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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

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 ( C ) = 200 / 300 and P ( O ) = 100 / 300 . The probability that a student selected is on campus and satisfied is taken to be P ( C S ) = 127 / 300 . The probability a student is either on campus and satisfied or off campus and not satisfied is

P ( C S O U ) = P ( C S ) + P ( O U ) = 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.

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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 ( A ) / P ( B ) . If not otherwise specified, B is taken to be A c and we speak of the odds favoring A

O ( A ) = P ( A ) P ( A c ) = P ( A ) 1 - P ( A )

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

P ( A ) = O ( A ) 1 + O ( A )

In particular, if O ( A ) = a / b then P ( A ) = a / b 1 + a / b = a a + b .

O ( A ) = 0 . 7 / 0 . 3 = 7 / 3 . If the odds favoring A is 5/3, then P ( A ) = 5 / ( 5 + 3 ) = 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

{ A i : i J } such that Ω = i J A i

A partition of event A is a mutually exclusive class

{ A i : i J } such that A = i J 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 { B i : ı J } is mutually exclusive and A B = i J B i , then the class { A B i : ı J } is a partition of A and A = i J A B i .

We may begin with a sequence { A 1 : 1 i } and determine a mutually exclusive (disjoint) sequence { B 1 : 1 i } as follows:

Questions & Answers

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Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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I think
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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LITNING Reply
What is meant by 'nano scale'?
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LITNING
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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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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what is Nano technology ?
Bob Reply
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
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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..
Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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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 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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