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A family has three children. Determine whether the following pair of events are mutually exclusive.

M = The family has at least one boy size 12{M= left lbrace "The family has at least one boy" right rbrace } {}
N = The family has all girls size 12{N= left lbrace "The family has all girls" right rbrace } {}

Although the answer may be clear, we list both the sets.

M = BBB , BBG , BGB , BGG , GBB , GBG , GGB size 12{M= left lbrace ital "BBB", ital "BBG", ital "BGB", ital "BGG", ital "GBB", ital "GBG", ital "GGB" right rbrace } {} and N = GGG size 12{N= left lbrace ital "GGG" right rbrace } {}

Clearly, M N = Ø size 12{M intersection N="Ø"} {}

Therefore, the events M size 12{M} {} and N size 12{N} {} are mutually exclusive.

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We will now consider problems that involve the union of two events.

If a die is rolled, what is the probability of obtaining an even number or a number greater than four?

Let E size 12{E} {} be the event that the number shown on the die is an even number, and let F size 12{F} {} be the event that the number shown is greater than four.

The sample space S = 1,2,3,4,5,6 size 12{S= left lbrace 1,2,3,4,5,6 right rbrace } {} . The event E = 2,4,6 size 12{E= left lbrace 2,4,6 right rbrace } {} , and the event F = 5,6 size 12{F= left lbrace 5,6 right rbrace } {}

We need to find P E F size 12{P left (E union F right )} {} .

Since P E = 3 / 6 size 12{P left (E right )=3/6} {} , and P F = 2 / 6 size 12{P left (F right )=2/6} {} , a student may say P E F = 3 / 6 + 2 / 6 size 12{P left (E union F right )=3/6+2/6} {} . This will be incorrect because the element 6, which is in both E size 12{E} {} and F size 12{F} {} has been counted twice, once as an element of E size 12{E} {} and once as an element of F size 12{F} {} . In other words, the set E F size 12{E union F} {} has only four elements and not five. Therefore, P E F = 4 / 6 size 12{P left (E union F right )=4/6} {} and not 5 / 6 size 12{5/6} {} .

This can be illustrated by a Venn diagram.

The sample space S size 12{S} {} , the events E size 12{E} {} and F size 12{F} {} , and E F size 12{E intersection F} {} are listed below.

S = 1,2,3,4,5,6 size 12{S= left lbrace 1,2,3,4,5,6 right rbrace } {} , E = 2,4,6 size 12{E= left lbrace 2,4,6 right rbrace } {} , F = 5,6 size 12{F= left lbrace 5,6 right rbrace } {} , and E F = 6 size 12{E intersection F= left lbrace 6 right rbrace } {} .

The figure shows that everything in the square equals S. In the Venn Diagram 2 and 4 equal E, while 5 equals F and 6 equals both E and F.

The above figure shows S , E , F , and E F .

Finding the probability of E F , is the same as finding the probability that E will happen, or F will happen, or both will happen. If we count the number of elements n ( E ) in E , and add to it the number of elements n ( F ) in F , the points in both E and F are counted twice, once as elements of E and once as elements of F . Now if we subtract from the sum, n ( E ) + n ( F ) , the number n ( E F ) , we remove the duplicity and get the correct answer. So as a rule,

n ( E F ) = n ( E ) + n ( F ) n ( E F )

By dividing the entire equation by n ( S ) , we get

n ( E F ) n ( S ) = n ( E ) n ( S ) + n ( F ) n ( S ) n ( E F ) n ( S )

Since the probability of an event is the number of elements in that event divided by the number of all possible outcomes, we have

P ( E F ) = P ( E ) + P ( F ) P ( E F )

Applying the above for this example, we get

P ( E F ) = 3 / 6 + 2 / 6 - 1 / 6 = 4 / 6

This is because, when we add P ( E ) and P ( F ) , we have added P ( E F ) twice. Therefore, we must subtract P ( E F ) , once.

This gives us the general formula, called the Addition Rule , for finding the probability of the union of two events. It states

P ( E F ) = P ( E ) + P ( F ) P ( E F )

If two events E and F are mutually exclusive, then E F = and P ( E F ) = 0 , and we get

P ( E F ) = P ( E ) + P ( F )
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If a card is drawn from a deck, use the addition rule to find the probability of obtaining an ace or a heart.

Let A size 12{A} {} be the event that the card is an ace, and H size 12{H} {} the event that it is a heart.

Since there are four aces, and thirteen hearts in the deck, P A = 4 / 52 size 12{P left (A right )=4/"52"} {} and P H = 13 / 52 size 12{P left (H right )="13"/"52"} {} . Furthermore, since the intersection of two events is an ace of hearts, P A H = 1 / 52 size 12{P left (A intersection H right )=1/"52"} {}

We need to find P A H size 12{P left (A union H right )} {} .

P A H = P A + P H P A H = 4 / 52 + 13 / 52 1 / 52 = 16 / 52 size 12{P left (A union H right )=P left (A right )+P left (H right )–P left (A intersection H right )=4/"52"+"13"/"52" - 1/"52"="16"/"52"} {} .

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Two dice are rolled, and the events F size 12{F} {} and T size 12{T} {} are as follows:

F = The sum of the dice is four size 12{F= left lbrace "The sum of the dice is four" right rbrace } {} and T = At least one die shows a three size 12{T= left lbrace "At least one die shows a three" right rbrace } {}

Find P F T size 12{P left (F union T right )} {} .

We list F size 12{F} {} and T size 12{T} {} , and F T size 12{F intersection T} {} as follows:

F = 1,3 , 2,2 , 3,1 size 12{F= left lbrace left (1,3 right ), left (2,2 right ), left (3,1 right ) right rbrace } {}
T = 3,1 , 3,2 , 3,3 , 3,4 , 3,5 , 3,6 , 1,3 , 2,3 , 4,3 , 5,3 , 6,3 size 12{T= left lbrace left (3,1 right ), left (3,2 right ), left (3,3 right ), left (3,4 right ), left (3,5 right ), left (3,6 right ), left (1,3 right ), left (2,3 right ), left (4,3 right ), left (5,3 right ), left (6,3 right ) right rbrace } {}
F T = 1,3 , 3,1 size 12{F intersection T= left lbrace left (1,3 right ), left (3,1 right ) right rbrace } {}

Since P F T = P F + P T P F T size 12{P left (F union T right )=P left (F right )+P left (T right ) - P left (F intersection T right )} {}

We have P F T = 3 / 36 + 11 / 36 2 / 36 = 12 / 36 size 12{P left (F union T right )=3/"36"+"11"/"36" - 2/"36"="12"/"36"} {} .

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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Google
da
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Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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da
Application of nanotechnology in medicine
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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.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
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
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
If March sales will be up from February by 10%, 15%, and 20% at Place I, Place II, and Place III, respectively, find the expected number of hot dogs, and corn dogs to be sold
Logan Reply
8. It is known that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events, wearing a seat belt and quitting smoking, independent?
William Reply
Mr. Shamir employs two part-time typists, Inna and Jim for his typing needs. Inna charges $10 an hour and can type 6 pages an hour, while Jim charges $12 an hour and can type 8 pages per hour. Each typist must be employed at least 8 hours per week to keep them on the payroll. If Mr. Shamir has at least 208 pages to be typed, how many hours per week should he employ each student to minimize his typing costs, and what will be the total cost?
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At De Anza College, 20% of the students take Finite Mathematics, 30% take Statistics and 10% take both. What percentage of the students take Finite Mathematics or Statistics?
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Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
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