The Monty Ball Problem (Re: Nest Ball and Great Ball)

The original article was written by Shintaro Ito. Please find the original Japanese-language article here. Note my translation is not a perfect translation and I have rearranged and reworded some parts of the original article for clarity and readability. 

Yesterday, a survey went around on Twitter:
[Translation: You are playing Lycanroc Zoroark. In your hand, you have a Zorua, 4 Zoroark GX, a Great Ball, a Nest Ball and a Double Colourless Energy. How do you use Nest Ball or Great Ball? Please select your option below.
  • Nest Ball first
  • Great Ball first
  • Nest Ball
  • Great Ball ]
Even amongst top players, opinions were divided.
Why did opinions vary so much? What kind of things do we need to consider in order to get the correct answer? I will breakdown the problem for you.
Not playing any cards down
Many people have misunderstood the nature of this problem. They think the problem is between two options: using Nest Ball and then Great Ball, or using Great Ball and then Nest Ball.
In fact, there are more than just these two options. We have the option to preserve all cards in our hand and not play down anything. This play actually has some merit to it. This is one reason why so many top players have differing opinions.
We will consider the options in depth in order to find the correct solution.
In order to answer this question, we must first consider all information that is given to us. Let us see what the survey says.

[Translation: The deck list is borrowed from Shimada's winning deck.]

[Translation: For people who are telling me to "be more specific!" your opponent opens with a Tapu Lele GX.]

It includes a deck list (the winning deck list from the Niigata Champions League), and we know that our opponent opens with a lone Tapu Lele GX.
As we are using the Niigata Champions League winner’s deck list, we will also assume the format we are playing is SM1-SM8b. That is, before the release of Tag Bolt.
This is what our board state looks like:
Under these conditions, we must consider what the optimal play is.
Our Goal
I will not include every gritty detail in this article. It is impossible to consider absolutely everything as there are too many possibilities. So in this article I will explain my train of thought.
Even just going through my thought process is hard work, but having a vague grasp on it will potentially improve your play. By doing this, you will have a better chance of using your intuition in game to lead you to make the correct play. I do not necessarily want you to understand everything, but I want you to be able to understand enough so that you can grasp these types of situations intuitively.
How do we find the “correct answer” for this problem? I will define this as “the play that gives us the highest chance of winning.” In other words, we need to think outside of just our first turn. We should consider our next turn, and the turn after that, and how to eventually win.
What deck is our opponent playing?
Firstly, we should consider what type of deck our opponent is playing. Depending on what they are playing, how we play out our turn will differ.
Although the only information we have about our opponent’s deck is that they play Tapu Lele GX, we can actually infer some things about their deck from this. Firstly, and most obviously, it means it is a deck that plays Tapu Lele GX, and it is highly likely that the number of Tapu Lele GXs in their deck is not low. That is, we can predict that our opponent plays a number of Tapu Lele GX and there are not so many other basic Pokémon in the deck.
Then, we should think about prevalent decks in this meta. The top 64 decks from the Niigata tournament are published in the following link, on a foreign website. This will be helpful to some extent:
