If you haven’t read part one, do that now.

This article is a direct predecessor and refers to terms I explained in the previous one. The reason for this second part is to continue where I left off and talk about additional interactions.

Here’s a short recap of the main points I talked about earlier:

- The most important step of sequencing is planning what you want to hit.
- If you want multiple cards, rank their importance and adjust accordingly.
- Some decks have unique sequencing patterns – practice makes perfect.
- Uncertainty should be considered, especially when many cards interact with each other.

There’s one final point I want to add here, and it’s one that I’ll cover in-depth to start. It can quantify different outcomes of a string of decisions, specifically when you have additional information from a card effect.

- Sequencing based on uncertainty should be considered as a second method, and when compared with pure digging can possibly raise your expected end of turn score.

The “end of turn score” is what’s new here. Its phrasing is similar to “lessening the probability of your bad outcomes” but removes the limitation of only the bad outcome. In other words, it’s taking the safe play – which calculates out to be the best play – objectively.

## Expected Score of a Turn

The “end of turn score” can be classified as a rating based on how well your turn went. If it went horribly, you’d say it was a one out of ten. If it went perfectly, a ten out of ten! In probability classes, you’d learn about the *expected value* of a probability distribution function, denoted as E[x], where x is the distribution. We can actually consider this rating of one through ten and determine the expected rating based on the probability of receiving any given rating and summing across all ratings.

E[x] = x*p(x) = 1*p(1)+2*p(2)+ … + 10*p(10)

It’s important to note that p(1)+p(2)+ … p(10) = 1 by the Law of Total Probability. Now this formulation is cool in theory, but impossible to apply over an entire turn. There isn’t a feasible way to calculate p(x) with multiple decisions, and our 1–10 scale is objectively inexact.

This formulation has use when looking at decisions individually. Let’s refer back to the example from last week involving Jirachi, Reset Stamp, Professor’s Research, and Marnie. You have Marnie and Professor’s Research in hand, a Jirachi in play, and a Reset Stamp in a 20-card deck. You have to disrupt the opponent’s hand, or you lose. What’s the objectively correct play?

First we must assign ratings to each outcome and calculate the probability of each outcome. In the cases of “Marnie” and “Professor’s Research,” you missed the Reset Stamp. Our two policies are Stellar Wish before a Supporter, and Stellar Wish after a Supporter. The rating stays the same for both, but the probabilities will be slightly different. Note that the probabilities for each Supporter pair are complementary: they sum to one. After calculating, we’ll separate the Professor’s Research probabilities from the Marnie probabilities, as they’re separate cases. With two cases and two policies, we’ll end up with four numbers to compare. **There’s also a fifth outcome that deals with the additional information from Stellar Wish.**

### Ratings

- Professor’s Research + Reset Stamp: 10
- Marnie + Reset Stamp: 8
- Marnie: 5
- Professor’s Research: 1

### Probabilities (Supporter first)

- 12/20 = 0.6
- 10/20 = 0.5
- 10/20 = 0.5
- 8/20 = 0.4

### Probabilities (Stellar Wish first)

- 1 – (15/20*12/19) = 10/19 = 0.5263
- 1 – (15/20*14/19) = 7/16 = 0.4474
- (15/20)*(14/19) = 21/38 = 0.5526
- (15/20)*(12/19) = 9/19 = 0.4737

Note: As a proof to the “Acro Bike before Trainers’ Mail when looking for a Trainer,” notice how the probability of success (#1 and #2) decreases when you Stellar Wish first.

### Expected Values (20-card deck)

- Professor’s Research then Stellar Wish = 10*(0.6)+1*(0.4) = 6.1
- Marnie then Stellar Wish = 8*(0.5)+5*(0.5) = 6.5
- Stellar Wish then Professor’s Research = 10*(0.5263)+1*(0.4737) = 5.7368
- Stellar Wish then Marnie = 8*(0.4474)+5*(0.5526) = 6.3421
**Stellar Wish into either = 15/20*((5*14/19)+(8*5/19))+5/20*(10) = 6.8421****p(J’)*E(Marnie) + p(J)*E(Prof)**

The expected value for #4 is slightly inaccurate, as it doesn’t consider hitting the Reset Stamp on the Stellar Wish. However, that’s not the point; the takeaway is from #5, which shows the value in the Stellar Wish information. p(J’) is the probably of **not** hitting the Stellar Wish, in which chase you’d play Marnie (it has a higher expected value assuming Stellar Wish failed: 5.75 vs. 4.15). If you hit the Stellar Wish, then you’d play Professor’s Research without worry and happily take the score of 10. The 1 becomes impossible because you’ve already fulfilled the Reset Stamp.

