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This module provides a review of the probability formulas, including the definitions of independent, complementary, and mutually exclusive events as well as the addition and multiplication rules.


If A and A' are complements then P(A) + P(A' ) = 1


Addition rule

P(A OR B) = P(A) + P(B) - P(A AND B)


Mutually exclusive

If A and B are mutually exclusive then P(A AND B) = 0 ; so P(A OR B) = P(A) + P(B) .


Multiplication rule

  • P(A AND B) = P(B) P(A|B)
  • P(A AND B) = P(A) P(B|A)


If A and B are independent then:

  • P(A|B) = P(A)
  • P(B|A) = P(B)
  • P(A AND B) = P(A) P(B)


Conditional probability

The likelihood that an event will occur given that another event has already occurred.

Contingency table

The method of displaying a frequency distribution as a table with rows and columns to show how two variables may be dependent (contingent) upon each other. The table provides an easy way to calculate conditional probabilities

Equally likely

Each outcome of an experiment has the same probability.


A subset in the set of all outcomes of an experiment. The set of all outcomes of an experiment is called a sample space and denoted usually by S. An event is any arbitrary subset in S. It can contain one outcome, two outcomes, no outcomes (empty subset), the entire sample space, etc. Standard notations for events are capital letters such as A, B, C, etc.


A planned activity carried out under controlled condition.

Independent event

The occurrence of one event has no effect on the probability of the occurrence of any other event. Events A and B are independent if one of the following is true:
(1). P (A | B) = P (A)
(2). P (B | A) = P (B)
(3). P (A and B) = P (A) P(B)

Mutually exclusive

An observation cannot fall into more than one class (category). Being in more than one category prevents being in a mutually exclusive category.


A particular result of an experiment.


A number between 0 and 1, inclusive, that gives the likelihood that a specific event will occur. The foundation of statistics is given by the following 3 axioms (by A. N. Kolmogorov, 1930’s): Let S denote the sample space and A and B are two events in S . Then:
(1). 0≤P(A)≤1
(2). If A and B are any two mutually exclusive events, then P(A or B)= P(A)+P(B)
(3). P(S)=1

Sample space

The set of all possible outcomes of an experiment.

Tree diagram

The useful visual representation of a sample space and events in the form of a “tree” with branches marked by possible outcomes simultaneously with associated probabilities (frequencies, relative frequencies).

Venn diagram

The visual representation of a sample space and events in the form of circles or ovals showing their intersection.

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
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