Basic Concepts of Probability

Read this section about basic concepts of probability, including spaces, and events. This section discusses set operations using Venn diagrams, including complements, intersections, and unions. Finally, it introduces conditional probability and talks about independent events.

LEARNING OBJECTIVES

To learn the concept of the sample space associated with a random experiment.
To learn the concept of an event associated with a random experiment.
To learn the concept of the probability of an event.

Sample Spaces and Events

Rolling an ordinary six-sided die is a familiar example of a random experiment, an action for which all possible outcomes can be listed, but for which the actual outcome on any given trial of the experiment cannot be predicted with certainty. In such a situation we wish to assign to each outcome, such as rolling a two, a number, called the probability of the outcome, that indicates how likely it is that the outcome will occur. Similarly, we would like to assign a probability to any event, or collection of outcomes, such as rolling an even number, which indicates how likely it is that the event will occur if the experiment is performed. This section provides a framework for discussing probability problems, using the terms just mentioned.

Definition

A random experiment is a mechanism that produces a definite outcome that cannot be predicted with certainty. The sample space associated with a random experiment is the set of all possible outcomes. An event is a subset of the sample space.

Definition

An event $E$ is said to occur on a particular trial of the experiment if the outcome observed is an element of the set $E$ .

EXAMPLE 1

Construct a sample space for the experiment that consists of tossing a single coin.

Solution:

The outcomes could be labeled $h$ for heads and $t$ for tails. Then the sample space is the set $S=\{h,t\}$ .

EXAMPLE 2

Construct a sample space for the experiment that consists of rolling a single die. Find the events that correspond to the phrases "an even number is rolled" and "a number greater than two is rolled".

Solution:

The outcomes could be labeled according to the number of dots on the top face of the die. Then the sample space is the set $S=\{1,2,3,4,5,6\}$ .

The outcomes that are even are 2, 4, and 6, so the event that corresponds to the phrase "an even number is rolled" is the set $\{2,4,6\}$ , which it is natural to denote by the letter $E$ . We write $E=\{2,4,6\}$ .

Similarly the event that corresponds to the phrase "a number greater than two is rolled" is the set $T=\{3,4,5,6\}$ , which we have denoted $T$ .

A graphical representation of a sample space and events is a Venn diagram, as shown in Figure 3.1 "Venn Diagrams for Two Sample Spaces" for Note 3.6 "Example 1" and Note 3.7 "Example 2". In general the sample space $S$ is represented by a rectangle, outcomes by points within the rectangle, and events by ovals that enclose the outcomes that compose them.

Figure 3.1 Venn Diagrams for Two Sample Spaces

EXAMPLE 3

A random experiment consists of tossing two coins.

a. Construct a sample space for the situation that the coins are indistinguishable, such as two brand new pennies.

b. Construct a sample space for the situation that the coins are distinguishable, such as one a penny and the other a nickel.

Solution:

a. After the coins are tossed one sees either two heads, which could be labeled $2h$ , two tails, which could be labeled $2t$ , or coins that differ, which could be labeled d. Thus a sample space is $S=\{2h,2t,d\}$ .

b. Since we can tell the coins apart, there are now two ways for the coins to differ: the penny heads and the nickel tails, or the penny tails and the nickel heads. We can label each outcome as a pair of letters, the first of which indicates how the penny landed and the second of which indicates how the nickel landed. A sample space is then $S'=\{hh,ht,th,tt\}$ .

A device that can be helpful in identifying all possible outcomes of a random experiment, particularly one that can be viewed as proceeding in stages, is what is called a tree diagram. It is described in the following example.

EXAMPLE 4

Construct a sample space that describes all three-child families according to the genders of the children with respect to birth order.

Solution:

Two of the outcomes are "two boys then a girl," which we might denote $bbg$ , and "a girl then two boys," which we would denote $gbb$ . Clearly there are many outcomes, and when we try to list all of them it could be difficult to be sure that we have found them all unless we proceed systematically. The tree diagram shown in Figure 3.2 "Tree Diagram For Three-Child Families", gives a systematic approach.

Figure 3.2

Tree Diagram For Three-Child Families

The diagram was constructed as follows. There are two possibilities for the first child, boy or girl, so we draw two line segments coming out of a starting point, one ending in a $b$ for "boy" and the other ending in a $g$ for "girl". For each of these two possibilities for the first child there are two possibilities for the second child, "boy" or "girl," so from each of the $b$ and $g$ we draw two line segments, one segment ending in a $b$ and one in a $g$ . For each of the four ending points now in the diagram there are two possibilities for the third child, so we repeat the process once more.

The line segments are called branches of the tree. The right ending point of each branch is called a node. The nodes on the extreme right are the final nodes; to each one there corresponds an outcome, as shown in the figure.

From the tree it is easy to read off the eight outcomes of the experiment, so the sample space is, reading from the top to the bottom of the final nodes in the tree,

$S=\{bbb,bbg,bgb,bgg,gbb,gbg,ggb,ggg\}$