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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

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?
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
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 to know photocatalytic properties of tio2 nanoparticles...what to do now
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Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
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
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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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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