<< Chapter < Page Chapter >> Page >

Reversal of conditioning

Students in a freshman mathematics class come from three different high schools. Their mathematical preparation varies. In order to group them appropriately in classsections, they are given a diagnostic test. Let H i be the event that a student tested is from high school i , 1 i 3 . Let F be the event the student fails the test. Suppose data indicate

P ( H 1 ) = 0 . 2 , P ( H 2 ) = 0 . 5 , P ( H 3 ) = 0 . 3 , P ( F | H 1 ) = 0 . 10 , P ( F | H 2 ) = 0 . 02 , P ( F | H 3 ) = 0 . 06

A student passes the exam. Determine for each i the conditional probability P ( H i | F c ) that the student is from high school i .

SOLUTION

P ( F c ) = P ( F c | H 1 ) P ( H 1 ) + P ( F c | H 2 ) P ( H 2 ) + P ( F c | H 3 ) P ( H 3 ) = 0 . 90 0 . 2 + 0 . 98 0 . 5 + 0 . 94 0 . 3 = 0 . 952

Then

P ( H 1 | F c ) = P ( F c H 1 ) P ( F c ) = P ( F c | H 1 ) P ( H 1 ) P ( F c ) = 180 952 = 0 . 1891

Similarly,

P ( H 2 | F c ) = P ( F c | H 2 ) P ( H 2 ) P ( F c ) = 590 952 = 0 . 5147 and P ( H 3 | F c ) = P ( F c | H 3 ) P ( H 3 ) P ( F c ) = 282 952 = 0 . 2962
Got questions? Get instant answers now!

The basic pattern utilized in the reversal is the following.

(CP3) Bayes' rule If E i = 1 n A i (as in the law of total probability), then

P ( A i | E ) = P ( A i E ) P ( E ) = P ( E | A i ) P ( A i ) P ( E ) 1 i n The law of total probability yields P ( E )

Such reversals are desirable in a variety of practical situations.

A compound selection and reversal

Begin with items in two lots:

  1. Three items, one defective.
  2. Four items, one defective.

One item is selected from lot 1 (on an equally likely basis); this item is added to lot 2; a selection is then made from lot 2 (also on an equally likely basis). This second itemis good. What is the probability the item selected from lot 1 was good?

SOLUTION

Let G 1 be the event the first item (from lot 1) was good, and G 2 be the event the second item (from the augmented lot 2) is good. We want to determine P ( G 1 | G 2 ) . Now the data are interpreted as

P ( G 1 ) = 2 / 3 , P ( G 2 | G 1 ) = 4 / 5 , P ( G 2 | G 1 c ) = 3 / 5

By the law of total probability (CP2) ,

P ( G 2 ) = P ( G 1 ) P ( G 2 | G 1 ) + P ( G 1 c ) P ( G 2 | G 1 c ) = 2 3 4 5 + 1 3 3 5 = 11 15

By Bayes' rule (CP3) ,

P ( G 1 | G 2 ) = P ( G 2 | G 1 ) P ( G 1 ) P ( G 2 ) = 4 / 5 × 2 / 3 11 / 15 = 8 11 0 . 73
Got questions? Get instant answers now!

Additional problems requiring reversals

  • Medical tests . Suppose D is the event a patient has a certain disease and T is the event a test for the disease is positive. Data are usually of the form: prior probability P ( D ) (or prior odds P ( D ) / P ( D c ) ), probability P ( T | D c ) of a false positive , and probability P ( T c | D ) of a false negative. The desired probabilities are P ( D | T ) and P ( D c | T c ) .
  • Safety alarm . If D is the event a dangerous condition exists (say a steam pressure is too high) and T is the event the safety alarm operates, then data are usually of the form P ( D ) , P ( T | D c ) , and P ( T c | D ) , or equivalently (e.g., P ( T c | D c ) and P ( T | D ) ). Again, the desired probabilities are that the safety alarms signals correctly, P ( D | T ) and P ( D c | T c ) .
  • Job success . If H is the event of success on a job, and E is the event that an individual interviewed has certain desirable characteristics, thedata are usually prior P ( H ) and reliability of the characteristics as predictors in the form P ( E | H ) and P ( E | H c ) . The desired probability is P ( H | E ) .
  • Presence of oil . If H is the event of the presence of oil at a proposed well site, and E is the event of certain geological structure (salt dome or fault), the data are usually P ( H ) (or the odds), P ( E | H ) , and P ( E | H c ) . The desired probability is P ( H | E ) .
  • Market condition . Before launching a new product on the national market, a firm usually examines the condition of a test market as anindicator of the national market. If H is the event the national market is favorable and E is the event the test market is favorable, data are a prior estimate P ( H ) of the likelihood the national market is sound, and data P ( E | H ) and P ( E | H c ) indicating the reliability of the test market. What is desired is P ( H | E ), the likelihood the national market is favorable, given the test market is favorable.
Got questions? Get instant answers now!

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller 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?

Ask