Download Math 1312 – Test II review

Math 1312 – Test II review Old Material – Review
1. Fill in the blank - definitions
The set that contains all objects in A or in B or in Both A and B is called ________________________________
The complement of A is the set that contains all objects in U that ________________________________
Two sets A and B are _________________________ if P(A  B ) = P(A )P(B)
A ________________________ experiment has only two outcomes.
A Binomial experiment where n = 5 and p =-.4 has a r.v. X . List all the values of X : ___________________________
The mean of a binomial r.v. is given by  = ______________ and its variance by ________________
A ___________________________________ curve has a mean of 0 and a standard deviation of 1
An inflection point of a normal curve with mean  and standard deviation s is found at _________ or at ___________
The area under a normal curve consists of ________ unit
2. Some sets : shade the set A  B/
3. Some counting :
a) 30 students passed exam I, 20 students passed exam : if the same 35 students took both exams then how many
passed both exams? __________________, how many passed neither exam ? _________
b) A sample of 50 students:
30 enrolled in a math class
20 enrolled in a science class
25 enrolled in a business class
___ enrolled in a math class and a sc. class
___ enrolled in a math class and a business class
_____ enrolled in a sc. class and a business class
______ enrolled in a math class but not
4. Some probabilities
Basic:
a) A card is drawn . What is the probability that the card is an ace or a king ? ___________
b) A coin is tossed four times. What is the probability that three heads come up ? __________
c) a die is rolled four times. What is the probability that you get three sixes ? _____________
Basic-Hard
a) Five cards are drawn from a standard deck of cards. What is the probability that
1)
all cards are diamonds ?
2) your hand consists of 2 aces and three kings ?
3) your hand consists of exactly three aces ?
How many five-card hands are possible (different ) ?
b) A club consists of 20 members; 8 female and 12 male members. three of the female members are under 30 and 5 of the
male members are under 30.
1) one person is selected at random. What is the probability that the person is under 30 ? ___________
under 30 if they are known to be female ? _______________
2) Four members are selected at random. What is the probability that
all of them are female? _____________
exactly 3 are female ? ______________
none are under 30 if all are known to be male ? ________________________
c) Let A and B be event with
probabilities )
P(A ) = 0.8
and P( B ) = 0.5 ( answer the question with respect to this situation – these
could A and B be mutually exclusive ? Why or why not ? If yes, tell under what conditions will they be mutually
exclusive ?
could A and B independent ? If yes, tell under what conditions will they be independent.
d) Suppose X is a binomial r.v. with n = 60 and p = 0.3
list all of the values of X : ______________________ find expected value of X = ___________________
Find the variance of X :
Other questions:
1. List the three types of r.v.: _____________________ _____________________-- _______________________
2. classify the type of r.v. in each of the following cases.
a) let the r.v. X represent the number of questions that you will answer correctly in a 10 problem quiz.
list the values of X and classify X.
b) Let the r.v. Y represent the number of cars you will test drive before you find one that you like.
List the values of Y and classify Y.
c) Let the r.v. Z represent the exact amount of oxygen that a person can use in terms of cubic inches during the next 50
seconds. List the values of Z and classify Z
3. Use the following r.v. and its distribution to answer the questions that follow
X=x
-2
-1
0
1
P( X = x )
0.1
0.2
0.4
0.3
a) find the expected value, the variance, and the standard deviation
b) sketch a histograph of the r.v.
c) Find P (X > 0 ) = __________
P (X =1.5 ) = _____________
4. Find the probability distribution of the binomial r.v. with n = 2 and p=0.2
5. A loaded coin is tossed three times. If the probability of a head coming up is 0.7, then what is the probability that in
five tosses you will get exactly three heads.
6. A company is to select a chairman for each of three committees. It is known that a man is three times more likely to be
chosen than a woman.
What is the probability that all three chairmen are women ? _____________
Exactly two are women ? _______________
At least one is a man ? ____________
7. Life insurance problem: see examples from class ( quiz )
8. Game problem: see example from class
9. Find the area to the left of 20 under a normal curve with mean 25 and variance = 4.
Should know
11. What is the standard deviation of a standard normal curve ? _________________
12.
A doctor finds that a person with a dark skin condition leads to skin cancer in 1 out of 50 cases.
The doctor examines 200 patients during the year.
What is the probability that
4 of them will have cancer ?
HOw many patients are expected to have cancer ?
13. A person drives down the road and passes five lights. The probability that a light is red is 0.2. What is the probability t
that he gets all red lights ?
at least one red light ?
exactly two red lights ?
