Download Probablity for General GRE

Probability for GRE (Level 4-5) /
Quantum CAT (Level 2) / GMAT
General GRE
Online Class By: Satyadhar Joshi
http://onlineclasses.nanotechbiz.org
Content
 Introduction to Probability
 Syllabus
 Type of questions
 Practice questions
 Conclusion
 References
Areas of Math often tested
Statistics (mean, mode, SD, range, ND, graphical
representation of ND)
Quadratic equations (roots, type of roots, number of roots,
positive and negative roots, etc.)
Series (AP, GP, series definition, nth term of a series, etc.)
Number theories (divisors, remainders, GCD, LCM, prime
factors, number line, etc.)
Probability (counting principle, basic probability, coin and
die tossing, arrangements, etc.)
Speed and work problems (relation between speed, distance
and time, rule of 3, rule of 5, etc.)
Some other concepts (ratios, inequalities, etc.)
Introduction
 We would be solving questions on probability from various
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sources
The toughest possible question will be solved
Target questions are 20-30 for the class with explanation and
discussion
Most comprehensive coverage and CAT level questions
All of three Exams easily covered
To get the list of question check my uploads or contact me at
[email protected]
Example: Flipping a coin
What’s the probability of getting heads when flipping a coin? Ans: There is only one way to get
heads in a coin toss. Hence, the top of the probability fraction is 1. There are two possible
results: heads or tails. Forming the probability fraction gives 1/2.
Example: Tossing a die
What’s the probability of getting a 3 when tossing a die? Ans: A die (a cube) has six faces,
numbered 1 through 6. There is only one way to get a 3. Hence, the top of the fraction is 1.
There are 6 possible results: 1, 2, 3, 4, 5, and 6. Forming the probability fraction gives 1/6.
Example: Drawing a card from a deck
What’s the probability of getting a king when drawing a card from a deck of cards? Ans: A deck of
cards has four kings, so there are 4 ways to get a king. Hence, the top of the fraction is 4.
There are 52 total cards in a deck. Forming the probability fraction gives 4/52, which reduces
to 1/13. Hence, there is 1 chance in 13 of getting a king.
Example: Drawing marbles from a bowl
What’s the probability of drawing a blue marble from a bowl containing 4 red marbles, 5 blue
marbles, and 5 green marbles? Ans: There are five ways of drawing a blue marble. Hence, the
top of the fraction is 5. There are 14 (= 4 + 5 + 5)possible results. Forming the probability
fraction gives 5/14.
Important Information you need to know
 Chess board is 8*8, selection of any block will use “C”
 Leap year has 366 days, and has 52 full weeks and 2 extra
days
 Considering things in one
 Conditional probability in drawing
Low scoring vs. High scoring exam
 Reduction of a problem in the closest option (CAT)
 Taking a go when you have more than 50% (not
recommended for GMAT and GRE because the questions are
easy)
Permutation
 Permutation: In mathematics, the notion of permutation is
used with several slightly different meanings, all related to
the act of permuting (rearranging in an ordered fashion)
objects or values. Informally, a permutation of a set of objects
is an arrangement of those objects into a particular order.
There are six permutations of the set {1,2,3}, namely
[1,2,3], [1,3,2], [2,1,3], [2,3,1], [3,1,2], and [3,2,1].
http://en.wikipedia.org/wiki/Permutation
Combination
 In mathematics a combination is a way of selecting several
things out of a larger group, where (unlike permutations)
order does not matter. In smaller cases it is possible to count
the number of combinations. For example given three fruit,
an apple, orange and pear say, there are three combinations of
two that can be drawn from this set: an apple and a pear; an
apple and an orange; or a pear and an orange.
http://en.wikipedia.org/wiki/Combination
Axiomatic Approach to Probability
Theorem
Definition
The sample space, denoted by , is the collection or totality of
all possible outcomes of a conceptual experiment.
Toss of a coin twice :
= {HH, HT, TH, TT}
Definition
An event, is a subset of the sample space. The class of all events
associated with a given experiment is defined to be the event
space. We usually denote the event space by F.
http://myweb.polyu.edu.hk/~majlee/AMA372/lec1_4.pdf
http://en.wikibooks.org/wiki/Probability/Introduction#Axiomatic_probability_theory
http://www.tutornext.com/axiomatic-approach-some-theorems-probability/1443
Conditional Probability
 Conditional probability is the probability of some event
A, given the occurrence of some other event B. Conditional
probability is written P(A|B), and is read "the (conditional)
probability of A, given B" or "the probability of A under the
condition B". When in a random experiment the event B is
known to have occurred, the possible outcomes of the
experiment are reduced to B, and hence the probability of
the occurrence of A is changed from the unconditional
probability into the conditional probability given B.
http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter4.pdf
http://en.wikipedia.org/wiki/Conditional_probability
http://en.wikipedia.org/wiki/Bayes%27_theorem
http://en.wikipedia.org/wiki/Bayesian_probability
Multiplication Theorem
The multiplication theorem is used to answer the following questions:
• What is the probability of two or more events occurring either simultaneously
or in succession?
• For two events A and B: What is the probability of event A and event B
occurring?
The individual probability values are simply multiplied to arrive at the answer. The
word
“and” is the key word that indicates multiplication of the individual probabilities. The
multiplication theorem is applicable only if the events are independent. It is not
valid
when dealing with conditional events. The product of two or more probability values
yields the intersection or common area of the probabilities. Mutually exclusive events
do not
have an intersection or common area. The probability of two or more mutually
exclusive
events is always zero.
For mutually exclusive events: • P(A) and P(B) = 0
For independent events: • Probability (A and B) = P(A) and P(B) = P(A) X P(B)
http://cqeweb.com/previews/chapter3_preview.pdf
Binomial Theorem in Probability
A binomial experiment is one that possesses the following
properties:
1. The experiment consists of n repeated trials;
2. Each trial results in an outcome that may be classified as a success
or a failure (hence the name, binomial);
3. The probability of a success, denoted by p, remains constant from
trial to trial and repeated trials are independent.
The number of successes X in n trials of a binomial experiment is
called a binomial random variable.
The probability distribution of the random variable X is called a
binomial distribution, and is given by the formula:
P(X) = Cnxpxqn−x where
Cnx is a combination
http://www.amscopub.com/%5Cimages%5Cfile%5CFile_671.pdf
In mathematics a combination is a way of selecting
several things out of a larger group, where (unlike
permutations) order does not matter. In smaller cases it
is possible to count the number of combinations. For
example given three fruit, an apple, orange and pear say,
there are three combinations of two that can be drawn
from this set: an apple and a pear; an apple and an
orange; or a pear and an orange. More formally a kcombination of a set S is a subset of k distinct elements
of S. If the set has n elements the number of kcombinations is equal to the binomial coefficient
which can be written using factorials as
whenever
, and which is zero when k > n.
Bayes Theorem
Thomas Bayes addressed both the case of discrete probability distributions of
data and the more complicated case of continuous probability distributions. In the
discrete case, Bayes' theorem relates the conditional and marginal probabilities of
events A and B, provided that the probability of B does not equal zero:
In Bayes' theorem, each probability has a conventional name:

