# 1.2 Interpretations  (Page 4/5)

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## Empirical probability based on survey data

A survey asks two questions of 300 students: Do you live on campus? Are you satisfied with the recreational facilities in the student center?Answers to the latter question were categorized “reasonably satisfied,” “unsatisfied,” or “no definite opinion.” Let C be the event “on campus;” O be the event “off campus;” S be the event “reasonably satisfied;” U be the event ”unsatisfied;” and N be the event “no definite opinion.” Data are shown in the following table.

Survey Data

 S U N C 127 31 42 O 46 43 11

If an individual is selected on an equally likely basis from this group of 300, the probability of any of the events is taken to be the relative frequency of respondents ineach category corresponding to an event. There are 200 on campus members in the population, so $P\left(C\right)=200/300$ and $P\left(O\right)=100/300$ . The probability that a student selected is on campus and satisfied is taken to be $P\left(CS\right)=127/300$ . The probability a student is either on campus and satisfied or off campus and not satisfied is

$P\left(CS\bigvee OU\right)=P\left(CS\right)+P\left(OU\right)=127/300+43/300=170/300$

If there is reason to believe that the population sampled is representative of the entire student body, then the same probabilities would be applied to any studentselected at random from the entire student body.

It is fortunate that we do not have to declare a single position to be the “correct” viewpoint and interpretation.The formal model is consistent with any of the views set forth. We are free in any situation to make the interpretation most meaningful and natural to the problem at hand . It is not necessary to fit all problems into one conceptual mold; nor is itnecessary to change mathematical model each time a different point of view seems appropriate.

## Probability and odds

Often we find it convenient to work with a ratio of probabilities. If A and B are events with positive probability the odds favoring A over B is the probability ratio $P\left(A\right)/P\left(B\right)$ . If not otherwise specified, B is taken to be A c and we speak of the odds favoring A

$O\left(A\right)=\frac{P\left(A\right)}{P\left({A}^{c}\right)}=\frac{P\left(A\right)}{1-P\left(A\right)}$

This expression may be solved algebraically to determine the probability from the odds

$P\left(A\right)=\frac{O\left(A\right)}{1+O\left(A\right)}$

In particular, if $O\left(A\right)=a/b$ then $P\left(A\right)=\frac{a/b}{1+a/b}=\frac{a}{a+b}$ .

$O\left(A\right)=0.7/0.3=7/3$ . If the odds favoring A is 5/3, then $P\left(A\right)=5/\left(5+3\right)=5/8$ .

## Partitions and boolean combinations of events

The countable additivity property (P3) places a premium on appropriate partitioning of events.

Definition. A partition is a mutually exclusive class

$\left\{{A}_{i}:i\in J\right\}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{such}\phantom{\rule{4.pt}{0ex}}\text{that}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\Omega =\phantom{\rule{0.166667em}{0ex}}\underset{i\in J}{\overset{}{\bigvee }}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{A}_{i}$

A partition of event A is a mutually exclusive class

$\left\{{A}_{i}:i\in J\right\}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{such}\phantom{\rule{4.pt}{0ex}}\text{that}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}A=\phantom{\rule{0.166667em}{0ex}}\underset{i\in J}{\overset{}{\bigvee }}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{A}_{i}$

Remarks .

• A partition is a mutually exclusive class of events such that one (and only one) must occur on each trial.
• A partition of event A is a mutually exclusive class of events such that A occurs iff one (and only one) of the A i occurs.
• A partition (no qualifier) is taken to be a partition of the sure event Ω .
• If class $\left\{{B}_{i}:ı\in J\right\}$ is mutually exclusive and $A\subset B=\underset{i\in J}{\overset{}{\bigvee }}{B}_{i}$ , then the class $\left\{A{B}_{i}:ı\in J\right\}$ is a partition of A and $A=\underset{i\in J}{\overset{}{\bigvee }}A{B}_{i}$ .

We may begin with a sequence $\left\{{A}_{1}:1\le i\right\}$ and determine a mutually exclusive (disjoint) sequence $\left\{{B}_{1}:1\le i\right\}$ as follows:

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
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Stoney Reply
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Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
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research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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absolutely yes
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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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
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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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.
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s.
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for screen printed electrodes ?
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s. Reply
of graphene you mean?
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or in general
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in general
s.
Graphene has a hexagonal structure
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how did you get the value of 2000N.What calculations are needed to arrive at it
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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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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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