<< Chapter < Page Chapter >> Page >

Probabilities of boolean combinations

As in the nonindependent case, we may utilize the minterm expansion and the minterm probabilities to calculate the probabilities of Boolean combinations of events. However,it is frequently more efficient to manipulate the expressions for the Boolean combination to be a disjoint union of intersections.

A simple boolean combination

Suppose the class { A , B , C } is independent, with respective probabilities 0.4, 0.6, 0.8. Determine P ( A B C ) . The minterm expansion is

A B C = M ( 3 , 4 , 5 , 6 , 7 ) , so that P ( A B C ) = p ( 3 , 4 , 5 , 6 , 7 )

It is not difficult to use the product rule and the replacement theorem to calculate the needed minterm probabilities. Thus p ( 3 ) = P ( A c ) P ( B ) P ( C ) = 0 . 6 0 . 6 0 . 8 = 0 . 2280 . Similarly p ( 4 ) = 0 . 0320 , p ( 5 ) = 0 . 1280 , p ( 6 ) = 0 . 0480 , p ( 7 ) = 0 . 1920 . The desired probability is the sum of these, 0.6880.

As an alternate approach, we write

A B C = A A c B C , so that P ( A B C ) = 0 . 4 + 0 . 6 0 . 6 0 . 8 = 0 . 6880

Considerbly fewer arithmetic operations are required in this calculation.

Got questions? Get instant answers now!

In larger problems, or in situations where probabilities of several Boolean combinations are to be determined, it may be desirable to calculate all minterm probabilities then use theminterm vector techniques introduced earlier to calculate probabilities for various Boolean combinations. As a larger example for which computational aid is highly desirable, consideragain the class and the probabilities utilized in [link] , above.

Consider again the independent class { E 1 , E 2 , , E 10 } with respective probabilities { 0 . 13 0 . 37 0 . 12 0 . 56 0 . 33 0 . 71 0 . 22 0 . 43 0 . 57 0 . 31 } . We wish to calculate

P ( F ) = P E 1 E 3 ( E 4 E 7 c ) E 2 ( E 5 c E 6 E 8 ) E 9 E 10 c

There are 2 10 = 1024 minterm probabilities to be calculated. Each requires the multiplication of ten numbers. The solution with MATLAB is easy, as follows:

>>P = 0.01*[13 37 12 56 33 71 22 43 57 31];>>minvec10 Vectors are A1 thru A10 and A1c thru A10cThey may be renamed, if desired.>>F = (A1|(A3&(A4|A7c)))|(A2&(A5c|(A6&A8)))|(A9&A10c);>>pm = minprob(P);>>PF = F*pm' PF = 0.6636

Writing out the expression for F is tedious and error prone. We could simplify as follows:

>>A = A1|(A3&(A4|A7c));>>B = A2&(A5c|(A6&A8));>>C = A9&A10c;>>F = A|B|C; % This minterm vector is the same as for F above

This decomposition of the problem indicates that it may be solved as a series of smaller problems. First, we need some central facts about independence ofBoolean combinations.

Got questions? Get instant answers now!

Independent boolean combinations

Suppose we have a Boolean combination of the events in the class { A i : 1 i n } and a second combination the events in the class { B j : 1 j m } . If the combined class { A i , B j : 1 i n , 1 j m } is independent, we would expect the combinations of the subclasses to be independent. It is importantto see that this is in fact a consequence of the product rule, for it is further evidence that the product rule has captured the essence of the intuitive notion of independence.In the following discussion, we exhibit the essential structure which provides the basis for the following general proposition.

Proposition . Consider n distinct subclasses of an independent class of events. If for each i the event A i is a Boolean (logical) combination of members of the i th subclass, then the class { A 1 , A 2 , , A n } is an independent class.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
Samson Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Applied probability' conversation and receive update notifications?

Ask