<< 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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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.
If March sales will be up from February by 10%, 15%, and 20% at Place I, Place II, and Place III, respectively, find the expected number of hot dogs, and corn dogs to be sold
Logan 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 Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play

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?