percentages are easy to understand when reading).
0.5×100=50%. (This is also the probability that you get tails too because tails is also 1 out of 2 equally likely outcomes.)
Another way of saying this is that I will flip heads “around half of the time.” In other words, when you flip a coin multiple times, you will get heads on about half of the number of flips you make. It may already be known to you that a “percentage” represents the probability that something will occur when compared to 100 trials.
To make this clear, “50%” means that the event should occur about 50 times every 100 tries; about half of the time. 60% means that out of every 100 trials, the event should occur about 60 times. We should note that the probability of a certain outcome NOT occurring is equal to one (1) minus the probability that it DOES happen, or; 100-the probability it DOES happen in percent (this may be useful later).
Another important note is that the probability that event “A” happens, or event “B” happens, is equal to the probability of “A” plus the probability of “B.” In other words, if P(A) and P(B) are the probabilities for the events “A” and “B,” then the probability of either “A” or “B” occurring is equal to P(A)+P(B).
Note that “A” and “B” must be distinct events in order for this to work. This means that neither “A” nor “B” should have any effect on the other or else the probability of either “A” or “B” occurring will change depending of which event comes first (but we won’t need to worry about that).
Another method that may be useful is determining the probability for both “A” and “B” to occur. The probability that both “A” and “B” occur is: P(A)*P(B), where (*) means multiplication. Please note that if you multiply a percent by another percent, then you will be multiplying two numbers that have each already been multiplied by 100.
So if you want to do a “P(A)*P(B)” calculation with the data from this article, you will need to divide the result by 100 to get the percent chance that “A” and “B” happen. Since I don’t have time to really explain why all of this works, and it is not necessary for you to know, then we will move on. Hopefully you now understand probability a bit better.
SECTION 2.0: Probabilities in the Pokémon TCG
Before we continue, there is a point to be addressed here: probabilities are ideal concepts, meaning that in the real world, results only tend toward the probability we calculate; they may not be exactly like the results. However, if we repeat a test for a large number of trials, the results will tend toward the actual probability.
This may be a little confusing, so let me clear it up with an example. Suppose you flipped a coin two times and you got two “heads.” The number of heads divided by the number of times you flipped is 2/2=1. But the probability that I get heads should be ½ (one out of every two times)! This is because the number of trials (coin flips) was relatively low.
To make this calculation exercise tend towards ½, simply increase the number of trials. This applies to the Pokémon TCG in the same way: The calculated probabilities will predict what will happen most of the time.
Now to get back to business: we will also be discussing probabilities of drawing certain cards from randomized decks. I understand that in real-life, people have certain shuffling habits that will affect the “measured,” or “real-life” probability, but it won’t stray too far from our predictions as long as decks are well randomized, so we can be sure that we’ll be accurate enough.
The real main goal of this article is to predict the occurrences of the game with mathematical models, and apply it to the game and make better decks and in-game decisions. However, doing this requires some numbers: So without further delay, let’s move to the next step.
SECTION 3.0: Data, data, data!
In this section we will talk about multiple probability models. These are specific formulas that tell you the chances for certain events (which I will explain in good detail). Please note that these formulas use “binomial coefficients,” which are ways to calculate a number of “combinations.”
You won’t need to read these things, but I wanted to mention it for those of you who want to know. To make things easier for you to read over, I’m going to organize section 3 into subparts.
Each of these subparts will be related to a subpart of section 4 which will give a few application examples of each formula. For example, 4.2 will contain applications of 3.2 Formula 2.
Here is the first of these subparts:
3.1: The Probability of NOT Having a Mulligan
BulbapediaThe first data set I will discuss with you is the data set for what I creatively named (not really): P(M,N,X). Let me break this down for you: P(M,N,X) is the probability that you will encounter at least one of a specific card upon drawing “N” many cards from a randomized deck containing “M” many cards. I will define “X” as the number of copies you have of the card you are trying to get.
