# Conditional probability  (Page 5/6)

 Page 5 / 6

## 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 , $1\le i\le 3$ . Let F be the event the student fails the test. Suppose data indicate

$P\left({H}_{1}\right)=0.2,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left({H}_{2}\right)=0.5,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left({H}_{3}\right)=0.3,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left(F|{H}_{1}\right)=0.10,\phantom{\rule{10pt}{0ex}}P\left(F|{H}_{2}\right)=0.02,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left(F|{H}_{3}\right)=0.06$

A student passes the exam. Determine for each i the conditional probability $P\left({H}_{i}|{F}^{c}\right)$ that the student is from high school i .

SOLUTION

$P\left({F}^{c}\right)=P\left({F}^{c}|{H}_{1}\right)P\left({H}_{1}\right)+P\left({F}^{c}|{H}_{2}\right)P\left({H}_{2}\right)+P\left({F}^{c}|{H}_{3}\right)P\left({H}_{3}\right)=0.90\cdot 0.2+0.98\cdot 0.5+0.94\cdot 0.3=0.952$

Then

$P\left({H}_{1}|{F}^{c}\right)=\frac{P\left({F}^{c}{H}_{1}\right)}{P\left({F}^{c}\right)}=\frac{P\left({F}^{c}|{H}_{1}\right)P\left({H}_{1}\right)}{P\left({F}^{c}\right)}=\frac{180}{952}=0.1891$

Similarly,

$P\left({H}_{2}|{F}^{c}\right)=\frac{P\left({F}^{c}|{H}_{2}\right)P\left({H}_{2}\right)}{P\left({F}^{c}\right)}=\frac{590}{952}=0.5147\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{10pt}{0ex}}P\left({H}_{3}|{F}^{c}\right)=\frac{P\left({F}^{c}|{H}_{3}\right)P\left({H}_{3}\right)}{P\left({F}^{c}\right)}=\frac{282}{952}=0.2962$

The basic pattern utilized in the reversal is the following.

(CP3) Bayes' rule If $E\subset \phantom{\rule{0.166667em}{0ex}}\underset{i=1}{\overset{n}{\bigvee }}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{A}_{i}\phantom{\rule{0.277778em}{0ex}}$ (as in the law of total probability), then

$P\left({A}_{i}|E\right)=\frac{P\left({A}_{i}E\right)}{P\left(E\right)}=\frac{P\left(E|{A}_{i}\right)P\left({A}_{i}\right)}{P\left(E\right)}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}1\le i\le n\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{The}\phantom{\rule{4.pt}{0ex}}\text{law}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{total}\phantom{\rule{4.pt}{0ex}}\text{probability}\phantom{\rule{4.pt}{0ex}}\text{yields}\phantom{\rule{4.pt}{0ex}}P\left(E\right)$

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 $P\left({G}_{1}|{G}_{2}\right)$ . Now the data are interpreted as

$P\left({G}_{1}\right)=2/3,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left({G}_{2}|{G}_{1}\right)=4/5,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left({G}_{2}|{G}_{1}^{c}\right)=3/5$

By the law of total probability (CP2) ,

$P\left({G}_{2}\right)=P\left({G}_{1}\right)P\left({G}_{2}|{G}_{1}\right)+P\left({G}_{1}^{c}\right)P\left({G}_{2}|{G}_{1}^{c}\right)=\frac{2}{3}\cdot \frac{4}{5}+\frac{1}{3}\cdot \frac{3}{5}=\frac{11}{15}$

By Bayes' rule (CP3) ,

$P\left({G}_{1}|{G}_{2}\right)=\frac{P\left({G}_{2}|{G}_{1}\right)P\left({G}_{1}\right)}{P\left({G}_{2}\right)}=\frac{4/5×2/3}{11/15}=\frac{8}{11}\approx 0.73$

• 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 $P\left(D\right)$ (or prior odds $P\left(D\right)/P\left({D}^{c}\right)$ ), probability $P\left(T|{D}^{c}\right)$ of a false positive , and probability $P\left({T}^{c}|D\right)$ of a false negative. The desired probabilities are $P\left(D|T\right)$ and $P\left({D}^{c}|{T}^{c}\right)$ .
• 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 $P\left(D\right)$ , $P\left(T|{D}^{c}\right)$ , and $P\left({T}^{c}|D\right)$ , or equivalently (e.g., $P\left({T}^{c}|{D}^{c}\right)$ and $P\left(T|D\right)$ ). Again, the desired probabilities are that the safety alarms signals correctly, $P\left(D|T\right)$ and $P\left({D}^{c}|{T}^{c}\right)$ .
• 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 $P\left(H\right)$ and reliability of the characteristics as predictors in the form $P\left(E|H\right)$ and $P\left(E|{H}^{c}\right)$ . The desired probability is $P\left(H|E\right)$ .
• 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 $P\left(H\right)$ (or the odds), $P\left(E|H\right)$ , and $P\left(E|{H}^{c}\right)$ . The desired probability is $P\left(H|E\right)$ .
• 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 $P\left(H\right)$ of the likelihood the national market is sound, and data $P\left(E|H\right)$ and $P\left(E|{H}^{c}\right)$ indicating the reliability of the test market. What is desired is $P\left(H|E\right),$ the likelihood the national market is favorable, given the test market is favorable.

Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive