Relations
Pamela Leutwyler
definition: Let A and B be sets. A binary relation from A to B is a subset of AxB.
example 1: Suppose A is a set of students cfw_Al, Beth, Carl, Donna, Ed
and B is the set of courses cfw_ French, Geology, History, Interpersonal
Pamela Leutwyler
An experiment is a procedure that yields one of a given set of results.
The possible results are called outcomes.
The set of all outcomes is called the sample space.
An event is a subset of the sample space.
An experiment is a procedure t
Pamela Leutwyle
example 1
An urn contains 10 marbles. 3 are red and 7 are blue.
Draw 3 without replacement.
What is the probability that they are all red ?
example 1
An urn contains 10 marbles. 3 are red and 7 are blue.
Draw 3 without replacement.
What is
Conditional Probability
Pamela Leutwyler
The notation p( B | A ) represents the probability that event B occurs IF event A occurs,
sometimes phrased the probability that event B occurs GIVEN THAT event A occurs.
definition: Let E and F be events with p (F
Finite Probability
Pamela Leutwyler
definition: If S is a finite nonempty sample space of equally likely outcomes,
and E is an event, that is, a subset of S, then the probability of E is p(E) =
|E|
|S|
example 1 An urn contains 5 red and 4 blue marbles. W
PRACTICE FINAL EXAM for DISCRETE MATHEMATICS solutions
1. Make a truth table for the statement (p r) (r p)
p
T
T
F
r
T
F
T
(p r) (r p)
T
T
T
F
F
T
2. Which of the following is equivalent to the statement p r ?
a.)p r,
b.)(r p),
c.) p r,
3. Determine the o
Complements and Unions of Events
Pamela Leutwyler
Roll a single die. S = cfw_ 1, 2, 3, 4, 5, 6
E = theeventthatthedielandsonanevennumber.E = cfw_2,4,6
E = theeventthatthediedoesNOTlandonanevennumber.E = cfw_1,3,5
theorem1: LetEbeaneventinasamplespaceS.
T
What is the probability that
AT LEAST
2 of the 4 dice
land on the same number
Roll 4 dice
6
x
6
x
6
x
6
= 1296 possible outcomes
What is the probability that at least two land on the same number?
Some of the possible outcomes:
2
5
4
1
3
1
.
.
3
2
4
6
4
2
Independence
Pamela Leutwyler
H
*
C
*
*
*
*
*
*
*
*
*
In this example, the probability that a woman
develops heart disease if she smokes is twice
the probability that she develops heart disease.
It implies that smoking causes heart disease.
*
P(H) = .25
P
Math 250 Exam #2 Key (Spring 2016)
1 Newtons Law of Cooling states that T 0 (t) = k[T (t) M ], where M is the temperature of
the oven. Here we have T (0) = 70, T (0.5) = 120, and T (1) = 160. Now,
Z
Z
dT
0
= k dt ln |T M | = kt + c M T = ekt+c ,
T = k(T M
Math 250 Exam #1 Key (Spring 2016)
1 The initial-value problem
(9 y 2 )y 0 = x2 ,
y(x0 ) = y0 ,
will have a unique solution if
x2
2x2 y
and
f
(x,
y)
=
y
9 y2
(9 y 2 )2
are continuous on some open rectangle containing (x0 , y0 ). This will be the case for
Math 250 Exam #4 Key (Spring 2016)
1 The IVT is
y 00 + 10y 0 + 16y = 0, y(0) = 1, y 0 (0) = 12.
The auxiliary equation r2 + 10r + 16 = 0 has roots 8 and 2, and so the general solution to
the ODE is
y(t) = c1 e2t + c2 e8t .
With the initial conditions we f
Math 250 Exam #2 Key (Spring 2015)
1 Newtons Law of Cooling states that T 0 (t) = k[T (t) Ta ]. Here we have Ta = 5, T (1) = 55,
and T (5) = 30. Now, noting that T (t) 5 for all t 0,
Z
Z
1
0
T = k(T 5)
dT = k dt ln |T 5| = kt + c T 5 = ekt+c ,
T 5
and so
Math 250 Exam #1 Key (Spring 2017)
1 We have
y0 = p
xy
, y(x0 ) = y0 ,
64
which is an IVP that will have a unique solution if
xy
f (x, y) = p
y 2 64
y2
and
fy (x, y) =
(2y 2 y 128)x
(y 2 64)3/2
are both continuous on an open rectangle containing (x0 , y0
Math 250 Exam #3 Key (Spring 2015)
1a We have y 00 + y 0 + 4y = et + et .
