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Complement and complementary events

A final notion that is important to understand is the notion of complement . Just as in geometry when two angles are called 'complementary' if they added upto 90 degrees, (the two angles 'complement' each other to make a right angle), the complement of a set of outcomes A is the set of all outcomes in the sample space but not in A . It is usually denoted A ' (or sometimes A C ), and called 'the complement of A ' or simply ' A -complement'. Because it refers to the set of everything outside of A , it is also often referred to as 'not- A '. Thus, by definition, if S denotes the entire sample space of possible outcomes, and A is any subset of outcomes that we are interested in (i.e. an event ), then A A ' = S is always true, (i.e. A ' complements A to form the entire sample space). So in the exercise above, P ' = { 1 , 4 , 6 , 8 , 9 } , while E ' = { 1 , 3 , 5 , 7 , 9 } . So n ( P ' ) = n ( E ' ) = 5

The probability of a complementary event refers to the probability associated with the complement of an event, i.e. the probability that something other than the event in question will occur. For example, if P ( A ) = 0 , 25 , then the probability of A not occurring is the probability associated with all other events in S occurring less the probability of A occurring.

In theory, it is very easy to calculate complements, since the number of elementsin the complement of a set is just the total number of outcomes in the sample space minus the outcomes in that set (in the example above, there were 9possible outcomes in the sample space, and 4 possible outcomes in each of the sets we were interested in, thus both complements contained 9-4 = 5 elements).Similarly, it is easy to calculate probabilities of complements of events since they are simply the total probability (e.g. 1 if our total measure is 1) minus the probability of the event in question. So,

P ( A ' ) = 1 - P ( A )

Sometimes it is much easier to decide the probability of an event occurring by instead calculating the probability that the complementary event will NOT occur. For example, if the process in question was rolling three dice, and the event we were interested in was that at least one of the faces is a one, it is definitely much easier to figure out the probability that not getting a one will not occur than to try to figure out all the possible combinations of three dice where a one does occur!

If you throw two dice, one red and one blue, what is the probability that at least one of them will be a six?

  1. To solve that kind of question, work out the probability that there will be no six.

  2. The probability that the red dice will not be a six is 5/6, and that the blue one will not be a six is also 5/6.

  3. So the probability that neither will be a six is 5 / 6 × 5 / 6 = 25 / 36 .

  4. So the probability that at least one will be a six is 1 - 25 / 36 = 11 / 36 .

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A bag contains three red balls, five white balls, two green balls and four blue balls:

1. Calculate the probability that a red ball will be drawn from the bag.

2. Calculate the probability that a ball which is not red will be drawn

  1. Let R be the event that a red ball is drawn:

    • P(R)-n(R)/n(S)=3/14
    • R and R' are complementary events
  2. P(R') = 1 - P(R) = 1 -3/14 = 11/14

    • Alternately P(R') = P(B) + P(W) + P(G)
    • P(R') = 4/14 + 5/14 + 2/14 = 11/14
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Questions & Answers

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
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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
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