This may seem a bit confusing, so to give you a good understanding, let’s do an example:
Suppose you have 4 copies of Darkrai EX from Dark Explorers in your deck. P(60,7,4) is the probability that you will get at least one of Darkrai EX when you draw seven (7) cards at the beginning of the game.
The “60” represents the number of cards left in the deck, the “7” is the number of cards you are drawing and the “4” is the number of copies of the specific card you want.
By using math, I have derived (found) a way to calculate this probability based on the numbers “M”,”N” and X” that we will input. For the purposes of this article, we will only talk about formulas where M=60 and N=7 because we are only concerned with the beginning of the game for now.
Alright, here is the formula:
3.1; Formula 1
Ok, take a deep breath… Relax; this is just a formula that you don’t have to understand (just like high-school math class right?). Please note that the equation utilizes what mathematicians call “combinatorics,” which would require a bit of background to understand, but for those of you who do want to know how the formula works, please just message me and I’ll be happy to tell you!
Now you may be asking: “Ok, so what’s all this ‘combinawhosit’ stuff got to do with my Keldeo EX deck?” Good question, but I will have to cover that in 4.1. Now, since I don’t expect all of you to know how to calculate that gibberish up there, I have provided you with a data table so you can read the results:
3.1; Table 1
X (Number of Copies of Desired Card) | P(60,7,X) |
1 | 11.67% |
2 | 22.15% |
3 | 31.54% |
4 | 39.95% |
5 | 47.46% |
6 | 54.14% |
7 | 60.09% |
8 | 65.36% |
9 | 70.02% |
10 | 74.14% |
11 | 77.76% |
12 | 80.94% |
13 | 83.72% |
14 | 86.14% |
15 | 88.25% |
16 | 90.08% |
17 | 91.66% |
18 | 93.01% |
19 | 94.18% |
20 | 95.17% |
21 | 96.02% |
22 | 96.73% |
23 | 97.33% |
24 | 97.84% |
25 | 98.26% |
26 | 98.61% |
27 | 98.89% |
28 | 99.13% |
29 | 99.32% |
30 | 99.47% |
(If you want more than 30 Basics… we need to get you some help.)
Let me tell you how to read the table (skip this step if you already know). To help you understand it, let me give an example:
Let’s say that you have 4 Basic Pokémon in your deck, the table says that when you have a 39.95% chance of getting a Basic when you draw to start the game.
As you see on the table, the corresponding P(60,7,X) for X=4 is 39.95%. The right side of the table is how many Basics, and the left side is the probability that you draw 7 cards out of 60 and get at least one of your Basics.
You can also use this model, P(M,N,X), for the probability that a no-Basic card is in the six Prizes you set aside at the beginning of the game. To do this, we just use: P(59,6,X), so that you are drawing 6 instead of 7, out of a deck with 1 Basic set aside (because it is necessary that there is one Basic in the opening hand that you keep, and the remaining 59 cards can get prized). Here is a table for this case:
3.1; Table 2
X (Number of Copies of Desired Card) | P(59,6,X) |
1 | 10.17% |
2 | 19.46% |
3 | 27.94% |
4 | 35.66% |
5 | 42.68% |
6 | 49.05% |
7 | 54.82% |
8 | 60.03% |
9 | 64.73% |
10 | 68.96% |
11 | 72.76% |
12 | 76.17% |
13 | 79.21% |
14 | 81.92% |
15 | 84.33% |
16 | 86.47% |
17 | 88.36% |
18 | 90.02% |
19 | 91.48% |
20 | 92.76% |
21 | 93.87% |
22 | 94.84% |
23 | 95.68% |
24 | 96.40% |
25 | 97.02% |
26 | 97.54% |
27 | 97.99% |
28 | 98.37% |
29 | 98.68% |
30 | 98.95% |
In addition to these two tables of 3.1, I have a third that includes both of these two, but is color-coded by percent: P(Not Mulligan).xlsx. I will introduce these colored “windows” later on, but for now we need to move on…
3.2: The Probability of Starting with at Least “A” Many Basics
Now before we cover this model, I need to give credit where credit is due. Adam Capriola (who is the head of this website) sent me an email with a link to this article by Daniel Lee.
