# A Theoretical Model for The Probability of Winning a Basketball Game – Part 2

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*This is the second 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 one of this series I presented a way to break down the probability of winning a basketball game from the (theoretically) known probability of winning down to the **points scored per 5-player unit per play**.

Recall that at the 5-player unit level, the total points scored for any given play is:

Points Scored = FTM x 1 + 2FGM x 2 + 3FGM x 3

**Expected Points Scored per 5-player Unit per Play**

To break this down further we must now use mathematical expectation. In totality, the expected number of points scored for a 5-player unit on a given play is given by:

E(Points) = Pr(FTM = 0) x 0 + Pr(FTM = 1) x 1 + Pr(FTM = 2) x 2 + Pr(FTM = 3) x 3

+ Pr(2FGM = 0) x 0 + Pr(2FGM = 1) x 2 + Pr(3FGM = 0) + Pr(3FGM = 1) x 3

Where E() denotes expectation and Pr() denotes probability.

This formula now allows us to think of the number of points scored per 5-player unit per play in terms of expected value, but these probabilities are not independent. There are a lot of underlying probabilities involved that must be decomposed.

**Breaking Down the Probabilities of E(Points)
**

First recognize that E(Points) can also be written as:

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

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

It is obvious that Pr(Turnover) x E(Points | Turnover) always equals 0 because the conditional expectation E(Points | Turnover) always equals 0. As such, we can simply focus on the composition of Pr(No Turnover) x E(Points | No Turnover).

**Focusing on E(Points | No Turnover)**

Now it’s time to identify the structure of E(Points | No Turnover).

For brevity I will try to keep this as simple as possible, as there are a lot of possible combinations.

E(Points | No Turnover) = Pr(2FGA) x E(Points | 2FGA) + Pr(3FGA) x E(Points | 3FGA)

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

+ Pr(All Other Events) x Pr(Points | All Other Events)

I have extracted out **all other events** here to recognize events that cause the shot clock to reset and start a new play (such as jump ball situations). These always give us 0 points, so we will ignore it for now and focus on E(Points | 2FGA), E(Points | 3FGA), and E(Points | Non-Shooting Foul).

**Definition of E(Points | 2FGA)**

When a team attempts a 2 point shot the following events can happen:

- No Foul and Shot Made
- No Foul and Shot Missed
- Foul and Shot Made
- Foul and Shot Missed

This means E(Points | 2FGA) can be written as:

E(Points | 2FGA) = Pr(No Foul and Shot Made) x E(Points | No Foul and Shot Made)

+ Pr(No Foul and Shot Missed) x E(Points | No Foul and Shot Missed)

+ Pr(Foul and Shot Made) x E(Points | Foul and Shot Made)

+ Pr(Foul and Shot Missed) x E(Points | Foul and Shot Missed)

Clearly:

E(Points | No Foul and Shot Made) = 2

E(Points | No Foul and Shot Missed) = 0

The other two expectations rely on the foul shots. Therefore:

E(Points | Foul and Shot Made) = 2 + E(One FTA)

E(Points | Foul and Shot Missed) = E(Two FTA)

**Definition of E(Points | 3FGA)**

This is very similar to E(Points | 2FGA), so I’ll simply state the differences:

E(Points | No Foul and Shot Made) = 3

E(Points | Foul and Shot Made) = 3 + E(One FTA)

E(Points | Foul and Shot Missed) = E(Three FTA)

**Definition of E(Points | Non-Shooting Foul)**

This expectation can be defined as:

E(Points | Non-Shooting Foul) =Pr(Bonus Situation) x E(Two FTA)

+ Pr(Non-Bonus Situation) x 0

Where Pr(Bonus Situation) is always 1 or 0 depending, of course, on the bonus situation. Pr(Non-Bonus Situation) is the complement, 1 – Pr(Bonus Situation). Also, unless the shot clock resets for some reason, the foul in a non-bonus situation **does not** lead to a new play.

**The Player Level**

With the underlying expectations for E(Points) defined at the 5-player unit level, let me go back and define E(Points) in terms of players:

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)

So I’ve decided to end this part of the series hanging on that last definition of E(Points) in terms of players. In the last part of this series I will expand on this definition of E(Points) and build the picture of what the theoretical model looks like at the player level.

**Summary**

I’ll admit that this part of the series is pretty boring (it’s mostly a bunch of definitions with ugly notation). I wanted to define the basic structure of the points scored at the 5-player unit level so that I didn’t lose anything when creating the theoretical model for each player. This also helps remind us that we have to worry about 5 players together, and the player piece of this theoretical model will be cognizant of that.

Oh, and in case you’re wondering, I haven’t lost sight of defense, imporance of shot location, etc. These factors will clearly affect the underlying player probabilities and will be defined in the player part of this model.

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