<< 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

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris 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
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele 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
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