Jirachi Zapdos, Lycanroc Zoroark, Blacephalon Naganadel and Rayquaza were all played quite widely.
Many Jirachi Zapdos decks did not even include a Tapu Lele GX, or they only ran one copy. Since there are also many other Basic Pokémon in the deck, the chances that it is a Jirachi Zapdos deck is low.
So it is reasonable to predict that it is highly likely our opponent is playing a popular deck other than Jirachi Zapdos (in the first rounds, however, it is necessary to consider that there is a larger variety of decks than in the top 64).
Your opponent’s first turn and your second turn
You should play with your goal in mind. What should you do in the first turn? In Lycanroc Zoroark’s first turn, it is generally accepted that it is best to get out many Basic Pokémon. This is correct in itself, but don’t let this stop you asking yourself, “why do we want to get many Basic Pokémon out?” As mentioned above, the goal is to increase your chances of winning the game. It is important that you can evolve multiple Pokémon in your next turn, and that you can have access to more options by increasing your hand size. That is, you play out your first turn so you can set up on your second turn. To think about what you want to do this turn, you must think about what you want to do in subsequent turns.
In order to think about we want to do in our second turn, we need to anticipate what our opponent will do in their turn. For example, if our opponent is playing Lycanroc Zoroark, they may try to get multiple Zorua/Rockruffs out on their bench. If they are playing Blacephalon, they may try to get out Blacephalon GX and multiple Poiples, and attach an Energy to Blacephalon GX. If they are playing Rayquaza Vikavolt, they will either set up a Rayquaza GX and Grubbins and pass, or they will attach an Energy to Tapu Lele GX and retreat, setting up for a  Tempest GX.  What is important is that there are very little ways in which our opponent can knock out our active Zorua.
In order to deal with the aforementioned potential moves, what should we aim to do in our second turn? If you look at our hand, we have Zoroark GX and Double Colourless Energy, so we can definitely use Riotous Beating. The target of our Riotous Beating is the Defending Tapu Lele GX or potentially a benched Basic Pokémon. If we are against Rayquaza, there is also a possibility that a Rayquaza GX will be in the Active position.
With these Pokémon in mind, how much damage can Riotous Beating do? Ideally, we would like to hit 170 damage to knock out Tapu Lele GX, or 180 for other Pokémon. However, to reach these numbers, we must fill up our entire bench, use a Professor Kukui and attach a Choice Band – this is not easy.
If we cannot hit this combination, 60 damage is still an important number to reach. If you hit for 60, the Tapu Lele GX has 110 HP left (or Rayquaza GX has 120 HP left), meaning we can knock it out in the following turn with a full bench.
Where our opponent lays out Zoruas/Rockruffs/Poiples/Grubbins, these can all be knocked out with 60 or 70 damage. We could even target Inkays in Ultra Necrozma or Hoppips in Lost March. In other words, in our second turn, if we have two or three benched Pokémon we can achieve our goal.
What to use on your first turn
There are five options:
  1. Not using anything
  2. Using only Nest Ball
  3. Using only Great Ball
  4. Using Nest Ball and then Great Ball
  5. Using Great Ball and then Nest Ball 
Using Nest Ball meanss that we are able to take out any Basic Pokémon that you like. Using Great Ball means that we can add Evolution cards to our hand and also still be able to use Tapu Lele GX’s Wonder Tag.
With this in mind, let’s consider the pros and cons of each move.

