# 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

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

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?

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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
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
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
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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