The outcome may change if you alter the parameters: the rating per outcome: x, or outs in deck: p(x).

### Expected Values (15-card deck)

**Professor’s Research then Stellar Wish = 8.2**- Marnie then Stellar Wish = 7
- Stellar Wish into either = 7.3810

It’s marvelous to identify the math that goes on inside our brains without us even realizing it. You can intuitively understand the correct play but never know why it’s correct, at least in mathematical terms. Though it’s cool, you won’t calculate probabilities like this in a game. Recognize the pattern, internalize it, and move on. **In uncertain scenarios (large deck), the additional information means more than the increase in probability.**

## Dedenne-GX and Professor’s Research

One common interaction nowadays is Dedenne-GX and Professor’s Research. Which should you play first? There are several factors to consider, and the answer isn’t as clear cut as Acro Bike vs. Trainers’ Mail. The overarching decision – all its pros and cons – falls outside the realm of sequencing, but the question of “how do I best dig” is inside. A success is indicated by drawing into the other piece, such as Professor’s Research from Dedenne-GX or vice versa. (That’s the best way to determine which card is better to play first, assuming you want to draw the most cards.) For simplicity’s sake, I won’t compound this table with Stellar Wish, Pokegear 3.0, Great Ball, and other potential outs.

These tables aren’t anything too fancy but give a pretty good idea of the differences between the two choices. There are a few things to draw from these comparisons. First, p(Dedenne-GX) > p(Research) for the same number of cards and outs left in deck, purely because Professor’s Research draws an additional card. This is a given.

The second takeaway can be seen in the Differences table. The difference in p(success) is greatest when there are approximately three–four outs remaining. The maximum difference increases as deck size increases, but it occurs with fewer outs remaining. You can see this by looking at the 30-card and 20-card columns; the max of 0.064 is with four outs, whereas the max of 0.068 is with three outs, respectively.

Next, you receive diminishing marginal returns for every out added. This can be seen in the Marginal Difference table, which is calculated by p(Dedenne-GX, n) – p(Dedenne-GX, n-1) for n{2,3,…,9}. Simply speaking, it’s the difference between two outs and one out, three outs and two outs, etc. all the way down. Your probability of success greatly goes up when you increase from one to two outs, but very little when you go from five to six outs.

Finally, look at the fourth table. It compares p(success) by playing Dedenne-GX first with one additional out over Professor’s Research with a 25-card deck. Notice that, despite there being an additional out to Professor’s Research over Dedenne-GX, playing Professor’s Research first is eventually better. This is because the additional out means less than the seventh card drawn. Considering the case of 20 cards in deck (the table isn’t shown), this effect happens marginally sooner. As deck size decreases, the additional card from Professor’s Research becomes more valuable.

It’s possible to extrapolate values for uncertain outs, such as Great Ball or Pokegear 3.0, and other Supporter cards. If you consider drawing Marnie to be 75% as good as drawing Professor’s Research, you can add 0.75*x, where x = number of Marnie remaining, to your total outs to Professor’s Research. This method has no mathematical basis but is an efficient way to depict relative value while factoring in other components. If Great Ball has a 50% chance of hitting Dedenne-GX, you can add 0.5*y, where y = number of Great Ball remaining. **It’s not perfect, but it’s a rough, heuristic solution applicable during a game.**

Crobat V vs. Dedenne-GX

Crobat V and Dedenne-GX are comparable, though I won’t delve into a mix of probabilities with this one. The main choice to make here is in considering the value of the cards already in your hand, assuming you’re given the choice to Quick Ball for one or the other. The secondary choice is in how many cards of the six you’re expected to burn if you play Dedenne-GX first. If that’s greater than or equal to the cards you’d draw off of Crobat V, then you’ll see more cards off by playing Dedenne-GX first.

In my opinion, leaving yourself with a better-than-trash hand is of utmost importance. I’d be scared to use Dark Asset for only a few cards without a follow-up Supporter on the current or following turn. In today’s format where speed is key, I wouldn’t hesitate to Dedechange even if it meant burning a key card or two.

Givens and Thinning

I realized that I hadn’t create a section purely dedicated to thinning, and instead peppered it throughout the articles. I figured it doesn’t hurt to include a short section describing good thinning practices, especially because it’s the most common form of sequencing in today’s Item-heavy Standard format.

Givens are cards that search for only one type of card. Quick Ball, Viridian Forest, and Tag Call are examples of givens. You have perfect knowledge with these cards, making them prime for perfect sequencing. If you wanted to maximize the probability of hitting a certain card from an uncertain effect (Stellar Wish), you’d play each of these cards first because it thins the deck. If you were focused on uncertainty, you’d Stellar Wish first so that you could alter your choices from the given search cards. In this scenario, if your choices were dependent on Stellar Wish’s outcome, then you should hold them until afterward. Because of their nature, givens will almost always be at the beginning or end of a sequence, never in the middle.

The principle of thinning remains the same when using givens or uncertain cards (recall Acro Bike and Trainers’ Mail). It’s about reducing deck size – the denominator of the fraction (cards drawn)/(cards remaining) – in aiming to hit a certain card or combination. As deck size decreases, thinning has a greater impact in increasing the probability of success. It’s a process you should weigh against resource management at all stages in the game. You shouldn’t overthin the deck by discarding important cards you might need later. My personal rule is to rarely thin a card that could possibly contribute to a win condition.

## Rapid-Fire Good Practices

I’m putting all of the general practices that came to mind as I was planning this article here. These are simple enough to explain in a sentence or two, so I’ll list them out.

- If you’re unlikely to thin the hand back down to below three cards, and don’t have many important resources to discard, use Oranguru‘s Instruct before drawing more cards with Set Up/Dedechange/Supporter.
- You should discard a Supporter card with Battle Compressor when one isn’t already in your discard pile. Doing so adds VS Seeker to the list of live outs to a Supporter card.
- If you have a pair of Jirachi and a switching effect in hand, send up the one least able to retreat first (without Escape Board/Energy) so that you can instantly switch out after the first Stellar Wish.
- Search for the Water Energy first in ADPZ so that you don’t decrease the probability of hitting a Metal Energy with Intrepid Sword (unless playing around Energy denial).
- If able, use Mawile-GX at the start of the turn. Knowledge of your opponent’s hand is useful and can guide better decision-making.
- Discard the first Metal Energy you draw so that you have a target for Metal Saucer.
- If you have a Pokemon Communication, Basic Pokemon, and Quick Ball in hand, you can put the Basic Pokemon back with Pokemon Communication first to guarantee a Quick Ball target.
- When you have multiple cards that can search the deck before you’ve done your first search, use the most non-committal card first. Cherish Ball is a great example because you don’t have to discard cards from your hand.
- Attach Basic Energy before Special Energy when your opponent plays Dangerous Drill or Giratina.
- Retreat your Active Pokemon as late in the turn as possible.

## Conclusion

I hope these past two articles have taught you a thing or two about sequencing. It’s an easy skill to identify, but difficult to master because no two situations are exactly the same. Though the parameters are slightly different, they may tip the probabilities in one way over another from how you’ve seen it previously. That’s why I’ve stressed the importance of the guidelines; they’re a better learning tool than memorizing each scenario. When new sets and new scenarios arise, what would you do then?

The teacher’s job is to teach, and one effective method is to break down previously accepted ideologies and rebuild them with a stronger foundation. I’ll admit that I manufactured some of these terms, like “sequencing based on uncertainty” and “end of turn score.” It’s possible to understand sequencing perfectly well without ever reading this article. However, I believe these to be important generalizations in understanding *why* certain sequencing patterns are correct. By being able to label the reason behind a correct sequencing pattern, you can learn quicker and better than you would’ve otherwise.

My main goal with these articles, aside from teaching how to sequence, was to highlight the beautiful intersection between math and Pokemon. Even if you don’t appreciate the math, quantifiable numbers aid in understanding of qualitative descriptions. Something that seems lucky, such as drawing a Supporter card from a dead hand, may not be so unlikely after all once you take a look behind the scenes.