The article contains the probability that you start with at least a certain number of Basic Pokémon in your hand. The author of this article calculated this before I did; however, I was unaware of this and I had already derived this next formula prior to reading the above link.
The only difference is that I call this probability: P(M,N,Y,A), and I have calculated it for a few more cases. I definitely suggest reading his article if time permits you. He used different methods than I do, and he did a great job overall.
Moving on: P(M,N,Y,A) is the probability that you will get at least “A” many Basics when you draw “N” many cards from the deck. Let’s break this one down too: “M” is (the same as it will be in this article) the number of cards in the deck, “N” is the number of cards to be drawn, “Y” is the number of Basics in the deck, and “A” is the number of Basics you want to get from this draw:
3.2; Formula 2
To help you understand this, let’s do an example: Suppose we have 60 cards in the deck, 12 Basic Pokémon in the deck, and we are drawing 7 cards to start the game. P(60,7,12,2) is the probability that you will get at least 2 Basics in your starting hand. Here is the data table for this one:
3.2; Table 1
Y (Number of Basics) | P(60,7,Y,2) | P(60,7,Y,3) |
1 | 0% | 0% |
2 | 5.36% | 0% |
3 | 10.64% | 0.32% |
4 | 15.82% | 0.97% |
5 | 20.91% | 1.93% |
6 | 25.89% | 3.21% |
7 | 30.75% | 4.78% |
8 | 35.48% | 6.65% |
9 | 40.06% | 8.79% |
10 | 44.50% | 11.20% |
11 | 48.78% | 13.85% |
12 | 52.89% | 16.73% |
13 | 56.83% | 19.82% |
14 | 60.58% | 23.09% |
15 | 64.15% | 26.51% |
16 | 67.53% | 30.07% |
17 | 70.72% | 33.74% |
18 | 73.71% | 37.48% |
19 | 76.51% | 41.28% |
20 | 79.11% | 45.10% |
21 | 81.52% | 48.92% |
22 | 83.74% | 52.71% |
23 | 85.78% | 56.44% |
24 | 87.63% | 60.09% |
25 | 89.31% | 63.64% |
26 | 90.82% | 67.07% |
27 | 92.17% | 70.36% |
28 | 93.37% | 73.49% |
29 | 94.43% | 76.45% |
30 | 95.36% | 79.23% |
Now to tell you how to read this one: “Y” is the number of Basics in the deck, P(60,7,Y,2) is the probability that you get at least 2 Basics in your starting hand, and P(60,77,Y,3) is the probability that you start with at least 3.
I did not include any with “A” being anything greater than 3, because the probability is so low, and it’s pretty self-explanatory. Here is the color-coded version for this one:
3.3: The Probability of Starting the Game with an Ideal Starter
This next one is also in Daniel Lee’s article, but I call it: P(Y,Yi), which is the probability that you will start with an ideal starter Pokémon, based on how many ideal starters there are, and how many Basic there are total. In this formula, “Y” is the total number of Basics (again) and “Yi” is the number of ideal starters you have (hence the “i” in “Yi”). It turns out that:
3.3; Formula 3
In this formula, P(M,N,Y,1) is the special case of the model in 3.2: P(M,N,Y,A) where A=1. Obviously P(M,N,Yi,1) is the same thing with “Yi” in place of “Y”. I don’t expect you guys to do any calculations yourselves, even though this probability is just a ratio of probabilities we already know. So I have included a data table for it as well (Hopefully it will save you from doing all that division!):
3.3; Table 1
3.4: The Probability of Starting the Game with a Certain Card in Hand
Time to discuss the final probability model in this article: This particular model is of serious value to deck-building, so it will probably be used frequently by competitive TCG players. I give you P(M,N,X,Y):
3.4 Formula 4
Whoa, that’s one big formula… Good thing you don’t need to know how to calculate with it! Let me give you the info for this model: P(M,N,X,Y) is the probability that you will draw a certain card if there are “M” many cards in the deck, you have “X” many copies of that desired card, there are “Y” many Basics in the deck and you are going to draw “N” many cards.
