Permutations and Combinations
Permutations and Combinations
The factorial of a positive integer, n, is denoted by n! and it is given by:
n! = (n)(n-1)(n-2). . . .(2)(1)
Important, by definition, 0!
Operations on Sets (Union, Intersection)
The Union of Two Sets (Definition)
The union of A and B, written A B and pronounced A union B, is defined as:
The set of all elements that belong to either A
Conditional Probability
Conditional Probability
Consider two events E and F.
The probability P(E | F) is read the probability of event E given F.
The conditional probability is the probability that
Independence
Independence
Two events E and F are said to be independent if:
The probability that event E occurs is not affected by the occurrence of F.
The events E and F are independent if and onl
Calculating Probabilities of Events
Basic Concepts
Probability: It is a numerical measure of the likelihood that a specific event will occur.
The probability can take any value from zero to one.
Pr
Tree Diagrams
Tree Diagrams
In solving many probability problems, it is helpful to represent the various events and their associated
probabilities by a tree diagram.
The tree diagram helps you to co
Venn Diagrams and Counting
Venn Diagrams
Use the Venn Diagram to solving counting problems. They are specially useful in analyzing survey data.
The Venn Diagram divides the universal set into a cert
Medullary ray (botany)
From Wikipedia, the free encyclopedia
Transverse section of white oak, Quercus robur. A ray appears diagonally, from top left to
bottom middle.
Medullary rays are cellular struc
Photosynthesis in a Jar
5E Lesson Plan, Author: Amber Palmeri-Miles, Yakima WATERS Project, CWU, Fall 2010
This lesson is intended to teach students about the process of photosynthesis through
backgro
Multiplication Principle
The Multiplication Principle
Suppose that a task is composed of two consecutive choices.
If choice 1 can be performed in m ways, for each of these, choice 2 can be performed
The Binomial Theorem
The Binomial Theorem
An alternative notation to the formula of combination, C(n, r), is
You read nr as n choose r.
The symbol, nr is called a binomial coefficient.
Consider
n
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Line. a, ne (a OLSETOF hiberno.
iOLOuwed ne or agimjgm \Ene. _ --
..%9\m.cs-\ glam}: ,prwlasmmumerwysq .
3Q \ine mull always mean. c1s+rtngm ne.
:9 \me mnbe on 0h\imi
Operation on Sets (Exercises)
Let U=cfw_1,2,3,4,5, R=cfw_1,3,5, S=cfw_3,4,5, and T=cfw_2,4. List the elements of the following sets:
a) R S T
R S T = (R S) T
(R S) = cfw_3,5
(R S) T =
R S T = R (
The Inclusion-Exclusion Principle, De Morgans Laws
The Inclusion-Exclusion Principle (Definition)
If S is any set, we will denote the number of elements in S by n(S).
For example, if S=cfw_1,7,11, t