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 in n ways,
then
the complete task can be performed in
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! = 1. The factorial of zero is one.
For example, 5! =
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 or B (or both).
In other words, the union of two sets
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 the event E occurs, given that the event F has
already
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 only if:
P(E|F) = P(E) and P(F|E) = P(F)
By definition,
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.
Probability of zero means impossible, and probability of
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 compute the probability of an event that results from per
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 certain number of regions. (see textbook figure 1 on
page 1
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=2: 20
n=3: 30
n=4: 40
n=5: 50
n
r
.
and look at the
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, then n(S) = 3.
If S = then n(S) = 0. The empty set cont
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 (S T )
(S T ) = cfw_4
R (S T ) =
b) R S T 0
R S T 0