____________________________
Name
Math 1312 - Exam II
March 25, 2002
Total Absences = _____________
Absences since last exam = __________
HW Bonus on Exam I = __________
1. What is the difference between mutually exclusive events
and independent events ? or is there any difference ?
HW Bonus on Exam II = __________
Missed points on exam II ________
Grade on Exam II = ___________
2. Can A  B ever be a subset of A  B ? If so, state when .
3. Complete the formula for two events that are not mutually exclusive:
P(A  B ) = ____________________________________________
4. A visiting group is to interview 1 member from each of the four colleges; College of Sciences ( 80
members), College of Business( 60 members), College of Professional Studies(100), and College
of Liberal Arts(60). How many different groups of four can be selected ?
A group of 12 is selected at random from the entire university to help the visiting committee.
What is the probability that 6 are from the College of Sc., 4 are from the College of Bus., and 1 is
from the College of P. S. ?
5. Ten models are interviewing for a magazine feature. Four are to be selected. Each one will represent
one of the four seasons of the year. How many different selections are possible ?
If 3 come from the Midwest , 2 from the east, 2 from the southwest, and 3 from the westcoast, then
what is the probability that one is selected from each region ?
6.
A review of past records show that a student that takes Exam I in a math 1312 class will have a
probability of 0.8 of passing the exam.
Further review of such classes show that
based on previous exams: if a student passes an exam, the probability that he/she will pass the
next exam is 0.7
if the student fails an exam, the probability that he/she will pass the
next exam is 0.5
Use the following matrices to find the probability that a student will pass the second exam.
[ 0.8
____ ]
| 0.7 _______
|
| 0.5 _______
P ( the student will pass the second exam ) = __________________________
8.
Where are the inflection points of a normal curve with mean 500 and variance = 49 ?
10. An insurance company sells $100,000 policies for $600/year. The probability of death of a person in
this category is 0.008
Find the expected value of this policy in terms of the company.
If the insurance sells 1000 such policies, what is the expected value of all the policies ? _________
11.
Find the mean, median, mode of the following sets of values
a) 2, 1, 0, 2  median ________ ,
mode _______,
mean= _________
b) 2 occurs with frequency 10
5 occurs with frequency 30
8 occurs with frequency 10
median = ________________
mode _________, mean = ___________
12. Use the following probability distribution of the r.v. X to find the expected value of X, and the
variance.
X = ____________
variance = ___________
X=x
P(X = x )
===================
-2
0.1
--------------------------------1
0.4
--------------------------------4
0.5
---------------------------------
13.
Sketch the corresponding histograph for the r.v. X above ( #12)
14. If two sets A and B are independent, then what does P ( A  B ) = ? _____________________
15. Complete the three types of random variables: continuous,
_____________________________________ , ___________________________________________
16. Classify and “list the values” of each of the following r.v.s
a) Let the r.v. X represent the number of men that a young lady will have to meet before she
meets the perfect man.
X is _____________________________ values : ____________________________
b) Let the r.v. Y represent the exact amount of rain that is possible to occur in any month of any
given year ?
Y is ____________________________ values: ________________________________
17. A game is played as follows: It costs $ 5 to play. A pair of dice is rolled if a double occurs you
win $100, anything else you lose.
Find the expected value of such a game. ______________________
If you played this game five times, what would be your expected value of the five games ? ______
18. What is the standard deviation of a standard normal curve ? _______________
19. Construct a Venn –Diagram and shade the following set: A  B /
20.
Five cards are selected from a standard deck. Each card is replaced before the next card is
selected. How many different five cards are possible ?
21.
A Pres. , a V. P. , and a Treasurer are to be selected from a group of 20. Anybody can serve in any
position, but a person can only hold one office. How many different selections are possible ?
22.
How many five card hands are possible ? ______________
What is the probability that your five card hand consists of all diamonds ? __________________
What is the probability that your hand consists of exactly three diamonds ? ___________________
23. A fair die is rolled six times. What is the probability that three of the six rolls were fours ?
24. A 12 problem multiple choice quiz is given with four possible answers to each question – only one of
which is correct.
What is the probability that the student will get at least 10 of them right?
25. A matching test is given. There are 5 questions and 10 answers. Each answer can only be used
once. What is the probability that you will not miss any question ?
26. Use the following histograph to answer the questions that follow.
Find each of the following probabilities.
a) P ( X = - 2 ) = ________________
b) P ( X = 2. 5 ) = __________________
c ) P ( X > 0 ) = _________________
27.
A doctor finds that a person with a dark skin condition( D ) leads to skin cancer in 2 out of 25
cases. The doctor examines 200 patients during the year. What is the probability that
8 of them will have cancer ?