



Consider the class
$\{A,\phantom{\rule{0.166667em}{0ex}}B,\phantom{\rule{0.166667em}{0ex}}C,\phantom{\rule{0.166667em}{0ex}}D\}$ 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
$P(E\cup F)=P\left(E\right)+P\left({E}^{c}F\right)$ and
${(A\cup C)}^{c}={A}^{c}{C}^{c}$ .
$$P(A\cup C\cup B\cup D)=P(A\cup C)+P\left({A}^{c}{C}^{c}(B\cup D)\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{so}\phantom{\rule{4.pt}{0ex}}\text{that}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left({A}^{c}{C}^{c}(B\cup D)\right)=0.900.75=0.15$$
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 Use minterm maps to show which of the
following statements are true for any class
$\{A,\phantom{\rule{0.166667em}{0ex}}B,\phantom{\rule{0.166667em}{0ex}}C\}$ :

$$\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}A\cup {\left(BC\right)}^{c}=A\cup B\cup {B}^{c}{C}^{c}$$

$$\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{(A\cup B)}^{c}={A}^{c}C\cup {B}^{c}C$$

$$\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}A\subset AB\cup AC\cup BC$$
 Repeat part (1) using indicator functions (evaluated on minterms).
 Repeat part (1) using the mprocedure 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 = AB(Bc&Cc);
disp([E;F])
1 1 1 0 1 1 1 1 % Not equal1 0 1 1 1 1 1 1
G = ~(AB);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
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Use (1) minterm maps, (2) indicator functions (evaluated on minterms),
(3) the mprocedure minvec3 and MATLAB logical operations to show that

$$\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}A(B\cup {C}^{c})\cup {A}^{c}BC\subset A(BC\cup {C}^{c})\cup {A}^{c}B$$

$$\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}A\cup {A}^{c}BC=AB\cup BC\cup AC\cup A{B}^{c}{C}^{c}$$
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&(BCc))(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
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Minterms for the events
$\{A,B,C,D\}$ , 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
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Minterms for the events
$\{A,B,C,D,E\}$ , 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
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Questions & Answers
Introduction about quantum dots in nanotechnology
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?
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it is a goid question and i want to know the answer as well
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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?
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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
what's the easiest and fastest way to the synthesize AgNP?
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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AMJAD
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
Source:
OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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