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Probability: Independent and Mutually Exclusive Events is part of the collection col10555 written by Barbara Illowsky and Susan Dean and explains the concept of independent events, where the probability of event A does not have any effect on the probability of event B, and mutually exclusive events, where events A and B cannot occur at the same time. The module has contributions from Roberta Bloom.

Independent and mutually exclusive do not mean the same thing.

Independent events

Two events are independent if the following are true:

  • P(A|B) = P(A)
  • P(B|A) = P(B)
  • P(A AND B) = P(A) ⋅ P(B)

Two events A and B are independent if the knowledge that one occurred does not affect the chance the other occurs. For example, the outcomes of two roles of a fair die are independent events. The outcome of the first roll does not change the probability for the outcome of the secondroll. To show two events are independent, you must show only one of the above conditions. If two events are NOT independent, then we say that they are dependent .

Sampling may be done with replacement or without replacement .

  • With replacement : If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.
  • Without replacement: : When sampling is done without replacement, then each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent.

If it is not known whether A and B are independent or dependent, assume they are dependent until you can show otherwise .

Mutually exclusive events

A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B) = 0 .

For example, suppose the sample space S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} . Let A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8} , and C = {7, 9} . A AND B = { 4 , 5 } . P(A AND B) = 2 10 and is not equal to zero. Therefore, A and B are not mutually exclusive. A and C do not have any numbers in common so P(A AND C) = 0 . Therefore, A and C are mutually exclusive.

If it is not known whether A and B are mutually exclusive, assume they are not until you can show otherwise .

The following examples illustrate these definitions and terms.

Flip two fair coins. (This is an experiment.)

The sample space is {HH, HT, TH, TT} where T = tails and H = heads. The outcomes are HH , HT , TH , and TT . The outcomes HT and TH are different. The HT means that the first coin showed heads and the second coin showed tails. The TH means that the first coin showed tails and the second coin showed heads.

  • Let A = the event of getting at most one tail . (At most one tail means 0 or 1 tail.) Then A can be written as {HH, HT, TH} . The outcome HH shows 0 tails. HT and TH each show 1 tail.
  • Let B = the event of getting all tails. B can be written as {TT} . B is the complement of A . So, B = A' . Also, P(A) + P(B) = P(A) + P(A') = 1 .
  • The probabilities for A and for B are P(A) = 3 4 and P(B) = 1 4 .
  • Let C = the event of getting all heads. C = {HH} . Since B = {TT} , P(B AND C) = 0 . B and C are mutually exclusive. ( B and C have no members in common because you cannot have all tails and all heads at the same time.)
  • Let D = event of getting more than one tail. D = {TT} . P(D) = 1 4 .
  • Let E = event of getting a head on the first roll. (This implies you can get either a head or tail on the second roll.) E = { HT , HH } . P(E) = 2 4 .
  • Find the probability of getting at least one (1 or 2) tail in two flips. Let F = event of getting at least one tail in two flips. F = { HT , TH , TT } . P(F) = 3 4
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Questions & Answers

How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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
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Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
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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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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
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
Akash Reply
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Maciej
characteristics of micro business
Abigail
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Anassong
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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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
1 It is estimated that 30% of all drivers have some kind of medical aid in South Africa. What is the probability that in a sample of 10 drivers: 3.1.1 Exactly 4 will have a medical aid. (8) 3.1.2 At least 2 will have a medical aid. (8) 3.1.3 More than 9 will have a medical aid.
Nerisha Reply
Practice Key Terms 2

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Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
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