Whew, that’s a lot of info.
To simplify things, the only case of this monster formula we will be discussing is for M=60 and N=8. This is because the formula’s one and only use (that I am aware of now) is to determine the probability that you start with a certain card at the beginning of the game (when you have 60 cards left and you draw 8, with the 8th being the draw for beginning your turn).
The formula was derived assuming that you have at least 1 Basic in hand already. An important note is that “X” is never going to be a Basic Pokémon, because Y is the number of Basics and “X” doesn’t fit into that category.
You may also note that we have already discussed the probability that you get a certain Basic Pokémon (starter) when you start the game: P(Y,Yi) from 3.3, so please use this model when you need a specific Basic at the game’s opening. Now that we’ve covered all of the preliminary stuff, here is the data table for P(M,N,X,Y):
3.4; Table 1
Opening Hand Probabilities.xlsx
WikipediaInterestingly, as you can see from the data table, having more Basics actually makes it slightly more probable that you will start with a specific card! This is because of the fact that when you have more Basics, there are more mulliganless hands that you can construct that contain the desired card.
Think of it this way: when you draw at the beginning of the game, the only opening hands you will keep are the hands with at least one Basic right? Well, when there are more Basics, there are more opening hands that you keep when you start the game, so there are more possible opening hands that contain the desired card.
You can see this effect if you read from left to right on 3.4; Table 1 (moving from low to high amounts of Basics). Okay, we’ve covered all the formulas that we will use for our analysis of the beginning of the game, so I think that wraps up section 3.
SECTION 4.0: A Bit of Application
Now that we have several probability models to use, let’s see how we can apply them! To make it simple, I’m also going to organize this section into subparts that correspond to the probabilities of section 3.
4.1: Application of P(M,N,X)
The applications of this model are really straightforward. We can use this one to determine how many Basics to put in the deck based on how often we want to NOT mulligan. To simplify the data from the list of 3.1; Table 1, we will analyze what I call the “10% windows.”
An example is having from 7 to 8 Basics in the deck puts you between 60%-70% window chance of NOT mulliganing. Each of the subparts of section 4 will have available tables with highlighted “10% windows.” Here is a list of the windows and there corresponding number of Basics for this model:
4.1; Table 1
Percent | Number of Basics |
10%-20% | 1 |
20%-30% | 2 |
30%-40% | 3-4 |
40%-50% | 4-5 |
50%-60% | 6-7 |
60%-70% | 7-8 |
70%-80% | 9-11 |
80%-90% | 12-15 |
90%-100% | 16-60 |
As you see, I have highlighted the ideal areas for Basic Pokémon numbers. These are ideal because they keep the number of Basics under 16 (because more than 15 Basics would start to overcrowd your deck, a topic which I will try to cover in the future), and they allow for higher percentages.
If the number of Basics in your deck fits within the “7-11” range, then you will have around a 60-80 percent chance of NOT mulliganing. This is high enough to keep you in the game!
Another application is to see the probability that a certain card is in the randomly selected Prizes at the beginning of the game. See 3.1; Table 1 for P(60,6,X) to see what the chances are for a card to be Prized.
Yet another application for this model would be to identify the probability that a certain card is drawn in a situation other that the beginning of the game. This can be done by replacing “M”,”N”, and “X” accordingly for the situation.
