# Probability: part 1

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We use $n\left(S\right)$ to refer to the number of elements in a set $S$ , $n\left(X\right)$ for the number of elements in $X$ , etc.

In a box there are pieces of paper with the numbers from 1 to 9 written on them. A piece of paper is drawn from the box andthe number on it is noted. Let $S$ denote the sample space, let $P$ denote the event 'drawing a prime number', and let $E$ denote the event 'drawing an even number'. Using appropriate notation, in how many ways is it possible to draw: i)any number? ii) a prime number? iii) an even number? iv) a number that is either prime or even? v) a number that is both prime and even?

• Drawing a prime number: $P=\left\{2;3;5;7\right\}$
• Drawing an even number: $E=\left\{2;4;6;8\right\}$
1. The union of $P$ and $E$ is the set of all elements in $P$ or in $E$ (or in both). $P\cup E=2,3,4,5,6,7,8$ .

2. The intersection of $P$ and $E$ is the set of all elements in both $P$ and $E$ . $P\cap E=2$ .

3. $\begin{array}{ccc}\hfill \therefore \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}n\left(S\right)& =& 9\hfill \\ \hfill n\left(P\right)& =& 4\hfill \\ \hfill n\left(E\right)& =& 4\hfill \\ \hfill n\left(P\cup E\right)& =& 7\hfill \\ \hfill n\left(P\cap E\right)& =& 2\hfill \end{array}$

100 people were surveyed to find out which fast food chain (Nandos, Debonairs or Steers) they preferred. The following results were obtained:

• 50 liked Nandos
• 66 liked Debonairs
• 40 liked Steers
• 27 liked Nandos and Debonairs but not Steers
• 13 liked Debonairs and Steers but not Nandos
• 4 liked all three
• 94 liked at least one
1. How many people did not like any of the fast food chains?
2. How many people liked Nandos and Steers, but not Debonairs?

1. The number of people who liked Nandos and Debonairs is 27, so this is the intersection of these two events. The number of people who liked Debonairs and Steers is 13, so the intersection of Debonairs and Steers is 13. We are told that 4 people like all three options, and so this means that there are 4 people in the intersection of all three options. So we can work out that the number of people who like just Debonairs is $66–4–27-13=22$ (This is simply the total number who like Debonairs minus the number of people who like Debonairs and Steers, or Debonairs and Nandos or all three). We draw the following diagram to represent the data:
2. We are told that there were 100 people and that 94 liked at least one. So the number of people that liked none is: $100–94=6$ . This is the answer to a).
3. We can redraw the part of the Venn diagram that is of interest: Total people who like Nandos: 50
Of these 27 like both Nandos and Debonairs, and 4 people like all three options. So we find that the total number of people who just like Nandos is: $50–27–4=19$
Total people who like Steers: 40
Of these 13 like both Steers and Debonairs, and 4 like all three options. So we find that the total number of people who like just Steers is: $40–13–4=23$
Now use the identity $\mathrm{n\left(N or S\right)}=\mathrm{n\left(N\right)}+\mathrm{n\left(S\right)}–\mathrm{n\left(N and S\right)}$ to find the number of people who like Nandos and Steers, but not Debonairs.
$\begin{array}{ccc}\hfill \mathrm{n\left(N or S\right)}& =& \mathrm{n\left(N\right)}+\mathrm{n\left(S\right)}–\mathrm{n\left(N and S\right)}\hfill \\ \hfill 28& =& 23+19–\mathrm{n\left(N and S\right)}\hfill \\ \hfill \mathrm{n\left(N and S\right)}& =& 14\hfill \end{array}$
The Venn diagram with that represents all this information is given:

## Activity: venn diagrams

Which cellphone networks have you used or are you signed up for (e.g. Vodacom, Mtn or CellC)? Collect this information from your classmates as well. Then use the information to draw a Venn diagram (if you have more than three networks, then choose only the three most popular or draw Venn diagrams for all the combinations). Try to see if you can work out the number of people who use just one network, or the number of people who use all the networks.

#### Questions & Answers

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
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
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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