Download Chapter 5: Probability

Chapter 5: Probability
Chapter Objectives
When you finish this chapter you should be able to
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describe the sample space of a random experiment.
distinguish among the three views of probability.
apply the definitions and rules of probability.
calculate odds from given probabilities.
determine when events are independent.
apply the concepts of probability to contingency tables.
Quiz Yourself
True/False Questions
T F 1. Joint probabilities refer to likelihoods of simple and multiple events.
T F 2.
Marginal probabilities refer to likelihoods of single events.
T F 3.
Joint probabilities never refer to more than two simultaneous events while marginal probabilities
do.
T F 4.
If A and B are mutually exclusive events, when P(A) = .2 and P(B) = .3, then P(A and B) is 0.0.
T F 5.
The probability of two events occurring independently must be equal 1.
T F 6.
The relative frequency approach to probability depends on the law of large numbers.
T F 7.
When it is impossible to perform an experiment one has no choice but to rely on the classic
approach.
Multiple Choice Questions
1.
If P(A) = 0.20, P(B) = 0.30 and P(A and B) = 0.06, then A and B are:
A.
mutually exclusive events.
B.
complementary events.
C.
independent events.
D.
dependent events.
The data in the following table are the amount of wealth possessed by current billionaires in the world,
sorted by location. Amounts are in dollars. Use this information to answer the following seven questions.
Wealth
in
Billions
↓
<1.5
1.5-3.5
3.5-5.5
5.5-7.5
>7.5
Total
2.
Location of Billionaires
Asia
Europe
10
19
8
29
9
0
38
1
80
Mideast
9
7
United
States
Other
24
5
1
12
14
2
1
64
29
1
2
Total
98
5
8
What is the probability of being a billionaire in Europe or the United States?
A.
0.09
B.
0.30
C.
0.62
D.
0.69
3.
What is the probability of being one of the “poorest” billionaires or one of the “wealthiest”
billionaires?
A.
4.
0.01
D.
Insufficient information
0.15
B.
0.08
C.
0.06
D.
0.03
0.14
B.
0.48
C.
0.52
D.
0.86
75
B.
40
C.
33
D.
14
Which is greater, the probability that, given a billionaire is located in Europe, a billionaire will
have 1.5 to 3.5 billion in wealth OR the probability that, given a billionaire is located in the United
States, a billionaire will have 1.5 to 3.5 billion in wealth?
A.
B.
C.
D.
8.
C.
If Location of Billionaires and Wealth in Billions are independent, how many billionaires would
you expect would be located in Europe and have wealth of as much as 1.5 billion dollars, rounded
to the nearest whole person?
A.
7.
0.25
What is the probability of being a Mideastern billionaire or having wealth of 1.5 to 3.5 billion
dollars?
A.
6.
B.
What is the probability of having wealth of 3.5 to 5.5 dollars and being from Asia?
A.
5.
0.44
The proportion of European billionaires with 1.5 to 3.5 billion is greater.
The proportion of United States billionaires with 1.5 to 3.5 billion is greater.
The probability of both is the same.
Insufficient information to answer the question.
Which is greater, the probability that, given a billionaire has 1.5 to 3.5 billion in wealth, the
billionaire will be located in Europe or the probability that, given a billionaire has 1.5 to 3.5
billion in wealth, the billionaire will be located in the United States?
A.
B.
C.
D.
The probability of both is the same.
The proportion of billionaires located in Europe is greater.
The proportion of billionaires located in United States is greater.
Insufficient information to answer the question.
9.
In a television advertisement for Fix-A-Flat, the announcer says, “In this world, there are two
kinds of people: those who have had a flat tire and those who will have.” Thinking as a
statistician, which of the following statements is correct with respect to the categories in this
statement?
A.
The categories are both mutually exclusive and exhaustive.
B.
The categories are neither mutually exclusive nor exhaustive.
C.
The categories are mutually exclusive, but not exhaustive.
D.
The categories are exhaustive, but not mutually exclusive.
10.
If two events are collectively exhaustive, what is the probability that one or the other occurs?
A.
0.25
B.
0.50
C.
0.75
D.
1.00
Worked Problems
5.2
a.
b.
S = {(S,L), (S,T), (S,B), (P,L), (P,T), (P,B), (C,L), (C,T), (C,B)}
There are different likelihoods of risk levels among the 3 types of business forms;
therefore the different elementary events will have different likelihoods.
5.4
a.
b.
S ={(1,H), (2,H), (3,H), (4,H), (5,H), (6,H), (1,T), (2,T), (3,T), (4,T), (5,T), (6,T)}
Yes, assuming that the die is fair and the coin is fair.
5.6
a.
b.
Subjective
Opinion of a group of telecommunication stock brokers.
5.8
a.
b.
Classical
There are 36 different outcomes from rolling two dice. There are 6 ways to roll a 7. The
6
likelihood of rolling one 7 is 36
 16 . If the dice are fair, each roll is independent of the
other and the probability of 3 sevens in a row is
5.12
a.
b.
5.16
5.20
 16  16 
1
216
 0.00463.
(Multiplication Law for Independent Events)
(.017) /(.983) /100 = .0173 to 1
98.3 / .017 = 57.83 to 1
P(A∩B). = P(A) * P(B) = .40*.50 =0.20
a.
b.
5.22
1
6
a.
b.
c.
d.
e.
f.
g.
h.
There is 25% chance that a clock will not ring (a failure, F). Both clocks would have to
fail in order to have him oversleep. Assuming independence: P(F1∩ F2) = P(F1) * P(F2)
= .25*.25 = .0625.
The probability that at least one of the clocks rings is 1 – (P(F1)*P(F2)*P(F3)) = 1(.25*.25*.25) = .9844, which is less than 99%.
P(A) = 100/200 = .50. There is a 50% chance that a student is an accounting student.
P(M) =102/200 = .51. There is a 51% chance that a student is male.
P(A ∩ M) = 56/200 = .28. There is a 28% chance that a student is a male accounting
major.
P(F ∩ S) = 24/200 = .12. There is a 12% chance that a student is a female statistics major.
P(A | M) = 56 /102 = .549. There is 54.9% chance that a male student is an accounting
major.
P(A | F) = P(F ∩ A) / P(F)= (44/200)/(98/200) = .4489. There is a 44.89% chance that a
female student is an accounting major.
P(F | S) = P(F ∩ S) / P(S) (24/200) /(40/200) = .60. There is a 60% chance that a statistics
student is female.
(E U F) = P(E) + P(F) - P(F ∩ E) 60/200 + 98/100 – 30/100 = 128/200 = 64%. There is
64% chance that a student is an economics major or a female.
Quiz Yourself Answers
True/False
1 F
2 T
3 F
4 T
5 F
6 T
7 F
1
2
3
4
5
Multiple Choice
C
6
C
7
A
8
D
9
B
10
C
B
A
B
D