Probability for Beginning Statistics

By — McGraw-Hill Professional
Updated on Aug 12, 2011

Practice problems for these concepts can be found at:

Experiment, Outcomes, and Sample Space

An experiment is any operation or procedure whose outcomes cannot be predicted with certainty. The set of all possible outcomes for an experiment is called the sample space for the experiment.

EXAMPLE 4.1   Games of chance are examples of experiments. The single toss of a coin is an experiment whose outcomes cannot be predicted with certainty. The sample space consists of two outcomes, heads or tails. The letter S is used to represent the sample space and may be represented as S = {H, T}. The single toss of a die is an experiment resulting in one of six outcomes. S may be represented as {1, 2, 3, 4, 5, 6}. When a card is selected from a standard deck, 52 outcomes are possible. When a roulette wheel is spun, the outcome cannot be predicted with certainty.

EXAMPLE 4.2   When a quality control technician selects an item for inspection from a production line, it may be classified as defective or nondefective. The sample space may be represented by S = {D, N}. When the blood type of a patient is determined, the sample space may be represented as S = {A, AB, B, O}. When the Myers-Briggs personality type indicator is administered to an individual, the sample space consists of 16 possible outcomes.

The experiments discussed in Examples 4.1 and 4.2 are rather simple experiments and the descriptions of the sample spaces are straightforward. More complicated experiments are discussed in the following section and techniques such as tree diagrams are utilized to describe the sample space for these experiments.

Tree Diagrams and The Counting Rule

In a tree diagram, each outcome of an experiment is represented as a branch of a geometric figure called a tree.

EXAMPLE 4.3   Figure 4.1 shows the sample spaces for three different experiments. For the experiment of tossing a coin twice, there are 4 outcomes. For the experiment of tossing a coin three times there are 8 outcomes. For the experiment of tossing a coin four times there are 16 outcomes. The outcomes are called branches of the tree and the device for showing the branches is called a tree diagram. Note that N tosses results in 2N outcomes or branches.

The counting rule for a two-step experiment states that if the first step can result in any one of n1 outcomes, and the second step in any one of n2 outcomes, then the experiment can result in (n1)(n2) outcomes. If a third step is added with n3 outcomes, then the experiment can result in (n1)(n2)(n3) outcomes. The counting rule applies to an experiment consisting of any number of steps. If the counting rule is applied to Example 4.3, we see that for two tosses of a coin, n1 = 2, n2 = 2, and the number of outcomes for the experiment is 2 × 2 = 4. For three tosses, there are 2 × 2 × 2 = 8 outcomes and so forth. The counting rule may be used to figure the number of outcomes of an experiment and then a tree diagram may be used to actually represent the outcomes.

EXAMPLE 4.4   For the experiment of rolling a pair of dice, the first die may be any of six numbers and the second die may be any one of six numbers. According to the counting rule, there are 6 × 6 = 36 outcomes. The outcomes may be represented by a tree having 36 branches. The sample space may also be represented by a two-dimensional plot as shown in Fig. 4-2. The following plot was created using EXCEL's chart wizard.

EXAMPLE 4.5   An experiment consists of observing the blood types for five randomly selected individuals. Each of the five will have one of four blood types A, B, AB, or O. Using the counting rule, we see that the experiment has 4 × 4 × 4 × 4 × 4 = 1,024 possible outcomes. In this case, constructing a tree diagram would be difficult.

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