Chapter 8
Example Bobs Toothpaste Problem
Markov Processes
8.1 The Transition Matrix
1
Example Transition Diagram Bobs Toothpaste
Bobs Toothpaste company has 10% of the toothpaste
market.
It plans to conduct an aggressive sales campaign.
A research firm d
Chapter 8
Regular Stochastic Matrix
A stochastic matrix is said to be regular if some
power of the matrix has all positive entries.
(i.e. no entries are zero.)
Markov Processes
8.2 Regular Stochastic Matrices
1
Example Regular Stochastic Matrix
2
Stable M
Permutation
5.5 Permutations and Combinations
Given a set of n objects, a permutation of n
objects taken r at a time is an arrangement of r of
the n objects in a specific order.
The notation for a permutation is P(n,r)
1
2
Formula for P(n,r)
Example Permu
Chapter 5
Example Tossing Coins
An experiment consists of tossing a coin 10 times and
observing the sequence of heads and tails.
Sets and Counting
a. How many different outcomes are possible?
b. How many different outcomes have exactly two heads?
5.6 Furt
Chapter 6
Experiments, Trials and Outcomes
6.2 Experiments, Outcomes, and Events
An experiment is an activity with an observable
outcome.
Each repetition of the experiment is called a
trial.
In each trial we observe the outcome of the
experiment.
Probabil
Chapter 6
Conditional Probability
Let E and F be events is a sample space S. The
conditional probability Pr(E | F) is the probability
of event E occurring given the condition that
event F has occurred.
In calculating this probability, the sample space
is
Chapter 6
Probability of Equally Likely Outcomes
Let S be a sample space consisting of N equally likely
outcomes. Let E be any event. Then
Probability
6.4 Calculating Probabilities of Events
1
Example Equally Likely Outcomes #1
2
Example Equally Likely Ou
Chapter 6
Tree Diagram
A tree diagram helps us represent the various
events and their associated probabilities.
The various outcomes of each experiment are
represented as branches emanating from a point.
Each branch is labeled with the probability of the
Chapter 2
Example Applied Problem - Continued
The TVs-R-Us company produces three types of color television sets: model X,
Y and Z. The time (in hours) that it takes to create a TV is given in the matrix.
Elec
TVs-R-Us Problem Continued
X
Y
Z
(making sens
06/04/2015
Chapter 8
Absorbing State
A state for which all objects that start in that state,
stay in that state is called an absorbing state. That
is, an absorbing state is a state that always leads
back to itself.
A state is absorbing if
1. the correspon
Chapter 6
Mathematical Probability
Many events in the world exhibit a random
character.
Yet, by repeated observations of such events we
can often discern long-term patterns that persist
despite random, short-term fluctuations.
Probability is the branch of
Chapter 5
Binominal Coefficient
Another notation for C(n,r) is
Sets and Counting
.
is called a binominal coefficient.
5.7 The Binomial Theorem (Abridged)
1
Example Binominal Coefficients
2
Pascals Triangle
3
4
Number of Subsets
Example Dessert Time
In how
\ .
HDVQ \ :3 LKAUVM (Sim
. \ \
9 A3 \ D
O a
D
> : Mao
COMCEWQ gt
H 116 r~ cur H
00
CGVCOWQ dw'vx
Mm? cg muvx
f Dd <
1 (C510 0ch fwd) W/w (Qs a WWW,
/\ .
{We} :0 0ka 6%) <0 Aeqm (is WEWWUW.
/\
LAWN?" (W0}r Ma %S LE VCQAD 04
K {TCW'DNQ . If f(c) = 0,
Math 1251 Quiz 1
17 January, 2008
1. Find the equation of the line through the points (3, 4) and (7, 12)
Solution:
The slope is given by m = (12 4) / (7 3) = 8/4 = 2
If we use the point slope form with the point (3,4) we obtain
y 4 = 2 (x 3), which can be
Math 1251
Name_
Quiz 3 (5 February, 2008)
Consider the following Linear Programming problem:
Maximize the function C = x + 2y
Subject to :
x+ y 4
3y
x
x, y
0
What are the vertices of the feasible region ? You need not attempt to maximize the
function
So
Math 1251
Name_
Quiz 5 (Thursday, 21 February, 2008
1. A group of 6 men and 5 women wish to form a coed volleyball team, which
will require that there be 3 men and 3 women on the team. How many
different teams consisting of 3 men and 3 women can created f
Math 1251
Name_
Quiz 7 (Thursday, 13 March, 2008
A box contains 8 balls: 3 are red and 5 are white. Two balls are drawn from the box,
one after the other. Find :
(a) prob( the first ball drawn is red )
Solution:
There are C(8,1) = 8 ways to draw one ball
Chapter 2
2D Possible Solutions
Types of Solutions to Systems of
Equations
Consistent: a system that has at least one solution.
Inconsistent: a system that has no solution.
Dependent: a system that has an infinite number of solutions.
Independent: a syste