Modelling other situations with probability
We often model a data set as a random sample from some population and probability was introduced as a way of describing the randomness of such data. Probability is also used to model a variety of other situations involving randomness.
The randomness of games of chance involving cards, dice or roulette wheels can often be expressed simply in probability terms. Sporting competitions can also often be modelled using fairly simple probability models. Such models usually simplify reality, but they may capture the essentials of behaviour.
Women's tennis match
A simple model for a tennis match between two players, A and B, will now be described. In this model, we will assume that:
Probability of A winning her serve = π1
Probability of B winning her serve = π2
We will also assume that the results of successive points are independent — winning one point does not make A more likely to win the next point too — and the standard rules of tennis are also part of the model. This is a simplification of a real tennis match but it could still be useful to determine how the probability of winning a whole match depends on π1 and π2.
Simulation
How can a probability model be used to find information about such a system? One way is to use the probabilities to generate an instance of the system. If the model was to specify that something happens with a probability of 0.5, then we could toss a coin to generate an instance (with say head meaning that the event happens). Events with other probabilities can be generated in a similar way on a computer.
Generating all 'events' in the model from the probabilities in this way is called a simulation of the model. (The mechanism will be clarified in the example below.)
Women's tennis match
The diagram below shows how randomly generated points can simulate a complete women's tennis match with 3 sets. Initially, both players are equally matched and have probability 0.75 of winning their serves.
Click Simulate Next Point to play a single point of the match — the computer randomly generates a result, based on the probability of the server winning the point. Click this button repeatedly to generate points until the match is completed. (Or hold it down to speed up the simulation.)
Click Start New Match to perform another simulation. Note that the precise sequence of points is unlikely to be repeated exactly in different simulations, even when the probabilities are the same in successive matches. (Before performing further simulations, you may use the sliders to adjust the probabilities winning individual points for the two competitors.)
In practice, we would rarely be interested in displaying as much detail in a simulation (except perhaps when checking that we have programmed the rules of the match properly!). We are usually interested in only one or two outcomes (such as the identity of the winner or the total number of sets played in the match) and only these summaries need be displayed from each run of the simulation.