P(A) is the prior probability (or "unconditional" or "marginal" probability)
of A. It is "prior" in the sense that it does not take into account any
information about B; however, the event B need not occur after event A. In
the nineteenth century, the unconditional probability P(A) in Bayes's rule
was called the "antecedent" probability;[3] in deductive logic, the
antecedent set of propositions and the inference rule imply consequences.
The unconditional probability P(A) was called "a priori" by Ronald A. Fisher.

P(A|B) is the conditional probability of A, given B. It is also called the
posterior probability because it is derived from or depends upon the
specified value of B.

P(B|A) is the conditional probability of B given A. It is also called the
likelihood.

P(B) is the prior or marginal probability of B, and acts as a normalizing
constant.
Conditional Probability
Conditional probability is the probability of some event A, given
the occurrence of some other event B. Conditional probability
is written P(A|B), and is read "the probability of A, given B". It is
defined by
If P(B) = 0 then
is undefined.
Addition Rule for Probability
If either event A or event B or both events occur on a single performance of an
experiment this is called the union of the events A and B denoted as
two events are mutually exclusive then the probability of either occurring is
. If
For example, the chance of rolling a 1 or 2 on a six-sided die is
If the events are not mutually exclusive then
For example, when drawing a single card at random from a regular deck of cards,
the chance of getting a heart or a face card (J,Q,K) (or one that is both) is
, because of the 52 cards of a deck 13 are hearts, 12 are face
cards, and 3 are both: here the possibilities included in the "3 that are both" are
included in each of the "13 hearts" and the "12 face cards" but should only be
counted once.
Questions A
 Quantum CAT Level 2 type
question (similar questions
with different data and
language)
Questions B
 GRE Nova Math
Bible (similar
questions with
different data and
language)
 (Buy the book which
is requisite for this
class at
http://novapress.ne
t/)
 Without getting this
book you are not
supposed to land in
my class
Conclusion
 All problems of all level illustrated
 Helpful in all major exams
 Email me at [email protected] for any doubts
 Do register for future classes
 http://onlineclasses.nanotechbiz.org/
Getting Details about my future
Classes
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References
 GRE Math Bible Nova
 GMAT Nova Bible
 Arihant Quantum CAT For Admission into IIMs
 Quantitative CAT Arun Sharma, TMH