1
15
Auxiliary equation: r + r + 4 = 0; roots:
i.
2
2
Start with equation y 00 + y 0 + 4y = et , with nonhomogeneity f (t) = et . Since = 1 is not
a root of the auxiliary equation,
Math 250 Exam #1 Key (Spring 2015)
1 Substitute (x) = emx for y to obtain
2m2 emx + 7memx 4emx = 0 2m2 + 7m 4 = 0 (2m 1)(m + 4) = 0,
so m cfw_1/2, 4, and therefore 1 (x) = ex/2 and 2 (x) = e4x are solutions to the ODE.
(Note that y 0 is a solution also.)
Math 250 Exam #3 Key (Spring 2016)
1 Put equation in standard form: y 00 + 2t1 y 0 6t2 y = 0, so P (t) = 2/t. Were given that
y1 (t) = t2 is a solution. From this we obtain
Z R P (t) dt
Z 2 ln |t|
Z
e
e
1
1
2
2
y2 (t) = y1 (t)
dt = t
dt = t
dt = t2 15 t5
Math 250 Exam #4 Key (Spring 2015)
1 We have
X
n(n 1)cn x
n2
5
n=2
X
cn x
n+2
X
X
n
=
(n + 1)(n + 2)cn+2 x 5
cn2 xn
n=0
n=0
n=2
= 2c2 + 6c3 x +
= 2c2 + 6c3 x +
X
n=2
X
n
(n + 1)(n + 2)cn+2 x
(n + 1)(n + 2)cn+2 5cn2 xn .
2 Substituting
y=
cn x
n
0
and
y =
Section 2.3
Mean : (Average) the sum of all values in a sample, divided by the number of value
Sample Mean
Population Mean
calculate by using sample data
calculate by using ALL values in the population.
You try
Example 3-2 Hospital Infections
The data
NAME:_SS#:_
WORKSHEET FOR FAFSA QUESTIONS 28, 29, 30, & 48
ACADEMIC YEAR 2016-2017
Question # 28
Will you have your first bachelors degree before July 1, 2016? Please circle your
answer.
Yes
No
Question # 29
Please indicate (circle) what your grade level
Chapter 1
Data information stemming from observations, numerations,
measurements, or survey responses
Statistics collecting, organizing, analyzing, and interpreting data in
order to arrive at a conclusion
Population collection of ALL possible outcomes,
Bucks County Community College
20162017 Verification Worksheet
Dependent Student - V1
Your 20162017 Free Application for Federal Student Aid (FAFSA) was selected for review in a process called verification.
The law says that before awarding Federal Studen
Name _Dhruv Patel_
Class _MOS 5_
Date _3/17/15_
Quarterly Review 3 (Practice)
1. What happens when you finish accepting/rejecting tracked changes from a
reviewer?
a. The accepted changes are incorporated into the original document
b. The change balloons r
FOR THE DAYS FEB. 1 12
Name:
Period:
My February
JournalPart
#1!
Punxsutawney Phil
Gus.the 2nd most famous groundhog in PA
Keep on scratchin!
Monday, February 1, 2016
3 Full Sentences [Answer in full sentence form]
Last Monday, we had off from school beca
Name _Dhruv Patel_
Class _MOS_
Date _3/13/15_
Chapter 5 Review (Practice)
1. You use the Customize dialog box to assign macros to the Quick Access toolbar.
a. True
b. False
2. Macro names can contain spaces.
a. True
b. False
3. You can assign macros to sh
FOR THE DAYS FEB. 16 - 29
You have a day off in our
honor.hope you enjoyed
your holiday!
Tuesday, February 16, 2016
Please answer in full sentence form
This past weekend, we celebrated Valentines Day.
1. Who would you like to be your Valentine?
2. What gi
Pareto Charts
Used to represent a frequency distribution for a categorical variable.
Frequencies are displayed by heights of vertical bars, arranged from
highest to lowest.
Time Series Graph
Represents data that occurs over a specific period of time
Pie
Section 2-1
Frequency Distribution is the organization of data in a table form
using the frequencies of the data
- Categorical Frequency Distributions are used for data that can
be placed is specific categories such as nominal or ordinal
- With frequency