Introduction to Probability

Sample Spaces for Multi-Stage Probability Experiments

We can perform several different probability experiments, one after another, and then consider the probability of the series of outcomes that result. For example, we could toss a coin and then toss a standard die. This is a 2-stage experiment because it consists of two separate experiments performed one after the other. Each outcome would also have two parts. Such outcomes are written as ordered pairs using parentheses to indicate that the outcomes must follow in the order they are written. For a two-stage experiment, the sample space is the set of all possible ordered pairs, that is, we form the Cartesian Product of the two stages. For experiments with more than two stages, we often generate the sample space by making a tree diagram. Examples of both of these methods follow.

Generate the Sample Space Using a Table to Form the Cartesian Product

When there are only two stages in an experiment, a common way to list the possible outcomes is to form a Cartesian Product. This is similar to when you formed sets of ordered pairs in algebra when you graphed on a Cartesian coordinate plane. In algebra, we used a horizontal and vertical axis where the horizontal axis represented the first values, x, and the vertical axis represented the second values, y, in ordered pairs, (x, y). Here, we create a table to aid us in forming the Cartesian Product where the rows represent the possible outcomes for the first experiment (down the side) and the columns represent the possible outcomes for the second experiment (across the top). Like this:

To form the Cartesian Product, we list in each interior cell of the table, the ordered pair that results from the outcome listed at the side followed by the outcomes listed at the top.

The sample space is the set of outcomes listed in the shaded cells,

S = {(H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T, 2), (T,3), (T,4), (T,5), (T,6)}.

Note that we have twelve possible outcomes in this sample space for this two-stage experiment, which follows from the Fundamental Principle of Counting 2 · 6 = 12..

Generate the Sample Space Using a Tree Diagram to Form the Cartesian Product

When a probability experiment involves more than two actions, we often use a tree diagram to find the sample space. For example, for the experiment "toss a coin three times and record the results from each toss", we could draw the following tree diagram.

The sample space for the problem is S = {(H,H,H), (H,H,T), (H,T,H), (H,T,T), (T,H,H), (T,H,T), (T,T,H), (T,T,T)}. Each outcome is an ordered triple and we usually write the set in the abbreviated form S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. Also, note that HHT, HTH, and THH are three distinct outcomes even though they both consist of two heads and one tail. Also, note that there are   2 · 2 · 2 = 8 outcomes, which follows from the Fundamental Principle of Counting.

Self-Check Problem

A boy and girl are to be randomly chosen to represent their class. The boy is chosen from Tom, Dick, and Harry. The girl is chosen from Jane, Kathy, Pam, and Sally. Find the sample space for this problem.

Solution

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