# 0.12 Probability  (Page 7/8)

 Page 7 / 8

Let us now develop a formula for the conditional probability $P\left(E\mid F\right)$ .

Suppose an experiment consists of $n$ equally likely events. Further suppose that there are $m$ elements in $F$ , and $c$ elements in $E\cap F$ , as shown in the following Venn diagram.

If the event $F$ has occurred, the set of all possible outcomes is no longer the entire sample space, but instead, the subset $F$ . Therefore, we only look at the set $F$ and at nothing outside of $F$ . Since $F$ has $m$ elements, the denominator in the calculation of $P\left(E\mid F\right)$ is m. We may think that the numerator for our conditional probability is the number of elements in $E$ . But clearly we cannot consider the elements of $E$ that are not in $F$ . We can only count the elements of $E$ that are in $F$ , that is, the elements in $E\cap F$ . Therefore,

$P\left(E\mid F\right)=\frac{c}{m}$

Dividing both the numerator and the denominator by $n$ , we get

$P\left(E\mid F\right)=\frac{c/n}{m/n}$

But $c/n=P\left(E\cap F\right)$ , and $m/n=P\left(F\right)$ .

Substituting, we derive the following formula for $P\left(E\mid F\right)$ .

For Two Events $E$ and $F$ , the Probability of $E$ Given $F$ is

$P\left(E\mid F\right)=\frac{P\left(E\cap F\right)}{P\left(F\right)}$

A single die is rolled. Use the above formula to find the conditional probability of obtaining an even number given that a number greater than three has shown.

Let $E$ be the event that an even number shows, and $F$ be the event that a number greater than three shows. We want $P\left(E\mid F\right)$ .

$E=\left\{2,4,6\right\}$ and $F=\left\{4,5,6\right\}$ . Which implies, $E\cap F=\left\{4,6\right\}$

Therefore, $P\left(F\right)=3/6$ , and $P\left(E\cap F\right)=2/6$

$P\left(E\mid F\right)=\frac{P\left(E\cap F\right)}{P\left(F\right)}=\frac{2/6}{3/6}=\frac{2}{3}$ .

The following table shows the distribution by gender of students at a community college who take public transportation and the ones who drive to school.

 Male(M) Female(F) Total Public Transportation(P) 8 13 21 Drive(D) 39 40 79 Total 47 53 100

The events $M$ , $F$ , $P$ , and $D$ are self explanatory. Find the following probabilities.

1. $P\left(D\mid M\right)$
2. $P\left(F\mid D\right)$
3. $P\left(M\mid P\right)$

We use the conditional probability formula $P\left(E\mid F\right)=\frac{P\left(E\cap F\right)}{P\left(F\right)}$ .

1. $P\left(D\mid M\right)=\frac{P\left(D\cap M\right)}{P\left(M\right)}=\frac{\text{39}/\text{100}}{\text{47}/\text{100}}=\frac{\text{39}}{\text{47}}$ .
2. $P\left(F\mid D\right)=\frac{P\left(F\cap D\right)}{P\left(D\right)}=\frac{\text{40}/\text{100}}{\text{79}/\text{100}}=\frac{\text{40}}{\text{79}}$ .
3. $P\left(M\mid P\right)=\frac{P\left(M\cap P\right)}{P\left(P\right)}=\frac{8/\text{100}}{\text{21}/\text{100}}=\frac{8}{\text{21}}$ .

Given $P\left(E\right)=\text{.}5$ , $P\left(F\right)=.7$ , and $P\left(E\cap F\right)=.3$ . Find the following.

1. $P\left(E\mid F\right)$
2. $P\left(F\mid E\right)$ .

We use the conditional probability formula $P\left(E\mid F\right)=\frac{P\left(E\cap F\right)}{P\left(F\right)}$ .

1. $P\left(E\mid F\right)=\frac{\text{.}3}{\text{.}7}=\frac{3}{7}$ .
2. $P\left(F\mid E\right)=\text{.}3/\text{.}5=3/5$ .

Given two mutually exclusive events $E$ and $F$ such that $P\left(E\right)=\text{.}4$ , $P\left(F\right)=\text{.}9$ . Find $P\left(E\mid F\right)$ .

Since $E$ and $F$ are mutually exclusive, $P\left(E\cap F\right)=0$ . Therefore,

$P\left(E|F\right)=\frac{0}{.9}=0$ .

Given $P\left(F\mid E\right)=\text{.}5$ , and $P\left(E\cap F\right)=\text{.}3$ . Find $P\left(E\right)$ .

Using the conditional probability formula $P\left(E\mid F\right)=\frac{P\left(E\cap F\right)}{P\left(F\right)}$ , we get

$P\left(F\mid E\right)=\frac{P\left(E\cap F\right)}{P\left(E\right)}$

Substituting,

$\text{.}5=\frac{\text{.}3}{P\left(E\right)}$ or $P\left(E\right)=3/5$

In a family of three children, find the conditional probability of having two boys and a girl, given that the family has at least two boys.

Let event $E$ be that the family has two boys and a girl, and let $F$ be the probability that the family has at least two boys. We want $P\left(E\mid F\right)$ .

We list the sample space along with the events $E$ and $F$ .

$S=\left\{\text{BBB},\text{BBG},\text{BGB},\text{BGG},\text{GBB},\text{GGB},\text{GGG}\right\}$

$E=\left\{\text{BBG},\text{BGB},\text{GBB}\right\}$ and $F=\left\{\text{BBB},\text{BBG},\text{BGB},\text{GBB}\right\}$

$E\cap F=\left\{\text{BBG},\text{BGB},\text{GBB}\right\}$

Therefore, $P\left(F\right)=4/8$ , and $P\left(E\cap F\right)=3/8$ .

And

$P\left(E\mid F\right)-\frac{3/8}{4/8}=\frac{3}{4}$ .

At a community college 65% of the students use IBM computers, 50% use Macintosh computers, and 20% use both. If a student is chosen at random, find the following probabilities.

1. The student uses an IBM given that he uses a Macintosh.
2. The student uses a Macintosh knowing that he uses an IBM.

Let event $I$ be that the student uses an IBM computer, and $M$ the probability that he uses a Macintosh.

1. $P\left(I\mid M\right)=\frac{\text{.}\text{20}}{\text{.}\text{50}}=\frac{2}{5}$
2. $P\left(M\mid I\right)=\frac{\text{.}\text{20}}{\text{.}\text{65}}=\frac{4}{\text{13}}$ .

what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
what is Nano technology ?
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?
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
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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
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
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?