# Conditional probability  (Page 3/6)

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The formulation of conditional probability assumes the conditioning event C is well defined. Sometimes there are subtle difficulties. It may not be entirelyclear from the problem description what the conditioning event is. This is usually due to some ambiguity or misunderstanding of the information provided.

## What is the conditioning event?

Five equally qualified candidates for a job, Jim, Paul, Richard, Barry, and Evan, are identified on the basis of interviews and told that they are finalists. Three ofthese are to be selected at random, with results to be posted the next day. One of them, Jim, has a friend in the personnel office. Jim asks the friend to tell him the name ofone of those selected (other than himself). The friend tells Jim that Richard has been selected. Jim analyzes the problem as follows.

ANALYSIS

Let ${A}_{i},1\le i\le 5$ be the event the i th of these is hired ( A 1 is the event Jim is hired, A 3 is the event Richard is hired, etc.). Now $P\left({A}_{i}\right)$ (for each i ) is the probability that finalist i is in one of the combinations of three from five. Thus, Jim's probability of being hired, before receiving the information about Richard, is

$P\left({A}_{1}\right)=\frac{1×C\left(4,2\right)}{C\left(5,3\right)}=\frac{6}{10}=P\left({A}_{i}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}1\le i\le 5$

The information that Richard is one of those hired is information that the event A 3 has occurred. Also, for any pair $i\ne j$ the number of combinations of three from five including these two is just the number of ways of picking one from the remaining three.Hence,

$P\left({A}_{1}{A}_{3}\right)=\frac{C\left(3,1\right)}{C\left(5,3\right)}=\frac{3}{10}=P\left({A}_{i}{A}_{j}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}i\ne j$

The conditional probability

$P\left({A}_{1}|{A}_{3}\right)=\frac{P\left({A}_{1}{A}_{3}\right)}{P\left({A}_{3}\right)}=\frac{3/10}{6/10}=1/2$

This is consistent with the fact that if Jim knows that Richard is hired, then there are two to be selected from the four remaining finalists, so that

$P\left({A}_{1}|{A}_{3}\right)=\frac{1×C\left(3,1\right)}{C\left(4,2\right)}=\frac{3}{6}=1/2$

Discussion

Although this solution seems straightforward, it has been challenged as being incomplete. Many feel that there must be information about how the friend chose to name Richard. Many would make an assumption somewhat as follows. Thefriend took the three names selected: if Jim was one of them, Jim's name was removed and an equally likely choice among the other two was made; otherwise, the friendselected on an equally likely basis one of the three to be hired. Under this assumption, the information assumed is an event B 3 which is not the same as A 3 . In fact, computation (see Example 5, below) shows

$P\left({A}_{1}|{B}_{3}\right)\phantom{\rule{4pt}{0ex}}=\frac{6}{10}=P\left({A}_{1}\right)\ne P\left({A}_{1}|{A}_{3}\right)$

Both results are mathematically correct. The difference is in the conditioning event, which corresponds to the difference in the information given (or assumed).

$\square$

## Some properties

In addition to its properties as a probability measure, conditional probability has special properties which are consequences of the way it is related to the original probability measure $P\left(\cdot \right)$ . The following are easily derived from the definition of conditional probability and basic properties of the prior probabilitymeasure, and prove useful in a variety of problem situations.

(CP1) Product rule If $P\left(ABCD\right)>0$ , then $P\left(ABCD\right)=P\left(A\right)P\left(B|A\right)P\left(C|AB\right)P\left(D|ABC\right)$ .

Derivation

The defining expression may be written in product form: $P\left(AB\right)=P\left(A\right)P\left(B|A\right)$ . Likewise

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nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
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Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
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what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
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Professor
I think
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Nasa has use it in the 60's, copper as water purification in the moon travel.
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scanning tunneling microscope
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what is differents between GO and RGO?
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what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
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if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
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A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive

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