So most teams are at the tail end of their season with just two, three or four games left in their schedule. Some might be led to believe that the number of possibilities are few. It’s not. The numbers are still quite staggering and frankly, a lot of crazy can still happen. In this edition of Let’s Talk Basketball, we break down the next few games and we try to make a scientific and calculated approach to answering the race to the Final Four.

#### Introduction

To start, we need to answer what the sample space is. What do we mean?

We need to know how many possibilities are still left. Learning how many possibilities are left can help us answer important questions.

“What are the odds that FEU totally flames out again and miss the Final Four all together?”

“What are the odds that Team X gets into the Final Four?”

“What are the odds that Team Y gets a twice-to-beat advantage? A twice-to-beat disadvantage?”

There are twelve games left in the schedule. Each game can end in either a win or a loss. What this means is there are still:

2 x 2 x 2 x 2 … x2 or 2 ^ 12 = 4096

Yes, there are still 4096 possible outcomes even at this point of the season. As sir Mico Halili tweeted: “The numerical possibilities!”

Knowing how to break down those 4096 possibilities gives us a glimpse at what the future holds for the league, for our favorite team/school or for the Final Four.

#### The 50/50 Experience

To begin, you guys need to understand what a binomial tree means.

Ok, you got scared when I said “binomial” – let’s skip that. Let’s just call it a “tree”.

No, I did not mean this kind of tree – but it kinda is. (Photo Credit: Wikimedia.org)

When a tree grows branches, the next bud can go a multitude of ways, right? and when I mean, a multitude, I really mean it. (If you consider all the XYZ combinations you can think of, your mind might explode. Yes, that’s a 3-dimension analysis.)

But that’s still too complicated. Let’s break it down even further.

Imagine you’re driving in a road that is exclusively either left or right when it decides to fork (there are no left, straight, right). Now, this road has exactly three forks along the way. Each fork, you have a decision to make: “Do I go left or right?”. At each fork, you’ll find yourself asking this question and each question (and answer) is defined by the one that proceeds it. A left on the second fork can get you in two different places depending on what your decision was on the first fork (left or right?). Turning left on the second fork could mean you end up in a supermarket if you turned left on the first fork while a right on the first fork would mean you could end up in church once you turn left on the second fork.

That means you could end up in four different places just based on your first two decisions:

1. A left on the first fork and a left on the second fork.
2. A left on the first fork and a right on the second fork.
3. A right on the first fork and a left on the second fork.
4. A right on the first fork and a right on the second fork.

It suddenly becomes more complicated on the third fork, which further pushes the number to eight — since you have two decisions to make on three turns (2 raised to 3 or 2 multiplied by itself, 3 times). That “decision” tree looks somewhat like this:

This looks like a decision tree. (Photo Credit: www.thulasidas.com)

Now, you’re probably asking: What does this have to do with basketball?!

Well, imagine those 12 games left as “forks” that can go either two ways: Team A wins (which also means Team B loses) or vice versa. Remember how two forks and two decisions (left or right in my example, in this case it’s a win or loss) mean there are four places you can go to? Extend that to 12 forks and what you get is 4096 possible places you can go to.

Now, this doesn’t actually mean that all those 4096 possibilities are unique. In the image above, if you go up on the first decision then go down on the first decision, you’ll still end up at the same point if you went down on the first decision then went up on the next. But the destination may be the same but the journey wasn’t. It’s the same for the race to the Final Four.

Example: Ateneo has three games left, they can get to a record of 8-6 three different ways – LWW, WLW, WWL. But we all know that losing your last game of the season going into the Final Four won’t have as much momentum as say winning your final two games. Same record, different momentum (not to mention countless other possibilities for the other seven teams).

In reality, there are only 2430 unique “record” combinations for the UAAP.

Erratum: What we do is we play those 4096 unique record outcomes for all 8 teams and then how many of those combinations did Team A have a Top 4 record. We’ll then divide this number by 4096 and we’ll get the probability that Team A can get into the Final Four. We’ll also do it for a twice-to-beat advantage (doing the same procedure). Here is the table (Table 1.1).

Explanation for Erratum: I realized something was wrong when I realized that the 2430 unique record combinations did not necessarily have equal probabilities of happening. The numbers barely move but for the sake of being correct, I decided to do this erratum.

