# 0.12 Probability  (Page 3/8)

 Page 3 / 8

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

$M=\left\{\text{The family has at least one boy}\right\}$
$N=\left\{\text{The family has all girls}\right\}$

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

$M=\left\{\text{BBB},\text{BBG},\text{BGB},\text{BGG},\text{GBB},\text{GBG},\text{GGB}\right\}$ and $N=\left\{\text{GGG}\right\}$

Clearly, $M\cap N=\text{Ø}$

Therefore, the events $M$ and $N$ are mutually exclusive.

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 $E$ be the event that the number shown on the die is an even number, and let $F$ be the event that the number shown is greater than four.

The sample space $S=\left\{1,2,3,4,5,6\right\}$ . The event $E=\left\{2,4,6\right\}$ , and the event $F=\left\{5,6\right\}$

We need to find $P\left(E\cup F\right)$ .

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

This can be illustrated by a Venn diagram.

The sample space $S$ , the events $E$ and $F$ , and $E\cap F$ are listed below.

$S=\left\{1,2,3,4,5,6\right\}$ , $E=\left\{2,4,6\right\}$ , $F=\left\{5,6\right\}$ , and $E\cap F=\left\{6\right\}$ .

The above figure shows $S$ , $E$ , $F$ , and $E\cap F$ .

Finding the probability of $E\cup F$ , 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 $n\left(E\right)$ in $E$ , and add to it the number of elements $n\left(F\right)$ 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, $n\left(E\right)+n\left(F\right)$ , the number $n\left(E\cap F\right)$ , we remove the duplicity and get the correct answer. So as a rule,

$n\left(E\cup F\right)=n\left(E\right)+n\left(F\right)–n\left(E\cap F\right)$

By dividing the entire equation by $n\left(S\right)$ , we get

$\frac{n\left(E\cup F\right)}{n\left(S\right)}=\frac{n\left(E\right)}{n\left(S\right)}+\frac{n\left(F\right)}{n\left(S\right)}–\frac{n\left(E\cap F\right)}{n\left(S\right)}$

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

$P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)–P\left(E\cap F\right)$

Applying the above for this example, we get

$P\left(E\cup F\right)=3/6+2/6-1/6=4/6$

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

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

$P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)–P\left(E\cap F\right)$

If two events E and F are mutually exclusive, then $E\cap F=\varnothing$ and $P\left(E\cap F\right)=0$ , and we get

$P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)$

If a card is drawn from a deck, use the addition rule to find the probability of obtaining an ace or a heart.

Let $A$ be the event that the card is an ace, and $H$ the event that it is a heart.

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

We need to find $P\left(A\cup H\right)$ .

$P\left(A\cup H\right)=P\left(A\right)+P\left(H\right)–P\left(A\cap H\right)=4/\text{52}+\text{13}/\text{52}-1/\text{52}=\text{16}/\text{52}$ .

Two dice are rolled, and the events $F$ and $T$ are as follows:

$F=\left\{\text{The sum of the dice is four}\right\}$ and $T=\left\{\text{At least one die shows a three}\right\}$

Find $P\left(F\cup T\right)$ .

We list $F$ and $T$ , and $F\cap T$ as follows:

$F=\left\{\left(1,3\right),\left(2,2\right),\left(3,1\right)\right\}$
$T=\left\{\left(3,1\right),\left(3,2\right),\left(3,3\right),\left(3,4\right),\left(3,5\right),\left(3,6\right),\left(1,3\right),\left(2,3\right),\left(4,3\right),\left(5,3\right),\left(6,3\right)\right\}$
$F\cap T=\left\{\left(1,3\right),\left(3,1\right)\right\}$

Since $P\left(F\cup T\right)=P\left(F\right)+P\left(T\right)-P\left(F\cap T\right)$

We have $P\left(F\cup T\right)=3/\text{36}+\text{11}/\text{36}-2/\text{36}=\text{12}/\text{36}$ .

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
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?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
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?
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?
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?