<< Chapter < Page Chapter >> Page >
This chapter covers principles of probability. After completing this chapter students should be able to: write sample spaces; determine whether two events are mutually exclusive; use the addition rule; calculate probabilities using tree diagrams and combinations; solve problems involving conditional probability; determine whether two events are independent.

Chapter overview

In this chapter, you will learn to:

  1. Write sample spaces.
  2. Determine whether two events are mutually exclusive.
  3. Use the Addition Rule.
  4. Calculate probabilities using both tree diagrams and combinations.
  5. Do problems involving conditional probability.
  6. Determine whether two events are independent.

Sample spaces and probability

If two coins are tossed, what is the probability that both coins will fall heads? The problem seems simple enough, but it is not uncommon to hear the incorrect answer 1 / 3 size 12{1/3} {} . A student may incorrectly reason that if two coins are tossed there are three possibilities, one head, two heads, or no heads. Therefore, the probability of two heads is one out of three. The answer is wrong because if we toss two coins there are four possibilities and not three. For clarity, assume that one coin is a penny and the other a nickel. Then we have the following four possibilities.


The possibility HT, for example, indicates a head on the penny and a tail on the nickel, while TH represents a tail on the penny and a head on the nickel.

It is for this reason, we emphasize the need for understanding sample spaces.

An act of flipping coins, rolling dice, drawing cards, or surveying people are referred to as an experiment .

Sample Spaces
A sample space of an experiment is the set of all possible outcomes.

If a die is rolled, write a sample space.

A die has six faces each having an equally likely chance of appearing. Therefore, the set of all possible outcomes S size 12{S} {} is

1,2,3,4,5,6 size 12{ left lbrace 1,2,3,4,5,6 right rbrace } {} .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

A family has three children. Write a sample space.

The sample space consists of eight possibilities.

BBB , BBG , BGB , BGG , GBB , GBG , GGB , GGG size 12{ left lbrace ital "BBB", ital "BBG", ital "BGB", ital "BGG", ital "GBB", ital "GBG", ital "GGB", ital "GGG" right rbrace } {}

The possibility BGB size 12{ ital "BGB"} {} , for example, indicates that the first born is a boy, the second born a girl, and the third a boy.

We illustrate these possibilities with a tree diagram.

The tree diagram illustrates the different possibilities for the gender of three children in a family.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Two dice are rolled. Write the sample space.

We assume one of the dice is red, and the other green. We have the following 36 possibilities.

Red 1 2 3 4 5 6
1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

The entry (2, 5), for example, indicates that the red die shows a two, and the green a 5.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Now that we understand the concept of a sample space, we will define probability.


For a sample space S size 12{S} {} , and an outcome A size 12{A} {} of S size 12{S} {} , the following two properties are satisfied.
  1. If A size 12{A} {} is an outcome of a sample space, then the probability of A size 12{A} {} , denoted by P A size 12{P left (A right )} {} , is between 0 and 1, inclusive.
    0 P A 1 size 12{0<= P left (A right )<= 1} {}
  2. The sum of the probabilities of all the outcomes in S size 12{S} {} equals 1.

If two dice, one red and one green, are rolled, find the probability that the red die shows a 3 and the green shows a six.

Since two dice are rolled, there are 36 possibilities. The probability of each outcome, listed in [link] , is equally likely.

Since (3, 6) is one such outcome, the probability of obtaining (3, 6) is 1 / 36 size 12{1/"36"} {} .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
8. It is known that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events, wearing a seat belt and quitting smoking, independent?
William Reply
Mr. Shamir employs two part-time typists, Inna and Jim for his typing needs. Inna charges $10 an hour and can type 6 pages an hour, while Jim charges $12 an hour and can type 8 pages per hour. Each typist must be employed at least 8 hours per week to keep them on the payroll. If Mr. Shamir has at least 208 pages to be typed, how many hours per week should he employ each student to minimize his typing costs, and what will be the total cost?
Chine Reply
At De Anza College, 20% of the students take Finite Mathematics, 30% take Statistics and 10% take both. What percentage of the students take Finite Mathematics or Statistics?
Chalton Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Applied finite mathematics' conversation and receive update notifications?