# 10.1 Key concepts  (Page 3/4)

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## Project idea

Perform an experiment to show that as the number of trials increases, the relative frequency approaches the probability of a coin toss. Perform 10, 20, 50, 100, 200 trials of tossing a coin.

## Probability identities

The following results apply to probabilities, for the sample space $S$ and two events $A$ and $B$ , within $S$ .

$P\left(S\right)=1$
$P\left(A\cap B\right)=P\left(A\right)×P\left(B\right)$
$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$

What is the probability of selecting a black or red card from a pack of 52 cards

1. P(S)=n(E)/n(S)=52/52=1. because all cards are black or red!

What is the probability of drawing a club or an ace with one single pick from a pack of 52 cards

1. $P\left(\mathrm{club}\cup \mathrm{ace}\right)=\mathrm{P}\left(\mathrm{club}\right)+\mathrm{P}\left(\mathrm{ace}\right)-\mathrm{P}\left(\mathrm{club}\cap \mathrm{ace}\right)$
2. $\begin{array}{ccc}& =& \frac{1}{4}+\frac{1}{13}-\left(\frac{1}{4},×,\frac{1}{13}\right)\hfill \\ & =& \frac{1}{4}+\frac{1}{13}-\frac{1}{52}\hfill \\ & =& \frac{16}{52}\hfill \\ & =& \frac{4}{13}\hfill \end{array}$

Notice how we have used $P\left(C\cup A\right)=P\left(C\right)+P\left(A\right)-P\left(C\cap A\right)$ .

The following video provides a brief summary of some of the work covered so far.

## Probability identities

Answer the following questions

1. Rory is target shooting. His probability of hitting the target is $0,7$ . He fires five shots. What is the probability that all five shots miss the center?
2. An archer is shooting arrows at a bullseye. The probability that an arrow hits the bullseye is $0,4$ . If she fires three arrows, what is the probability that all the arrows hit the bullseye?
3. A dice with the numbers 1,3,5,7,9,11 on it is rolled. Also a fair coin is tossed. What is the probability that:
1. A tail is tossed and a 9 rolled?
2. A head is tossed and a 3 rolled?
4. Four children take a test. The probability of each one passing is as follows. Sarah: $0,8$ , Kosma: $0,5$ , Heather: $0,6$ , Wendy: $0,9$ . What is the probability that:
1. all four pass?
2. all four fail?
5. With a single pick from a pack of 52 cards what is the probability that the card will be an ace or a black card?

## Mutually exclusive events

Mutually exclusive events are events, which cannot be true at the same time.

Examples of mutually exclusive events are:

1. A die landing on an even number or landing on an odd number.
2. A student passing or failing an exam
3. A tossed coin landing on heads or landing on tails

This means that if we examine the elements of the sets that make up $A$ and $B$ there will be no elements in common. Therefore, $A\cap B=\varnothing$ (where $\varnothing$ refers to the empty set). Since, $P\left(A\cap B\right)=0$ , equation [link] becomes:

$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)$

for mutually exclusive events.

## Mutually exclusive events

Answer the following questions

1. A box contains coloured blocks. The number of each colour is given in the following table.
 Colour Purple Orange White Pink Number of blocks 24 32 41 19
A block is selected randomly. What is the probability that the block will be:
1. purple
2. purple or white
3. pink and orange
4. not orange?
2. A small private school has a class with children of various ages. The table gies the number of pupils of each age in the class.
 3 years female 3 years male 4 years female 4 years male 5 years female 5 years male 6 2 5 7 4 6
If a pupil is selceted at random what is the probability that the pupil will be:
1. a female
2. a 4 year old male
3. aged 3 or 4
4. aged 3 and 4
5. not 5
6. either 3 or female?
3. Fiona has 85 labeled discs, which are numbered from 1 to 85. If a disc is selected at random what is the probability that the disc number:
1. ends with 5
2. can be multiplied by 3
3. can be multiplied by 6
4. is number 65
5. is not a multiple of 5
6. is a multiple of 4 or 3
7. is a multiple of 2 and 6
8. is number 1?

#### 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
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