<< Chapter < Page Chapter >> Page >

Definition. If C is an event having positive probability, the conditional probability of A , given C is

P ( A | C ) = P ( A C ) P ( C )

For a fixed conditioning event C , we have a new likelihood assignment to the event A . Now

P ( A | C ) 0 , P ( Ω | C ) = 1 , and P ( j A j | C ) = P j A j C P ( C ) = j P ( A j C ) / P ( C ) = j P ( A j | C )

Thus, the new function P ( | C ) satisfies the three defining properties (P1), (P2), and (P3) for probability, so that for fixed C , we have a new probability measure , with all the properties of an ordinary probability measure.

Remark . When we write P ( A | C ) we are evaluating the likelihood of event A when it is known that event C has occurred. This is not the probability of a conditional event A | C . Conditional events have no meaning in the model we are developing.

Conditional probabilities from joint frequency data

A survey of student opinion on a proposed national health care program included 250 students, of whom 150 were undergraduates and 100 were graduate students.Their responses were categorized Y (affirmative), N (negative), and D (uncertain or no opinion). Results are tabulated below.

U 60 40 50
G 70 20 10

Suppose the sample is representative, so the results can be taken as typical of the student body. A student is picked at random. Let Y be the event he or she is favorable to the plan, N be the event he or she is unfavorable, and D is the event of no opinion (or uncertain). Let U be the event the student is an undergraduate and G be the event he or she is a graduate student. The data may reasonably be interpreted

P ( G ) = 100 / 250 , P ( U ) = 150 / 250 , P ( Y ) = ( 60 + 70 ) / 250 , P ( Y U ) = 60 / 250 , etc.


P ( Y | U ) = P ( Y U ) P ( U ) = 60 / 250 150 / 250 = 60 150

Similarly, we can calculate

P ( N | U ) = 40 / 150 , P ( D | U ) = 50 / 150 , P ( Y | G ) = 70 / 100 , P ( N | G ) = 20 / 100 , P ( D | G ) = 10 / 100

We may also calculate directly

P ( U | Y ) = 60 / 130 , P ( G | N ) = 20 / 60 , etc.
Got questions? Get instant answers now!

Conditional probability often provides a natural way to deal with compound trials carried out in several steps.

Jet aircraft with two engines

An aircraft has two jet engines. It will fly with only one engine operating. Let F 1 be the event one engine fails on a long distance flight, and F 2 the event the second fails. Experience indicates that P ( F 1 ) = 0 . 0003 . Once the first engine fails, added load is placed on the second, so that P ( F 2 | F 1 ) = 0 . 001 . Now the second engine can fail only if the other has already failed. Thus F 2 F 1 so that

P ( F 2 ) = P ( F 1 F 2 ) = P ( F 1 ) P ( F 2 | F 1 ) = 3 × 10 - 7

Thus reliability of any one engine may be less than satisfactory, yet the overall reliability may be quite high.

Got questions? Get instant answers now!

The following example is taken from the UMAP Module 576, by Paul Mullenix, reprinted in UMAP Journal, vol 2, no. 4. More extensivetreatment of the problem is given there.

Responses to a sensitive question on a survey

In a survey, if answering “yes” to a question may tend to incriminate or otherwise embarrass the subject, the response given may beincorrect or misleading. Nonetheless, it may be desirable to obtain correct responses for purposes of social analysis. Thefollowing device for dealing with this problem is attributed to B. G. Greenberg.By a chance process, each subject is instructed to do one of three things:

  1. Respond with an honest answer to the question.
  2. Respond “yes” to the question, regardless of the truth in the matter.
  3. Respond “no” regardless of the true answer.

Let A be the event the subject is told to reply honestly, B be the event the subject is instructed to reply “yes,” and C be the event the answer is to be “no.” The probabilities P ( A ) , P ( B ) , and P ( C ) are determined by a chance mechanism (i.e., a fraction P ( A ) selected randomly are told to answer honestly, etc.). Let E be the event the reply is “yes.” We wish to calculate P ( E | A ) , the probability the answer is “yes” given the response is honest.


Since E = E A B , we have

P ( E ) = P ( E A ) + P ( B ) = P ( E | A ) P ( A ) + P ( B )

which may be solved algebraically to give

P ( E | A ) = P ( E ) - P ( B ) P ( A )

Suppose there are 250 subjects. The chance mechanism is such that P ( A ) = 0 . 7 , P ( B ) = 0 . 14 and P ( C ) = 0 . 16 . There are 62 responses “yes,” which we take to mean P ( E ) = 62 / 250 . According to the pattern above

P ( E | A ) = 62 / 250 - 14 / 100 70 / 100 = 27 175 0 . 154
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
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
Samson Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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