Download 18 Problems for Exam 2 on Basic Probability And Statistics | MATH 1105 and more Exams Mathematical Statistics in PDF only on Docsity!
Review Problems for Exam # 2
1.) Draw a tree diagram to represent all the outcomes of tossing a coin 4 times. Then
list all the possible outcomes.
H HHHH
H T HHHT
H H HHTH
T T HHTT
H H HTHH
H T HTHT
T H HTTH
T T HTTT
H THHH
H T THHT
H H THTH
T T THTT
T H TTHH
H T TTHT
T H TTTH
T T TTTT
2.) An experiment has four steps in which three outcomes are possible in the first
step, four outcomes are possible in the second step, five outcomes are possible in
the third step, and 2 outcomes are possible in the fourth step. How many
experimental outcomes exist for the entire experiment?
3.) Give the sample space for the experiment of rolling a die then flipping a coin.
(Be sure to use proper notation!)
{(1,H), (1,T), (2,H), (2,T), (3,H), (3,T), (4,H), (4,T), (5,H), (5,T), (6,H), (6,T)}
4.) Suppose you are selecting a 3-person leadership committee from a class of 30
people. How many different committees are possible?
30 3 ^
C
5.) Suppose in your 3-person leadership committee, each person has a specific role…
president, treasurer, and secretary. Now how many different leadership
committees are possible? (Note: the same three people holding different offices
counts as a different committee for this problem.)
30 3 ^
P
6.) Suppose you are given an unfair die to toss. After 100 0 tosses, you have come up
with the relative frequencies as shown:
Relative Frequency
a.) What is the probability that the next time you roll the die you will land
on a 4?
f (4) 0.
b.) What is the probability that the next time you roll the die you will land
on a 6?
f
f
If A is the event that you land on an even number, and B is the event that you land
on a 1, 2, or 3, find the following:
c.) P A ( ) d.) P B ( )
0.12 0.22 0.16 0.50 0.05^ ^ 0.12^ ^ 0.31^ 0.
e.) P A ( B ) f.) P A ( B )
P x ( 2) 0.
P A P B P A B
g.) ( )
C P A
1 P A ( ) 1 .50 .
7.) If P A ( ) 0.66, P A ( B ) 0.44, P A ( B ) 0.80, find
a.) P B ( )
P A B P A P B P A B
P B
P B
b.) ( )
C P B
1 P B ( ) 1 0.58 0.
c.) P A B ( | )
P A B
P B
10.) The following table shows the distribution of jobs types for 300 people surveyed:
Teacher Doctor Lawyer Businessman
Men^90 27 15
Women 12 6 9 60
a.) Create a probability table from the data in the table above.
Teacher Doctor Lawyer Businessman Total
Men 0 .30 0 .09 0 .05 0 .27 0.
Women 0 .04 0 .02 0 .03 0 .20 0.
Total 0 .34 0 .11 0 .08 0 .47 1
b.) Find P Teacher ( Women ). State whether this is considered a joint
probability, a marginal probability or neither.
P Teacher ( Women ) 0.04 , joint probability
c.) Find P Lawyer ( ). State whether this is considered a joint probability, a
marginal probability, or neither.
P Lawyer ( ) 0.08 , marginal probability
d.) Find the probability that a person interviewed was a man.
P Man ( ) 0.
e.) Find the probability that a person will be a doctor if you know they are male
P Doctor Man P Doctor Man P Man
f.) Find the probability that a person be a male if you know they are a doctor?
P Man Doctor P Man Doctor P Doctor
g.) What is the probability that a married couple will both be businessmen?
Married couples have careers independent of each other…so
(Business|Man) (Business|Woman)
(Man Business) (Woman Business) .27.
( ) ( ) .71.
P P
P P
P Man P Woman
h.) What is the probability that the person interviewed was either a woman or a
teacher?
P Woman ( Teacher ) .29 .34 .04 .
11 .) Which of the following random variables are discrete and which are continuous?
a.) Number of hours on the internet in one week.
Continuous
b.) Number of emails received in one week.
Discrete
c.) Number of online purchases made in one month.
Discrete
d.) Your height (in inches)
Continuous
e.) The distance from your house to campus
Continuous
f.) The length of time you spend studying for exam 2
Continuous
g.) The number of questions you get completely correct on exam 2
Discrete
12 .) Given an experiment with the probability distribution function
x f x
for
values of x 1, 2,3, 4.
Notice that the distribution is:
x f(x)
a.) Find f (2) and f (3.5)
f f (3.5) 0 since this is not "reasonable"
b.) Find the expected value for the random variable.
E X
c.) Find the variance for the random variable.
(^) (^) (^) (^)
2 1 2 3 2 5 2 7 ( ) (1 3.125) (2 3.125) (3 3.125) (4 3.125) 0. 16 16 16 16
Var X
d.) Find the standard deviation for the random variable.
SD X ( ) Var X ( ) 0.859 0.
c.) Find the probability you win exactly 4 out of 6 times.
^ ^
4 2 (^6 18 ) (4) 0. 4 38 38
f
d.) Find the probability you win at least 4 times.
P x ( 4) f (4) f (5) f (6)
^ ^
5 1 (^6 18 ) (5) 0. (^5 38 )
f^ and
^ ^
6 0 (^6 18 ) (6) 0. (^6 38 )
f
So P x ( 4) 0.209 0.075 0.011 0.
e.) Find the probability you win less than 4 times.
P x ( 4) 1 P x ( 4) 1 0.295 0.
f.) What is the expected value of the random variable? What does this number
represent?
E X np.
If you place 6 bets you would expect to win about 2.84 2 of them.
h.) What is the variance?
Var X np p
i.) What is the standard deviation?
SD X ( ) Var X ( ) 1.496 1.
15 .) Suppose the average number of people living along 1 mile of road is 9. Let X
represent the number of people in living along 1 mile of road.
a.) Is X discrete or continuous?
Discrete, since the values of X are 0, 1, 2, 3, …
b.) Explain how this is a Poisson experiment.
(1) The probability of occurrence is the same for any two miles
of road
(2) The occurrence in one mile is independent from the occurrence in
any other mile
c.) What is the probability distribution function?
9 9 ( ) !
x e f x x
d.) Find the probability that exactly 5 people live in one mile.
5 9 9 (5) 0. 5!
e f
e.) Find the probability that at least 3 people live in one mile.
P x ( 3) 1 P x ( 3) 1 f (0) f (1) f (2)
0 9 9 (0) 0. 0!
e f ,
1 9 9 (1) 0. 1!
e f , and
2 9 9 (2) 0. 2!
e f
P x ( 3) 1 (0.0001 0.0011 0.0050) .
f.) Find the expected value of x.
E X ( ) 9
g.) Find the variance of x.
Var X ( ) 9
h.) Find the standard deviation of x.
SD X ( ) Var X ( ) 3
i.) Find the probability that 9 people live in two miles.
Notice that we need a new value for and a new function.
2(9) 18 and
18 18 ( ) !
x e f x x
, so we can now answer the
question:
9 18 18 ( 9) (9) 0. 9!
e P x f
16 .) One ticket will be drawn at random from the box below. Are color and number
independent? Explain.
Yes, number and color are independent. Consider the following:
P ,
P black ,
P red and these are all equal
Also,
P black ,
P black ,
P black are
equal. You can see similar results if you consider the number 8 or the
color red.
Since P A ( | B ) P A ( )and P B ( | A ) P B ( ) for any combination of color
and number as events A and B, then they are independent.
