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

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.