# A Theoretical Model for The Probability of Winning a Basketball Game – Part 3

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*This is the third in a 3 part series where I will present a theoretical model for the probability of winning a basketball game. The 3 parts will break this model down at the team, unit, and player level.*

In part two of this series I presented a way to break down the probability of winning a basketball game from the **points scored per 5-player unit per play** to the actual players themselves.

At the end of part two, I finished the post with the following formula:

E(Points) = Pr(O1) x E(Points | O1) + Pr(O2) x E(Points | O2) + Pr(O3) x E(Points | O3)

+ Pr(O4) x E(Points | O4) + Pr(O5) x E(Points | O5)

This formula is how I’ve chosen to break down the *points scored per 5-player unit per play* into the player-level components.

**The Components of This Formula for E(Points)**

Specifying E(Points) in this way allows us to understand how each player impacts the expectation at the 5-player unit level, which in turn affects the team’s probability of winning.

In this formula Pr(OX) represents the probability that offensive player X is responsible for the event that leads to the end of the play. Referring to the 5-player unit level, this is an event such as a field goal attempt, turnover, etc. Clearly Pr(O1) + Pr(O2) + Pr(O3) + Pr(O4) + Pr(O5) must be equal to 1.

**What Affects Pr(OX)?**

One of the most important pieces of this formula is how Pr(OX) is constructed. First, I’ll start with a quick list of things that affect a player’s probability of using any given play (that’s what I’m calling Pr(OX), for those of you that fell asleep during the last section):

- Game situation
- Fatigue
- Focus

**Game situation** affects the offensive and defensive team’s priorities. Thus any given player’s role within that context will change, hence changing Pr(OX) for every player. **Fatigue** is what I will refer to as each player’s physical state, and **focus** is what I will refer to as each player’s mental state. The offensive and defensive player’s fatigue and focus will also affect Pr(OX). It’s worth pointing out that game situation can affect focus, so these are certainly not independent factors.

**Breaking Down E(Points | OX)**

Now that we know what will affect Pr(OX), lets now derive what E(Points | OX) looks like. To prevent redundancy, I won’t bother to re-define the specific player level pieces of E(Points), as it is very similar to the breakdown at the 5-player unit level that started off with the definition of:

E(Points) = Pr(No Turnover) x E(Points | No Turnover)

+ Pr(Turnover) x E(Points | Turnover)

Instead of regurgitating the list of possibilities given specific shots with or without a shooting foul, I will focus on plays that lead directly to points. The rates of events like turnovers will increase with lower levels of fatigue and focus, but hopefully that is obvious.

**Focusing on Point Producing Events**

Every shooting event can be broken down into the following:

E(Points) = Pr(No Shooting Foul) x E(Points | No Shooting Foul)

+ Pr(Shooting Foul) x E(Points | Shooting Foul)

So if we know player X takes a shot, we know that some percentage of the time they are not fouled, and some percentage of the time they are fouled.

**Shots Without a Foul**

First lets examine E(Points | No Shooting Foul), the expected points for a shot given a shooting foul did not take place. For any given shot, E(Points) is given by:

E(Points) = Pr(Make) x E(Points | Make) + Pr(Miss) x 0

Because E(Points | Make) can only be 2 or 3 points (depending on the shot type), we simply need to focus on the probability Pr(Make).

Pr(Make) is affected by the same factors as Pr(OX). We can go a little further, though, and say that a player’s Pr(Make) for a shot is really conditional on these factors at the time of the shot, and can be written as:

Pr(Make) = Pr(Make | Shot Location, Own Fatigue, Own Focus, Opponent State)

In words, this means that a player’s probability of making a shot is conditional on the shot location, the player’s fatigue and focus, and the opponent state. For clarity, **opponent state** signifies the defensive pressure on the shot attempt.

Trying to understand the structure of how the shot location, own fatigue and focus, and opponent state is more of a practical model issue, as in “How would we create a model to generate a probability when we take these factors into account?”. It’s certainly something I think about often, and I would be lying if I had the perfect model in mind. Hopefully this will evolve over time as I work to construct these models.

**Shots With a Foul**

For completeness, I will make a comment about shots with a foul. Shots that have a foul really only change the opponent state in the specification of a shot without a foul given above. Clearly Pr(Make) will decrease in this situation. How we model this is again a practical issue.

**The Next Step**

The next step is to start building practical models of the pieces derived in this theoretical picture of the probability of winning a basketball game. As I build these practical pieces, the plan is to make assumptions about the missing components and work with the pieces I’ve got to create more important questions and generate new insight about what a good practical model for the probability of winning a basketball game looks like.

Overall I feel this is a good theoretical model to start with. It’s not perfect, it doesn’t capture everything, but it seems to capture the important stuff. If you made it through all 3 parts of this series, thanks for hanging in there with me! This isn’t the most entertaining aspect of this work, but (for me, at least) it helps focus future effort, which is very valuable.

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