3.1: Events, Sample Spaces, and Probability
Learning Objectives
 Know basic statistical terminology.
 List simple events and sample space.
 Know the basic rules of probability.
An event is something that occurs or happens. Flipping a coin is an event. Walking in the park and passing by a bench is an event. Anything that could possibly happen is an event.
Every event has one or more possible outcomes. Tossing a coin is an event but getting a tail is the outcome of the event. Walking in the park is an event and finding your friend sitting on a bench is an outcome of the event.
In statistics, the process of taking a measurement or making an observation is called an experiment. For example, tossing a coin and recording the up face in a table of data is an experiment because a measurement is taken.
Experiment
The process of taking a measurement or making an observation.
Keep in mind that the definition of an experiment in statistics is broader than the one used in science. A scientific experiment involves scientific instrumentations such as thermometers, microscopes, telescopes and tubes. A statistical experiment may involve all these items but it mainly involves recording data and measurements. For example, we may conduct an experiment to learn which brand of coffee a customer may prefer among three brands, recording a voter’s opinion on a particular political issue, or measuring the amount of carbon monoxide present in a certain environment. Any kind of observation, measuring, and recording that you may conduct can be considered a statistical experiment.
Suppose a coin is tossed once. There are two possible outcomes, either a head
Simple Event
The simplest outcome of an experiment.
Example:
Suppose that two pennies are tossed simultaneously. We could have both pennies land heads up (which we write as
Figure: The possible outcomes of flipping two coins.
What we have accomplished so far is a listing of all the possible simple events of an experiment. This collection is called the sample space of an experiment.
Sample Space
The set of all possible outcomes of an experiment, or the collection of all the possible simple events of an experiment. We will denote a sample space by
Example:
Experiment: We want to investigate the sample space of throwing a die and the sample space of tossing a coin.
Solution:
As we know, there are
Similarly, the sample space of tossing a coin is either head
Example:
Experiment: Suppose a box contains three balls, one red, one blue and one white. One ball is selected, its color is observed, and then the ball is placed back in the box. The balls are scrambled and again a ball is selected and its color is observed. What is the sample space of the experiment?
Solution:
It is probably best if we draw a diagram to illustrate all the possible drawings.
As you can see from the diagram, it is possible that you will get the red ball
Each pair in the set above gives the first and second drawings, respectively. That is,
We can also represent all the possible drawings by a table or a matrix:
Figure: Table representing the possible outcomes diagrammed in the previous figure
Where the first column represents the first drawing and the first row represents the second drawing.
Example:
Experiment: Consider the same experiment as in the example before last but this time we will draw one ball and record its color but we will not place it back into the box. We will then select another ball from the box and record its color. What is the sample space in this case?
Solution:
The diagram below illustrates this case:
You can clearly notice that when we draw, say, a red ball, there will remain blue and white balls. So on the second selection, we will either get a blue or a while ball. The sample space in this case is:
Now let us return to the concept of probability and relate it to the concepts that we have just studied. You may be familiar with the meaning of probability and may have used the term as a synonym with informal words like “chance” and “odds.” For the time being, we will begin our treatment of probability using these informal concepts and then later, we will solidify these meanings into formal mathematical definitions.
As you probably know from your previous math courses, if you toss a fair coin, the chance of getting a tail
The probability,
Property 1
The probability of an event
In addition, the probabilities of possible outcomes of an event must all add up to
Property 2
The sum of the probabilities of all possible outcomes must add up to
Notice that tossing a coin or throwing a dice results in outcomes that are all equally probable, that is, each outcome has the same probability as the other outcome in the same sample space. Getting a head or a tail from tossing a coin produces equal probability for each outcome,
Probability is usually denoted by
The following examples show you how to use this formula.
Example:
When tossing two coins, what is the probability of getting headhead
Solution:
Since there are
Notice that each of these
Example:
What is the probability of throwing a dice and getting either
Solution:
The sample space for a fair dice has a total of
So, there is a probability of
Example:
Consider an experiment of tossing two coins. Assume the coins are not balanced. The design of the coins is to produce the following probabilities shown in the table:
Sample Space  Probability 









Figure: Probability table for flipping two weighted coins.
What is the probability of observing exactly one head and the probability of observing at least one head?
Solution:
Notice that the simple events
Similarly, the probability of observing at least one head is:
Lesson Summary
 An event is something that occurs or happens with one or more outcomes.
 An experiment is the process of taking a measurement or making an observation.
 A simple event is the simplest outcome of an experiment.
 The sample space is the set of all possible outcomes of an experiment, typically denoted by
S .
Review Questions
 Consider an experiment composed of throwing a die followed by throwing a coin.
 List the simple events and assign a probability for each simple event.
 What are the probabilities of observing the following events?
A:{2on the die, H on the coin}B:{Even number on the die, T on the coin}C:{Even number on the die}D:{T on the coin}  The Venn diagram below shows an experiment with six simple events. Events
A andB are also shown. The probabilities of the simple events are:P(1)=P(2)=P(4)=2/9P(3)=P(5)=P(6)=1/9  Find
P(A)  Find
P(B)
 Find
 A box contains two blue marbles and three red ones. Two marbles are drawn randomly without replacement.
 Refer to the blue marbles as
B1 andB2 and the red ones asR1,R2, andR3 . List the outcomes in the sample space.  Determine the probability of observing each of the following events:
A:{2blue marbles are drawn}B:{1red and 1 blue are drawn}C:{2red marbles are drawn}  Refer to the blue marbles as
Review Answers

{1T,1H,2T,2H,3T,3H,4T,4H,5T,5H,6T,6H}  A:
1/12 B:1/4 C:1/2 D:1/2

4/9 
1/3

{B1B2,B1R1,B1R2,B1R3,B2B1,B2R1,B2R2,B2R3,R1B1,R1B2,R1R2,R1R3,R2B1,R2B2,R2R1,R2R3,R3B1,R3B2,R3R1,R3R2}  A:
1/10 B:3/5 C:3/10
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