Random experiment, outcomes, and sample space

Chapter 2. Probability

Introduction

The idea of probability is a familiar one to everyone. Statements such as the following are heard frequently: “You have better to take an umbrella because it is likely to rain”, “It is not likely to snow today”, “He will probably read at least three books during the next two weeks”, “ I am almost certain that we will go home for the holidays”.

These examples illustrate that most of us use the concept of probability in our everyday speech. They also illustrate that there is a great deal of imprecision involved in such statement. For instance, a family has been arguing about whether they will go to the wife’s home for holidays, and have not reached a decision. Even though the wife may say, “It is almost certain that we will go home for the holidays”, the husband might not even think that such an occurrence is likely. This sort of imprecision is intolerable in mathematics. These difficulties can be avoided if we restrict our discussion of probability to events which are outcomes of experiments that can be repeated, and if we deal with idealized situations.

One can think of probability as the language in which we discuss uncertainty. Before we can communicate with one another in this language, we need to acquire a common vocabulary. Moreover, as in any other language, rules of grammar are needed so that a clear statement can be made with our vocabulary.

Random experiment, outcomes, and sample space

Definition:

A random experiment is a process that, when performed, results in one and only one of at least two observations.

Definition:

The possible outcomes of a random experiment are called the basic outcomes.

Definition:

The set of all basic outcomes is called the sample space.

A sample space will be denoted by S.

Table 2.1 illustrates list of a few examples of random experiments, their outcomes, and their sample spaces.

Table 2.1

Experiment	Outcomes	Sample space
Toss a coin once Roll a die once Toss a coin twice Take a test Select a student	Head, Tail 1,2,3,4,5,6 HH,HT,TH,TT Pass, Fail Male, Female	S ={Head, Tail} S ={1,2,3,4,5,6} S ={ HH,HT,TH,TT} S ={ Pass, Fail } S ={Male, Female}

The sample space for an experiment can also be described by drawing a

Venn diagram. A Venn diagram is a picture (a closed geometric shape such as a rectangle, a square, or circle) that depicts all the possible outcomes for an experiment.

Example:

Draw the Venn diagram for the experiment of tossing a coin.

Solution:

This experiment has two possible outcomes: head and tail. Sample space is given by S = { H, T };

where H =head, T=Tail.

To draw Venn diagram for this example, we draw a rectangle and mark two points inside this rectangle that represent two outcomes, head and tail.

The rectangle is labelled by S because it represents the sample space. (Fig. 2.1)

Example:

Suppose we randomly select two employees from a company and observe whether the employee selected each time is a male or female. Write all the outcomes for this experiment and draw the Venn diagram.

Solution:

Let us denote selection of male by M and that of female by F. There are four final possible outcomes: MM, MF, FM, and FF.

We can write sample space as

S= { MM, MF, FM, FF }

The Venn diagram is given in Fig.2.2

Fig.2.2.

Definition:

An event is a collection of one or more of the outcomes from the sample space.

Usually, an event is denoted by , and so forth. We can denote it by any other letters too, that is, by A, B, C and so forth.

Definition:

An event that includes one and only one of the outcomes for an experiment is called a simple event.

Definition:

An event is called a compound event if it contains more than one outcome for an experiment.

Definition:

Let A and B be two events defined in sample space S. The intersection of A and B represents the collection of all outcomes that are common to both A and B and is denoted by either or AB.

Hence, or AB occurs if and only if both A and B occur.

More generally, given N events , their intersection, , is the set of all basic outcomes that belong to every , where .

Example:

Let A= the event that a family owns washing machine

B= the event that family owns a VCR. Then intersection of these events includes all the families who own both washing machine and VCR.

Definition:

Let A and B be two events defined in sample space S. Their union, denoted by is the set of all basic outcomes in S that belong to at least one of these two events. The union occurs if and only if either A or B (or both) occurs.

More generally, given N events , their union, , is the set of all basic outcomes that belong to at least one of these N events.

Definition:

Two events A and B are called mutually exclusive events if they have no common basic outcomes and their intersection is said to be empty set.

Definition:

Let be N events in the sample space S.

If = S, then these N events are said to be collectively exhaustive events.

Definition:

Let A be an event in the sample space. The complement of event A, denoted by and read as “A bar” or “A complement” is the event that includes all the outcomes for an experiment that are not in A.

Example:

A statistical experiment has eight equally likely outcomes that are denoted by 1, 2, 3, 4, 5, 6, 7, and 8. Let A= {2, 5, 7} and B= {2, 4, 8}

a) Find

b) Find

c) Are events A and B mutually exclusive events?

d) Are events A and B collectively exhaustive events?

e) Find and

Solution:

a) These two events have only one common element ={2}

b) ={2, 4, 5, 7, 8}

c) Events A and B are not mutually exclusive events, because they have common element, and their intersection is not empty set.

d) Events A and B are not collectively exhaustive events, because their union does not equal to sample space.

e) = {1, 3, 4, 6, 8}; = {1, 3, 5, 6, 7}

Example:

At a busy international airport arriving planes land a first come, first served basis. Let A, B, and C be the events that there are at least five, at most three, and exactly two planes waiting to land, respectively. Then

1. is the event that at most four planes are waiting to land.

2. is the event that at least four planes are waiting to land.

3. A is a subset of ; that is if A occurs, then occurs.

Therefore, .

4. C is a subset of B; that is if C occurs, then B occurs.

Therefore, .

5. A and B are mutually exclusive; that is, .

A and C also mutually exclusive since .

6. is the event that number of planes waiting to land is zero, one or three.

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