Unfortunately, I don’t have time to cover this in-depth. To do a quick example calculation, we will analyze this deck list for Blastoise/Keldeo EX:
Pokémon – 12 4 Squirtle BCR | via: https://sixprizes.com/?p=39577 via: https://sixprizes.com/?p=39577 Trainers – 32 4 N
| Energy – 16 12 Water |
To begin analyzing the deck, we will start by counting the number of Basic Pokémon. We see that there are 9 Basics in the deck, which puts this deck in the middle of the “7-11” range of Basics so that the probability of NOT getting a mulligan is in the 70%s. It is not surprising that most of the top decks fit into the “7-11” window of Basics (aside from quad Sigilyph).
It’s sort of fun to sift through the top decks and see how many fit into the optimal probabilities. We’ll come back to this deck list with the other formulas and sum it up in the conclusion.
4.2: Application of P(M,N,Y,A)
You may have noticed that Table 2 was about as small as Table 1, and you may have guessed that the applications for this model are short and to-the-point as well. This model is used to determine the chance that you start with more than one Basic Pokémon at the start of the game.
However, this is not the only application of this model. If we knew the probability that your opponent’s deck could get the ideal setup for a turn 1 KO, then we could use this model to calculate the probability that a donk will actually happen.
This is how the calculation might go:
(100-P(M,N,Y,A))*Q = The probability of a donk in percent
Where “Q” is the probability that your opponent has the ideal setup to KO your active Pokémon in the beginning of the game.
Other than this, you can just use P(M,N,Y,A) to find out how to optimize your chances of NOT being donked. My next article may contain a general study of the probabilities behind the donk scenario so that we can analyze the current format in tournaments.
To apply this model to your deck, simply locate the number on 3.2; Table 1 to find out how probable it is for you to get at least 2 (or 3) Basics when you start the game. The 10% windows will also help you choose the right number of starters more easily.
If we look at the Keldeo deck above, we see that the number of Basics is still 9 (go figure), and the corresponding probability for starting with 2 Basics is: P(60,7,9,2)=40.06%, which is below half a chance. This may seem negative, but we will soon see why this isn’t so bad.
4.3: Application of P(Y,Yi)
BulbapediaObviously, this one is self-explanatory as well. This is a way to find the probability that you start with an Ideal Basic Pokémon if “Y” is the number of Basics in the deck, and “Yi” is the number of ideal Basics that you want to start with.
Note that the data from 3.3; Table 1, the probability that you get an ideal starter is not found by dividing the number of ideal starters by the number of total Basics and converted to percent.
Instead, this model considers the possibility of encountering both an ideal starter and another Basic, making P(Y,Yi) slightly higher that just 100*(Yi/Y), unless Yi=Y.
To use this model on your deck, count the number of total Basics, and the number of ideal starters, then refer to 3.3; Table 1 to see what your chances are of getting the right start.
Choosing the right number of starters in a deck can be crucial, as a lot of starters become obsolete late game, and thus optimizing the chances of starting with one can be very important. We will see that this model also can give insight on how often donks happen.
To apply this model to the Keldeo deck, let’s say that we want to start with one of: Keldeo EX, Bouffalant, or Mewtwo EX. The probability that we start with one of these is: P(9,5)=67.77%. Since each of the Pokémon that we chose to be ideal is essentially impossible to KO on the first turn, we see that the probability of being donked is quite low.
We can’t easily calculate the chance that you start with at least 2 Basics or an ideal starter, as the ideal starters also occur in hands with 2 Basics. So P(Y,Yi) and P(M,N,Y,A) influence each other. This may be a topic to come back to.
4.4: Application of P(M,N,X,Y)
Alright, this is by far one of the most useful probability models in all of the Pokémon TCG. This is because this model can tell you how many Trainer cards (or any other non-Basic card like Energy) are ideal to have in your deck. This is crucial as many games are determined by the amount of support that a deck has throughout the match.
Notice that I specified “throughout the match.” This is because after the game begins and the players start using Trainers and playing Pokémon and Energies from their hand, which lowers the number of cards in the deck, thus making it more probable that they encounter more support that they haven’t used yet!
This is obviously assuming that the player has chosen ideal Trainer/Supporter cards for the deck.
