Probability
Originate from the study of games of chance and gambling during the
16th century.
Rank of Poker Hands
God of gamblers
http:/www.pagat.com/vying/pokerrank.html
Probability
Originate from the study of games of chance and gambling during the
16

T1S/MATH1013/2015-16/2nd
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1013 University Mathematics II
Tutorial 1 Solution
1. (In general, there can be more than one way to describe the same set with the set-builder notation.)
(a) cfw_n Z : n <

1
Ch6/MATH1013/YMC/2015-16/2nd
Chapter 6. Integration
6.1. The Fundamental Theorem of Calculus
The principal theorem of this section is the Fundamental Theorem of Calculus, which is the central
theorem of integral calculus. It provides a connection betwee

1
Ch5/MATH1013/YMC/2015-16/2nd
Chapter 5. Exponential and Logarithmic Functions
5.1. Review
Exponential Functions
An exponential function is a function of the form
f (x) = ax
where a > 0 and a 6= 1 is a constant. The number a is called the base of the ex

1
Ch4/MATH1013/YMC/2015-16/2nd
Chapter 4. Dierentiation and Its Applications
4.1. Review of derivatives
Dierence quotient and Derivatives
The dierence quotient of a function f (x) at x = a with increment h is given by
f (a + h)
h
f (a)
and the derivative

1
Ch3/MATH1013/YMC/2015-16/2nd
Chapter 3. Limits and Continuity
3.1. Limits and Continuity
The concept of a limit lies at the foundation of calculus. The idea involves the notion of getting
closer and closer to something, but yet not touching it. Limits a

1
Ch2/MATH1013/YMC/2015-16/2nd
Chapter 2. Functions
2.1. Functions and Graphs
The idea of a function expresses the dependence between two quantities, one of which is given and
the other is the output. A function associates a unique output with every input

MATH1013 University Mathematics II
Dr. Yat-Ming Chan
Department of Mathematics
The University of Hong Kong
Second Semester 2015-16
(Class 2C and 2D)
Content Outline
1. Pre-Calculus Topics
Functions and Graphs, Composite and Inverse Functions, Limits and C

1
Ch7/MATH1013/YMC/2015-16/2nd
Chapter 7. Matrices and Determinants
7.1. Matrix Arithmetic and Operations
This section is devoted to developing the arithmetic of matrices. We will see some of the differences
between arithmetic of real numbers and matrices

Independence
Irrelevant information
Based on a new information, we can update the sample space and
events to be new ones, so that we can calculate a conditional probability
of the updated event.
However, sometimes, the new information may be irrelevant.
T

Conditional probability
Probability of an event with a finite sample space
If the sample space for an experiment contains
finite, say N, elements,
all of which are equally likely to occur,
then the probability of any event A, denoted by P(A),
containing

Probability of an event
with a finite sample space having
equally likely outcomes
Example
Consider the experiment of tossing a coin. Suppose that a
head is as likely to appear as a tail. Then what is the
probability that a head occurs?
Denote the probabil

Circular permutation
There are n! different permutations of n distinct
(
objects. arranged in a line ).
How about other kinds of arrangement?
Say, if the people are randomly arranged in a
circle, how many different ways do we have?
There are 5! different

Counting sample points
What is the number of different ways to order distinct objects?
For example, suppose that there are 5 people, say
A B C D E
Now, we want to arrange them in a row randomly,
then how many ways are possible?
Consider the first seat. Wh

Review (2)
Probability
be a measure of the possibility that a random phenomenon occurs,
to find the regularity of these random phenomena.
Basics of probability theory
Sample space
Complement
Element (or sample point)
Intersection: Disjoint, mutually

T2/MATH1013/2015-16/2nd
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1013 University Mathematics II
Tutorial 2 (Feb 15 19)
1. Let f : R R be defined by f (x) = dxe where dxe denotes the smallest integer greater
than or equal to x. This functi