1. Not playing anything
An advantage of not playing any Ball cards is that we are able to decide our actions after our opponent has finished their first turn, so we can respond to them. However, it is necessary to consider how our actions will change depending on what deck our opponent is playing.  
Also, on our second turn we can play Nest Ball and Great Ball after we have already drawn a card, giving us a higher chance of hitting a useful Pokémon. If we use our Ball cards before drawing, there is a lower chance we will draw into a Basic Pokémon in our next turn.
A disadvantage of not playing any cards is that there is a possibility of our Zorua being knocked out and us losing the game. Some combinations leading to this scenario are: Double Colourless Energy and Professor Kukui, Buzzwole and Switch, etc. Also, compared to the option where we do play cards, we will have less Trades available in our following turn.

2. Playing only Nest Ball
This is what happens when we hold the Great Ball. The advantage is that we can maximise the chances of drawing a Tapu Lele GX next turn, as we have access to more Trades. We could use our two Trades and then use Great Ball afterwards to increase our chances of getting Tapu Lele GX. This is the highest possibility of finding a Tapu Lele GX compared to all the other options. The ability to use a Supporter like Guzma, Judge, etc. on our second turn may have a huge influence on the game.

We may also draw into an Ultra Ball from our Trades, which is another advantage, and we can pay for part of the cost with the Great Ball.
The disadvantage is that our board state will likely be weaker than if we use both Balls in the first turn. For example, compared to the case where we hit a Zorua with our Great Ball, the amount of times we can Trade is decreased by one. Or, if we hit a Tapu Lele GX with our Great Ball and can use a Supporter. There is a chance that this could affect the way the game is played from turn three onward.
3. Only using Great Ball
This is what happens when we hold the Nest Ball. How we play our turn out will change depending on what card we hit from our Great Ball, so this plan should be considered as a derived form of the ‘Great Ball, then Nest Ball’ plan. For example, if we find a Zorua from our Great Ball, we can then change which card we get from Nest Ball after observing our opponent’s plays, making it easier for us to set up any Evolved Pokémon in the third turn.
The disadvantage is our board is very fragile compared to when using both Balls, just like when we only play the Nest Ball.
4. Nest Ball, then Great Ball
From here on, we will discuss the turn in which we play both Balls. The biggest advantage of playing Nest Ball first is that this increases our probability of hitting a Tapu Lele GX with Great Ball the most. By reducing the number of cards in our deck with Nest Ball, we increase the chances of finding a Tapu Lele GX. Currently, there is no Supporter in our hand so this could be a big advantage.
It is also beneficial to check our deck before we use Great Ball to figure out what is in our Prize Cards. It is necessary to figure out how our Prize Cards may change what Pokémon we choose from Great Ball.
The disadvantage is that by playing Nest Ball first, we decrease the number of Pokémon in our deck. This means if we search for a Zorua with Nest Ball, there is a lower chance of us drawing a Zorua off Great Ball.
5. Great Ball, then Nest Ball
The advantage of using Great Ball first is that we improve our chances of having three Pokémon out in the first turn. The probability of hitting a basic Pokémon with Great Ball is high as we have not taken any out of our deck yet. Also, we may change our mind of what to get with Nest Ball depending on what Pokémon you hit with Great Ball. For example, if we hit a Lycanroc GX with Great Ball, we may get a Rockruff or Ditto Prism instead of a Zorua with Nest Ball.
Compared to the previous option, the chances of hitting a Tapu Lele GX decreases slightly.
Probability Calculation
So, how can we select the correct answer?
In game, of course we don’t really have the time to go through complex calculations, and what move we choose depends largely on the intuition that comes from experience. Here is a comparison between options 4 and 5 to explain how probability can be used to help determine the optimal play.
Being able to check your Prizes thanks to Nest Ball
Firstly, I will consider the point that you can confirm what is in your Prizes by using Nest Ball first. I would like to confirm this before using Great Ball. This is due to, for example, if Evolution cards are in my Prizes. While I have 4 Zoroark GX in hand, I have no idea what else is not in my Prize Cards. If you use Great Ball first, you may judge the situation incorrectly and take the incorrect Pokémon. So I will consider the following situations:
You may make a mistaken call when you see multiple Pokémon with Great Ball. For example, I could see a Rockruff and Sneasel from Great Ball and take Rockruff. But then I would use Nest Ball and discover all my Lycanrocs were Prized. If I had used Nest Ball first I would have known this and made a different decision.
At what probability does the above occur? If you can see Zorua and Rockruff, take the Zorua as you already have Zoroark GX in hand, and then after using Nest Ball you can check if you have Lycanroc Prized and then decide what you want to get. So, the condition of the above event occurring is as follows:
A.    In addition to Rockruff, you see a Pokémon other than Zorua and Lycanroc GX
B.    Both Lycanroc GX are Prized.
The occurrence of these simultaneously will lead you to use Nest Ball first.
There are two copies of Rockruff in the deck. Excluding Zorua, Zoroark GX, Rockruff, Lycanroc GX, and Ditto Prism there are seven Pokémon in the deck. Also, you have the 8 cards you have seen from your hand and board. So, you have 52 cards that you have not seen. The probability of A can be obtained by the following formula.
The probability of either Rockruff being revealed, multiplied by the probability in which any one of the seven 'other' Pokémon is also revealed multiplied by the probability of the remaining five cards all not being Pokémon, divided by  the total number of seven-card combinations gives us:

 \$\frac{2C1 \times 7C1}{52C1} = 2 \times 7 \times 435897 \div 133784560 = 4.56\%\$
The probability of B occurring can be obtained by the following:

