Understanding the Possible Outcomes of 3 Football Games

Introduction

In each football game, there are typically three possible outcomes: a win, a loss, or a draw. This article explores how to calculate the total number of possible combinations of outcomes when considering multiple games. The focus will be on non-US football (soccer) and US football, where ties are generally allowed, except for playoffs or championships.

Calculating Combinations

When calculating the total number of combinations of outcomes for a series of football games, we start by considering the basic principle that each game can have 3 possible results. If we have 3 games, then the total number of combinations can be calculated using the formula:

Total combinations 3number of games 33 27

This means that there are 27 possible combinations of outcomes when considering 3 football games.

Alternative Notation

Another way to represent these outcomes is by assigning numerical values to the results:

L (loss) 0 D (draw) 1 W (win) 2

By using a ternary number system (base-3), we can represent all the possible outcomes. For example:

LLL, LLD, LLW, LDL, LDD, LDW, LWL, LWD, LWW DLN, DLD, DLS, DWD, DWL, DWW WLL, WLD, WLS, WDL, WDD, WWL, WWD, WWW

Each of these combinations represents a unique outcome for the three games.

Independence of Match Results

The independence of the match results is a crucial concept here. The result of one match does not affect the results of the other matches. For clarity, let's consider an example with three teams: A, B, and C. The matches would be:

AB (A vs B) BC (B vs C) CA (C vs A)

Each match has three possible outcomes:

A win for the home team (W) A draw (D) A loss for the home team (L)

Since each match is independent, the total number of possible combinations of outcomes is:

33 27

This can be visualized with the following combinations:

HHH, HHD, HHW, HDD, HDW, HDW, HWH, HWL, HWW, DDD, DDH, DDL, DWD, DWL, DWW, WDD, WDH, WDL, WWD, WWW, WWH, WHL, WWL, WWW

As can be seen, there are indeed 27 unique combinations of outcomes.

Conclusion

The total number of possible combinations of outcomes for any number of football games, where each game can have three outcomes, can be calculated using the formula:

Total combinations 3number of games

For 3 games, this results in 27 combinations. Whether it's a simple scenario with three teams playing each other or a more complex setup with multiple matches, the principle remains the same.