Ideal Trainer/Supporter: A Trainer/Supporter card that has an effect that ensures the retrieval of one or more cards in the deck.
With the right number of Ideal Trainers/Supporter cards, the probability that you will have options for support throughout the game. This is because of the effect explained above; the more cards that are expended, the better the chances are that you run into more support upon drawing. You can hopefully read 3.4; Table 1 easily now after all the others.
Since we can’t cover all of the different cards in the Keldeo EX deck with this model in this small subsection, we will just do an example. Since many of the Supporter cards chain into each other (N, Bianca and Juniper can draw other Supporters, while Skyla and Random Receiver can search for one), it is important to know the chance that you will start with a Supporter in hand at the start of the game.
If we count Random Receiver as a “Supporter” (because it gets you one), then the probability of having at least 1 Supporter in hand at the start of the game for the Keldeo EX deck is: P(60,7,15,9)=90.36%. This is quite high, which ensures that the deck should do very well in this area.
Note that since Supporters chain into each other, a high probability of starting with a Supporter, implies a high probability of having a Supporter throughout the game (if you don’t waste/discard them).
Conclusion
To conclude, I want to do a quick review of the Keldeo EX deck:
Probability of NOT having a mulligan: 70.02%
Probability of starting with at least 2 Basics: 40.06%
Probability of starting with an EX or Bouffalant: 67.77%
Probability of starting with at least 1 Supporter: 90.36%
These numbers show us that most of the time: this deck will not have a mulligan, will not get donked, and will start with a Supporter. You may be wondering why there are Double Colorless Energies in the deck.
This is because the probability that you start with a Supporter, an Energy, and Mewtwo EX, Keldeo EX or Bouffalant is:
P(60,7,9,15)*P(60,7,9,16)*P(9,5)=(90.36*92*67.77)/(100^(2))=56.34%.
This means that I have more than a 50-50 chance of starting with a good Pokémon, Energy to attach turn 1 and a Supporter to get the remaining Energy to attack by turn 2. I have found that attacking with this deck turn 2 gives a very nice edge to the deck especially when facing decks like Darkrai EX/Hydreigon, because getting a KO on a Sableye/Deino turn 2 can be devastating for them.
To finish off, I want to show you one more application of the models to a deck list, because analyzing deck lists is one of the best ways to get insight about the game.
Excadrill Hammertime
Pokémon – 12 3 Drilbur EPO 54 | via: https://sixprizes.com/?p=39577 via: https://sixprizes.com/?p=39577 Trainers – 36 4 N
3 Dark Patch 3 Level Ball | Energy – 12 |
Let’s analyze this deck:
Probability of NOT having a mulligan: 70.02%
Probability of starting with at least 2 Basics: 40.06%
Probability of starting with at least one Sableye or Darkrai: 77.32%
Probability of starting with at least 1 Supporter: 86.18% (Again, we count Random Receiver as a “Supporter.”)
We see that this deck actually has a better chance of getting a good starter, and thus not getting donked. I want to point out that this rogue deck has good probabilities for getting the right cards; however, this does not imply that it can contend with the top decks.
The aspects that separate the winning decks from the losing decks are not just the probabilities of getting certain cards, but also the value of those cards in the game. For example, the probability of getting a Hydreigon on the field quickly is low, but the value of Hydreigon is so high that building a deck that gets around this low chance (Darkrai EX/Hydreigon) wins a lot!
This also has to do with how many cards work together, how many cards become stagnant in your hand, etc… But discussing the numbers behind these things will have to wait until later, so we will have to conclude our analysis for now.
I know that was a lot, and you guys probably had to read it one section at a time. So thank you very much for reading! If anyone has any questions or problems that need solving (preferable ones that involve math) then feel free to message me any time.
As I hinted at earlier, I will most likely be making several articles on probability, which will be a bit more specific than this one. Congratulations if you made it through this intro to probability in the Pokémon TCG, it means a lot. See you next time!