 \$\frac{2}{52} \times \frac{1}{51} \times 6C2 =1.13\%\$
Therefore, there is about a 0.05% chance that both A and B occur simultaneously. I have missed some things such as if B has occurred then A is slightly more likely, or if you see three or more Pokémon, but this will have an effect of less than 0.1 percent. [Note: Shintaro also could have considered the situation of seeing a Sneasel from the Great Ball where all Weaviles were Prized, which would have increased the odds slightly.]
Therefore, it is concluded that the probability that you are in a situation that benefits from confirming Prize Cards by using Nest Balls first is 1% or less.
Pokémon you want to find and probabilities
Next, consider the Pokémon we want to find with each Ball. Although we have 4 Zoroark GX in hand, we do not have a Supporter. There is also a possibility that our opponent will knock out a Zorua during their turn.
So, we can confirm that the Pokémon we want to find with Nest Ball is Zorua. Pokémon which we will choose from Great Ball is divided into four situations, where A is the strongest and D the weakest:
A: Tapu Lele GX
B: Zorua or Ditto Prism
C: Anything else
D: No Pokémon
For each pattern, consider the winning percentage from your own experience. In order to make the problem easier to understand, look at the following table:
Tapu Lele GX
Zorua or Ditto Prism
Other Pokémon
No Pokémon
Likelihood of winning
Next, for each situation, consider the probabilities of obtaining each card when using Nest Ball first and when using Great Ball first.
In the unknown 52 cards, there are two Tapu Lele GXs, 4 Zorua/Ditto Prism, and 9 Pokémon other than that. Cards which are not Pokémon comprise of 37 cards in the deck. The probabilities can be calculated as below:
Not hitting a Pokémon: \$\frac{37}{51} \times \frac{36}{50} \times \frac{35}{49} \times \frac{34}{48} \times \frac{33}{47} \times \frac{32}{46} \times \frac{31}{45} = 8.89\%\$
Hitting some 'other' Pokémon: \$\frac{46}{51} \times \frac{45}{50} \times \frac{44}{49} \times \frac{43}{48} \times \frac{42}{47} \times \frac{41}{46} \times \frac{40}{45} \text{-} (\mbox{Not hitting a Pokemon}) = 37.34\%\$
Hitting a Zorua/Ditto: \$\frac{49}{51} \times \frac{48}{50} \times \frac{47}{49} \times \frac{46}{48} \times \frac{45}{47} \times \frac{44}{46} \times \frac{43}{45} \text{ -} (\mbox{Not hitting a Pokemon}) \text{ -} (\mbox{Hitting some 'other' Pokemon})\$ = 27.96%
Hitting a Tapu Lele: \$ 1 \text{ -} (\mbox{Hitting a Zorua/Ditto}) \text{ -} (\mbox{Hitting some other Pokemon}) \text{ -} (\mbox{Not hitting a Pokemon}) =\$ 25.80%
Then the stats on using Great Ball and then Nest Ball are as follows:
Tapu Lele GX
Zorua or Ditto Prism
Other Pokémon
No Pokémon
Likelihood of winning
Playing Nest Ball First
Playing Great Ball First
With all this compiled, the percentage of each occurrence is multiplied by the winning percentage of that play. Calculating the winning percentage of using the Nest Ball first is 54.14%. Playing the Great Ball first gives a winning percentage of 55.53%.
Even if you set the winning percentage of drawing a Tapu Lele GX to 80%, playing the Nest Ball first still gives an overall lower winning percentage. Even if you do draw into a Tapu Lele GX I don’t think the winning percentage would be that high, so in actuality the difference of the winning rates of Nest Ball and Great Ball should be larger.
The advantage of confirming Prizes with Nest Ball is also not so large, considering there is a possibility that the winning percentage can increase by playing the Great Ball first (which may even change the card that is found with Nest Ball).
So we can conclude that using Great Ball before Nest Ball is better.
I will not include the calculation for comparing the two situations where you only play one Ball, but it can be calculated in a similar fashion.
The second option, where you play the Nest Ball but keep the Great Ball is actually quite a good move. You can compare the effectiveness of playing only the Nest Ball against playing both by considering how the differing plays affect the probability of drawing Guzma, etc. in the following turn.
Of course, there is no such time to do these calculations in the actual match. However, being aware of these kinds of things during playing or after the match can help you to make better choices as a player.
The accumulation of realising these small details can lead to you winning matches that you otherwise may have been just short of winning.