# 2.2 Problems on minterm analysis

Consider the class $\left\{A,\phantom{\rule{0.166667em}{0ex}}B,\phantom{\rule{0.166667em}{0ex}}C,\phantom{\rule{0.166667em}{0ex}}D\right\}$ of events. Suppose the probability that at least one of the events A or C occurs is 0.75 and the probability that at least one of the four events occurs is 0.90.Determine the probability that neither of the events A or C but at least one of the events B or D occurs.

Use the pattern $P\left(E\cup F\right)=P\left(E\right)+P\left({E}^{c}F\right)$ and ${\left(A\cup C\right)}^{c}={A}^{c}{C}^{c}$ .

$P\left(A\cup C\cup B\cup D\right)=P\left(A\cup C\right)+P\left({A}^{c}{C}^{c}\left(B\cup D\right)\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{so}\phantom{\rule{4.pt}{0ex}}\text{that}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left({A}^{c}{C}^{c}\left(B\cup D\right)\right)=0.90-0.75=0.15$
1. Use minterm maps to show which of the following statements are true for any class $\left\{A,\phantom{\rule{0.166667em}{0ex}}B,\phantom{\rule{0.166667em}{0ex}}C\right\}$ :
1. $\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}A\cup {\left(BC\right)}^{c}=A\cup B\cup {B}^{c}{C}^{c}$
2. $\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\left(A\cup B\right)}^{c}={A}^{c}C\cup {B}^{c}C$
3. $\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}A\subset AB\cup AC\cup BC$
2. Repeat part (1) using indicator functions (evaluated on minterms).
3. Repeat part (1) using the m-procedure minvec3 and MATLAB logical operations.

We use the MATLAB procedure, which displays the essential patterns.

minvec3 Variables are A, B, C, Ac, Bc, CcThey may be renamed, if desired. E = A|~(B&C); F = A|B|(Bc&Cc); disp([E;F]) 1 1 1 0 1 1 1 1 % Not equal1 0 1 1 1 1 1 1 G = ~(A|B);H = (Ac&C)|(Bc&C); disp([G;H]) 1 1 0 0 0 0 0 0 % Not equal0 1 0 1 0 1 0 0 K = (A&B)|(A&C)|(B&C); disp([A;K]) 0 0 0 0 1 1 1 1 % A not contained in K0 0 0 1 0 1 1 1

Use (1) minterm maps, (2) indicator functions (evaluated on minterms), (3) the m-procedure minvec3 and MATLAB logical operations to show that

1. $\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}A\left(B\cup {C}^{c}\right)\cup {A}^{c}BC\subset A\left(BC\cup {C}^{c}\right)\cup {A}^{c}B$
2. $\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}A\cup {A}^{c}BC=AB\cup BC\cup AC\cup A{B}^{c}{C}^{c}$

We use the MATLAB procedure, which displays the essential patterns.

minvec3 Variables are A, B, C, Ac, Bc, CcThey may be renamed, if desired. E = (A&(B|Cc))|(Ac&B&C); F = (A&((B&C)|Cc))|(Ac&B); disp([E;F]) 0 0 0 1 1 0 1 1 % E subset of F0 0 1 1 1 0 1 1 G = A|(Ac&B&C); H = (A&B)|(B&C)|(A&C)|(A&Bc&Cc); disp([G;H]) 0 0 0 1 1 1 1 1 % G = H0 0 0 1 1 1 1 1

Minterms for the events $\left\{A,B,C,D\right\}$ , arranged as on a minterm map are

0.0168 0.0072 0.0252 0.0108 0.0392 0.0168 0.0588 0.02520.0672 0.0288 0.1008 0.0432 0.1568 0.0672 0.2352 0.1008

What is the probability that three or more of the events occur on a trial? Of exactly two? Of two or fewer?

We use mintable(4) and determine positions with correct number(s) of ones (number of occurrences). An alternate is to use minvec4 and express theBoolean combinations which give the correct number(s) of ones.

npr02_04 Minterm probabilities are in pm. Use mintable(4) a = mintable(4);s = sum(a); % Number of ones in each minterm position P1 = (s>=3)*pm' % Select and add minterm probabilities P1 = 0.4716P2 = (s==2)*pm' P2 = 0.3728P3 = (s<=2)*pm' P3 = 0.5284

Minterms for the events $\left\{A,B,C,D,E\right\}$ , arranged as on a minterm map are

0.0216 0.0324 0.0216 0.0324 0.0144 0.0216 0.0144 0.0216 0.0144 0.0216 0.0144 0.0216 0.0096 0.0144 0.0096 0.01440.0504 0.0756 0.0504 0.0756 0.0336 0.0504 0.0336 0.0504 0.0336 0.0504 0.0336 0.0504 0.0224 0.0336 0.0224 0.0336

What is the probability that three or more of the events occur on a trial? Of exactly four? Of three or fewer? Of either two or four?

We use mintable(5) and determine positions with correct number(s) of ones (number of occurrences).

npr02_05 Minterm probabilities are in pm. Use mintable(5) a = mintable(5);s = sum(a); % Number of ones in each minterm position P1 = (s>=3)*pm' % Select and add minterm probabilities P1 = 0.5380P2 = (s==4)*pm' P2 = 0.1712P3 = (s<=3)*pm' P3 = 0.7952P4 = ((s==2)|(s==4))*pm' P4 = 0.4784

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive