4.2 Composite trials

Composite trials and component events

Often a trial is a composite one. That is, the fundamental trial is completed by performing several steps. In some cases, the steps are carried outsequentially in time. In other situations, the order of performance plays no significant role. Some of the examples in the unit on Conditional Probability involve such multistep trials. We examine more systematically how to model composite trials in terms of events determinedby the components of the trials. In the subsequent section, we illustrate this approach in the important special case of Bernoulli trials, in which each outcomeresults in a success or failure to achieve a specified condition.

We call the individual steps in the composite trial component trials . For example, in the experiment of flipping a coin ten times, we refer the i th toss as the i th component trial. In many cases, the component trials will be performed sequentially in time. But we may have an experiment in which ten coins areflipped simultaneously. For purposes of analysis, we impose an ordering— usually by assigning indices. The question is how to model these repetitions.Should they be considered as ten trials of a single simple experiment? It turns out that this is not a useful formulation. We need to consider thecomposite trial as a single outcome— i.e., represented by a single point in the basic space Ω .

Some authors give considerable attention the the nature of the basic space, describing it as a Cartesian product space, with each coordinate correspondingto one of the component outcomes. We find that unnecessary, and often confusing, in setting up the basic model. We simply suppose the basic space has enoughelements to consider each possible outcome. For the experiment of flipping a coin ten times, there must be at least $2^{10}=1024$ elements, one for each possible sequence of heads and tails.

Of more importance is describing the various events associated with the experiment. We begin by identifying the appropriate component events . A component event is determined by propositions about the outcomes of the corresponding component trial.