Repetitions of a simulation
Repeating a simulation and observing the variability in the results can give insight into the randomness of the system's behaviour.
Sport leagues
In many sports, teams are grouped into leagues, with each team playing every other team one or more times throughout during the year. Teams gain points for wins and draws and their total points are usually tabulated each week in newspapers. We will use a simulation to investigate how much the points in a league table reflect the randomness of individual matches and how much they depend on the abilities of the different teams.
In this page, we will consider a league with 10 teams in which each team plays each other twice and:
Points from a match = | 3 if team wins 1 if team draws 0 if team loses |
Model
The first stage in any simulation is to produce a model for the process. In the league table example, such a model defines the probabilities of winning, drawing and losing for each match during the season. A good model would express these probabilities in terms of different abilities for the various teams (perhaps based on their results from the previous year), a home-team advantage and changes during the season. However a much simpler model can still provide useful insight.
We initially assume that the two teams in each match are equally likely to win. More precisely, in any match between teams i and j, we assume that
P ( team i wins ) = P ( team j wins ) = 0.4
P ( draw ) = 0.2
Click Run League to perform a simulation in which each pair of teams plays two matches (one at each team's home ground).
Is the best team likely to be top of the league?
We will now concentrate on a single team, Team A, and examine how its skill level affects its league placing at the end of the season. This is shown by its rank at the end of the season on the dot plot at the right of the above diagram. (A rank of 1 means that the team was top or top equal in the league.)
With team A still equally likely to win and lose each match, click Accumulate and run the simulation several more times. Observe that Team A has (almost) the same chance of being in any position in the league at the end of the season.
The slider under the diagram allows us to adjust the probability of Team A winning its matches. (The other teams remain evenly matched.) Give Team A a probability of 0.55 of winning its matches — more than double its probability of losing — then repeat the simulation 100 times.
Observe that Team A often wins the league, but not always.
Even with over double the the chance of winning than losing each match, Team A only ends the season on top of the league in about half of the simulations.
Indeed, you will probably have observed that Team A's final placing was in the bottom half of the league in several simulated seasons!
A simple probability model can often give valuable and perhaps surprising insight into a system through a simulation.