# 2

# Sequential Experiments

Many applications of probability refer to sequential experiments in which the procedure consists of many actions performed in sequence, with an observation taken after each action. Each action in the procedure together with the outcome associated with it can be viewed as a separate experiment with its own probability model. In analyzing sequential experiments we refer to the separate experiments in the sequence as *subexperiments*.

## 2.1 Tree Diagrams

Tree diagrams display the outcomes of the subexperiments in a sequential experiment. The labels of the branches are probabilities and conditional probabilities. The probability of an outcome of the entire experiment is the product of the probabilities of branches going from the root of the tree to a leaf.

Many experiments consist of a sequence of *subexperiments*. The procedure followed for each subexperiment may depend on the results of the previous subexperiments. We often find it useful to use a type of graph referred to as a *tree diagram* to represent the sequence of subexperiments. To do so, we assemble the outcomes of each subexperiment into sets in a partition. Starting at the root of the tree,^{1} we represent each event in the partition of the first subexperiment as a branch and we label the branch with the probability of the event. Each branch leads to a node. The events in the partition of the second subexperiment appear as branches growing from every node at the end of the first subexperiment. The labels ...

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