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A tree diagram is a tool that is useful not only in describing the sample points but also in listing them in a systematic way. The following example illustrates this technique.

Figure 3.4.1 Tree diagram for the experiment in Example 3.3.1

Example 3.4.1 (Constructing a tree diagram) Consider a random experiment consisting of three trials. The first trial is testing a chip taken from the production line, the second is randomly selecting a part from the box containing parts produced by six different manufacturers, and the third is, again, testing a chip off the production line. The interest in this experiment is in describing and listing the sample points in the sample space of the experiment.

Solution: A tree‐diagram technique describes and lists the sample points in the sample space of the experiment consisting of three trials. The first trial in this experiment has two possible outcomes: the chip could be defective (D) or nondefective (N); the second trial has six possible outcomes because the part could come from manufacturer 1, 2, 3, 4, 5, or 6; and the third, again, has two possible outcomes (D, N). The problem of constructing a tree diagram for a multitrial experiment is sequential in nature: that is, corresponding to each trial, there is a step of drawing branches of the tree. The tree diagram associated with this experiment is shown in Figure 3.4.1.

The number of sample points in a sample space is equal to the number of branches corresponding to the last trial. For instance, in the present example, the number of sample points in the sample space is equal to the number of branches corresponding to the third trial, which is 24 (). To list all the sample points, start counting from o along the paths of all possible connecting branches until the end of the final set of branches, listing the sample points in the same order as the various branches are covered. The sample space S in this example is

The tree diagram technique for describing the number of sample points is extendable to an experiment with a large number of trials, where each trial has several possible outcomes. For example, if an experiment has n trials and the ith trial has possible outcomes (), then there will be branches at the starting point o, branches at the end of each of the branches, branches at the end of the each of branches, and so on. The total number of branches at the end would be , which represents all the sample points in the sample space S of the experiment. This rule of describing the total number of sample points is known as the Multiplication Rule.