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Use of a venn diagram

We can also use Venn diagrams to check whether events are dependent or independent.

Independent events

Events are said to be independent if the result or outcome of one event does not affect the result or outcome of the other event. So P(A/C)=P(A), where P(A/C) represents the probability of event A after event C has occured.

Dependent events

Two events are dependent if the outcome of one event is affected by the outcome of the other event i.e. P ( A / C ) P ( A )

. Also note that P ( A / C ) = P ( A C ) P ( C ) . For example, we can draw a Venn diagram and a contingency table to illustrate and analyse the following example.

A school decided that its uniform needed upgrading. The colours on offer were beige or blue or beige and blue. 40% of the school wanted beige, 55% wanted blue and 15% said a combination would be fine. Are the two events independent?

  1. Beige Not Beige Totals
    Blue 0,15 0,4 0,55
    Not Blue 0,25 0,2 0,35
    Totals 0,40 0,6 1
  2. P(Blue)=0,4, P(Beige)=0,55, P(Both)=0,15, P(Neither)=0,20

    Probability of choosing beige after blue is:

    P ( B e i g e / B l u e ) = P ( B e i g e B l u e ) P ( B l u e ) = 0 , 15 0 , 55 = 0 , 27
  3. Since P ( B e i g e / B l u e ) P ( B e i g e ) the events are statistically dependent.

Applications of probability theory

Two major applications of probability theory in everyday life are in risk assessment and in trade on commodity markets. Governments typically apply probability methods in environmental regulation where it is called “pathway analysis”, and are often measuring well-being using methods that are stochastic in nature, and choosing projects to undertake based on statistical analyses of their probable effect on the population as a whole. It is not correct to say that statistics are involved in the modelling itself, as typically the assessments of risk are one-time and thus require more fundamental probability models, e.g. “the probability of another 9/11”. A law of small numbers tends to apply to all such choices and perception of the effect of such choices, which makes probability measures a political matter.

A good example is the effect of the perceived probability of any widespread Middle East conflict on oil prices - which have ripple effects in the economy as a whole. An assessment by a commodity trade that a war is more likely vs. less likely sends prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are not assessed independently nor necessarily very rationally. The theory of behavioral finance emerged to describe the effect of such groupthink on pricing, on policy, and on peace and conflict.

It can reasonably be said that the discovery of rigorous methods to assess and combine probability assessments has had a profound effect on modern society. A good example is the application of game theory, itself based strictly on probability, to the Cold War and the mutual assured destruction doctrine. Accordingly, it may be of some importance to most citizens to understand how odds and probability assessments are made, and how they contribute to reputations and to decisions, especially in a democracy.

Another significant application of probability theory in everyday life is reliability. Many consumer products, such as automobiles and consumer electronics, utilize reliability theory in the design of the product in order to reduce the probability of failure. The probability of failure is also closely associated with the product's warranty.

End of chapter exercises

  1. In each of the following contingency tables give the expected numbers for the events to be perfectly independent and decide if the events are independent or dependent.
    1. Brown eyes Not Brown eyes Totals
      Black hair 50 30 80
      Red hair 70 80 150
      Totals 120 110 230
    2. Point A Point B Totals
      Busses left late 15 40 55
      Busses left on time 25 20 45
      Totals 40 60 100
    3. Durban Bloemfontein Totals
      Liked living there 130 30 160
      Did not like living there 140 200 340
      Totals 270 230 500
    4. Multivitamin A Multivitamin B Totals
      Improvement in health 400 300 700
      No improvement in health 140 120 260
      Totals 540 420 960
  2. A study was undertaken to see how many people in Port Elizabeth owned either a Volkswagen or a Toyota. 3% owned both, 25% owned a Toyota and 60% owned a Volkswagen. Draw a contingency table to show all events and decide if car ownership is independent.
  3. Jane invested in the stock market. The probability that she will not lose all her money is 0,32. What is the probability that she will lose all her money? Explain.
  4. If D and F are mutually exclusive events, with P(D ' )=0,3 and P(D or F)=0,94, find P(F).
  5. A car sales person has pink, lime-green and purple models of car A and purple, orange and multicolour models of car B. One dark night a thief steals a car.
    1. What is the experiment and sample space?
    2. Draw a Venn diagram to show this.
    3. What is the probability of stealing either a model of A or a model of B?
    4. What is the probability of stealing both a model of A and a model of B?
  6. The probability of Event X is 0,43 and the probability of Event Y is 0,24. The probability of both occuring together is 0,10. What is the probability that X or Y will occur (this inculdes X and Y occuring simultaneously)?
  7. P(H)=0,62, P(J)=0,39 and P(H and J)=0,31. Calculate:
    1. P(H ' )
    2. P(H or J)
    3. P(H ' or J ' )
    4. P(H ' or J)
    5. P(H ' and J ' )
  8. The last ten letters of the alphabet were placed in a hat and people were asked to pick one of them. Event D is picking a vowel, Event E is picking a consonant and Event F is picking the last four letters. Calculate the following probabilities:
    1. P(F ' )
    2. P(F or D)
    3. P(neither E nor F)
    4. P(D and E)
    5. P(E and F)
    6. P(E and D ' )
  9. At Dawnview High there are 400 Grade 12's. 270 do Computer Science, 300 do English and 50 do Typing. All those doing Computer Science do English, 20 take Computer Science and Typing and 35 take English and Typing. Using a Venn diagram calculate the probability that a pupil drawn at random will take:
    1. English, but not Typing or Computer Science
    2. English but not Typing
    3. English and Typing but not Computer Science
    4. English or Typing

Questions & Answers

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Other chapter Q/A we can ask
Moahammedashifali Reply

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Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
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