As of 9/1/13

Final Four 60.9% 0.0% 91.6% 100% 100% 40.6% 0.0% 60.4%
Top 2 17.0% 0.0% 53.4% 83.0% 89.1% 9.4% 0.0% 16.9%

As of 9/4/13

Final Four 61.3% 0.0% 100% 100% 100% 20.5% 0.0% 61.3%
Top 2 12.1% 0.0% 69.5% 72.7% 96.9% 0.6% 0.0% 12.9%

A couple of notes:

1. This assumes that all 4096 combinations are equally likely to happen. That means that by extension, in each of the 12 games, each team is likely to win as they are to lose. We know that that’s not the case. UP doesn’t have a fifty-fifty chance against NU. However, for the sake of argument and theoretical discourse, we’ll assume this first (and then move towards adjusting the probabilities later).

2. A “Final Four” doesn’t mean you’re in – it just means you have a record that ranks fourth best. Again, it doesn’t mean you’re in. It can also mean that you’re tied for the last spot in the Final Four. Here is a sample record sheet at the end of the season:

Team Wins Losses
NU Bulldogs 11 3
FEU Tamaraws 10 4
De La Salle Green Archers 9 5
Ateneo De Manila Blue Eagles 7 7
UST Growling Tigers 7 7
UE Red Warriors 7 7
UP Fighting Maroons 1 13

In this case, Ateneo, UST and UE are tied for the third best record. According to my sources (read: the awesome source), this will be decided by a quotient system. Which means this will be settled by their margin of victory. You get the point, if you wanted to know the chances of your team getting into the Final Four for sure (and a twice-to-beat advantage), here are the numbers (Table 1.2):

As of 9/1/13

Sure Final Four 26.8% 0.0% 49.2% 56.2% 56.2% 12.5% 0.0% 24.0%
Top 2 2.5% 0.0% 16.9% 35.9% 42.5% 1.2% 0.0% 2.7%

As of 9/4/13

Sure Final Four 30.9% 0.0% 63.1% 63.1% 63.1% 3.9% 0.0% 28.3%
Top 2 0.8% 0.0% 24.2% 26.6% 46.9% 0.0% 0.0% 1.6%

Time to eat the numbers up:

1. First, let’s give a big warm applause to the Adamson Soaring Falcons and the UP Fighting Maroons . Both teams are out of the Final Four race for good. I think for Adamson, the final nail in the coffin happened against Ateneo (their seventh loss). For UP, it happened in the first round, which means they’re playing for pride right now and (maybe) some momentum (ANY type of momentum) they can carry to next year’s team.

2. Looking at the numbers, their’s a clear delineation: there’s FEU and NU as clear front-runners for the Top 2 seed then there’s DLSU just lurking in the shadow against in case one of FEU and NU screws up, there’s Ateneo and UST jockeying for that final Final Four spot and then UE praying against hope that they still have a shot at getting in.

3. Question: Why is Ateneo more sure to get into the Final Four (26.4 percent) than UST (24 percent) but they have equal chances at a Top 2 seed?

That’s because Ateneo has a lot of “double-valued games.” What do I mean? Well, Ateneo still has games against two teams that are clearly trying to take their precarious spot at fourth – one against UE on September 8 and another one against UST on a yet to be determined date and they have another date against NU. A win there would not only a “W” in their column but also an “L” on the columns of those teams that matter.

Meanwhile, UST has only two such games: against Ateneo and La Salle. Their other game? A date with bottom feeders UP.

A harder schedule does have its perks, you know?

4. FEU and NU are almost safely into the Final Four. A 56.2 percent chance to get in and a 43.8 percent chance to at least have a chance to get in? It would take a massive meltdown (losing all their remaining games) and bad luck to get them out of the Final Four.

5. Ateneo and UST  have the same odds of not getting into the Final Four as UE’s odds of getting into the Final Four. UE better bring their A-game (without Mammie and Olivares) or they’ll be out of it, quick.

#### 9/4/13 Update:

1. As you can see, the numbers clearly reflect what everyone thinks. At least now, we have a numerical value we can place on this belief. La Salle just made a huge leap both in their chances at a Final Four slot and a Top 2 slot with their huge win against UE. Consequently, UE took a big hit – with their chances at a Final Four dwindling to just 4.3 percent and a 16.8 percent chance to tie for a Final Four worthy record. Odds are now against them and they don’t hold all the cards.

2. FEU received a slight hit on their Top 2 seed — now that La Salle has a tied record with them. I’d even go so far as to say that La Salle has a bigger chance considering they have a higher margin of victory.

3. The NU win just further cemented their bid for a Top 2 slot – a 47.4 percent chance at a Top 2 seed and a 49 percent chance to at least tie for a Top 2 seed. One more win would probably push this up to 60~70 percent.

4. Ateneo and UST received mixed results due to the DLSU/UE battle. Their Final Four chances got a bump up due to the UE loss but their bid for a twice-to-beat advantage took a shot with the La Salle win.

If you’re a fan of the other team, you better hold on to those numbers, tight. I’m sure I will.