Download Chapter 6: Probability: The Study of Randomness

Chapter 6: Probability and Simulation: The Study of Randomness
1. I toss a penny and observe whether it lands heads up or tails up. Suppose the penny is
fair; that is, the probability of heads is ½ and the probability of tails is ½. This means
A) that every occurrence of a head must be balanced by a tail in one of the next two or
three tosses.
B) that if I flip the coin many, many times, the proportion of heads will be
approximately ½, and this proportion will tend to get closer and closer to ½ as the
number of tosses increases.
C) that regardless of the number of flips, half will be heads and half tails.
D) generally, the flips will alternate between heads and tails.
E) all of the above.
2. A phenomenon is observed many, many times under identical conditions. The
proportion of times a particular event A occurs is recorded. This proportion represents
A) the probability of the event A.
D) the proportionality of the event A.
B) the distribution of the event A.
E) the variance of the event A.
C) the correlation of the event A.
3. A basketball player makes 160 out of 200 free throws. We would estimate the
probability that the player makes his next free throw to be
A) 0.16. B) 50-50; either he makes it or he doesn’t. C) 0.80. D) 1.2.
E) 80.
4. If the individual outcomes of a phenomenon are uncertain, but there is nonetheless a
regular distribution of outcomes in a large number of repetitions, we say the
phenomenon is
A) random. B) predictable. C) deterministic. D) probabilistic.
E) none of the above.
5. I toss a thumbtack 60 times and it lands point up on 35 of the tosses. The approximate
probability of landing point up is
A) 35. B) 0.35. C) 0.58. D) 0.65. E) 58.
6. Suppose we have a loaded die that gives the outcomes 1–6 according to the following
probability distribution:
X
P(X)
1
0.1
2
0.2
3
0.3
4
0.2
5
0.1
6
0.1
Note that for this die all outcomes are not equally likely, as they would be if the die
were fair. If this die is rolled 6000 times, the number of times we get a 2 or a 3 should
be about
A) 1000. B) 2000. C) 3000. D) 4000.
E) The answer cannot be determined since the probabilities are only approximate.
Chapter 6: Probability and Simulation: The Study of Randomness
7. Suppose we roll a red die and a green die. Let A be the event that the number of spots
showing on the red die is 3 or less and B be the event that the number of spots showing
on the green die is more than 3. The events A and B are
A) disjoint. B) complements. C) independent. D) reciprocals.
E) dependent.
8. In a particular game, a fair die is tossed. If the number of spots showing is either 4 or 5,
you win $1; if the number of spots showing is 6, you win $4; and if the number of spots
showing is 1, 2, or 3, you win nothing. If it costs you $1 to play the game, the
probability that you win more than the cost of playing is
A) 0. B) 1/6. C) 1/3. D) 2/3. E) 5/6.
9. I select two cards from a deck of 52 cards and observe the color of each (26 cards in the
deck are red and 26 are black). Which of the following is an appropriate sample space S
for the possible outcomes?
A) S = {red, black}.
B) S = {(red, red), (red, black), (black, red), (black, black)}, where, for example, (red,
red) stands for the event “the first card is red and the second card is red.”
C) S = {0, 1, 2}.
D) All of the above.
E) The results will vary since the cards are drawn without replacement.
10. A game consists of drawing three cards at random from a deck of playing cards. You
win $3 for each red card that is drawn. It costs $2 to play. For one play of this game, the
sample space S for the net amount you win (after deducting the cost of play) is
A) S = {$0, $1, $2, $3}
D) S = { –$2, $3, $6, $9}
B) S = {−$6, −$3, $0, $6}
E) S = {$0, $3, $6, $9}
C) S = { –$2, $1, $4, $7}
Use the following to answer questions 11 through 14:
If you draw an M&M candy at random from a bag of M&M’s, the candy you draw will have one
of six colors. The probability of drawing each color depends on the proportion of each color
among all candies made. Assume the table below gives the probability that a randomly chosen
M&M has each color.
Color
Probability
Brown
.3
Red
.3
Yellow
?
Green
.1
Orange
.1
11. The probability of drawing a yellow candy is
A) 0.1. B) 0.2. C) 0.3. D) 3.
E) impossible to determine from the information given.
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Chapter 6: Probability and Simulation: The Study of Randomness
12. The probability of not drawing a red candy is
A) 0.1. B) 0.3. C) 0.6. D) 0.7. E) 0.9.
13. The probability of drawing neither a brown nor a green candy is
A) 0.1. B) 0.3. C) 0.4. D) 0.6. E) 0.7.
14. If two M&M’s are selected and the colors are independent, then the probability that both
are the same color is
A) 0.01. B) 0.03. C) 0.09. D) 0.22. E) 0.25.
15. Event A occurs with probability 0.3. If event A and B are disjoint, then
A) P(B)  0.3. B) P(B)  0.3. C) P(B)  0.7. D) P(B)  0.7.
E) P(B) = 0.21.
16. Students at University X must have one of four class ranks—freshman, sophomore,
junior, or senior. At University X, 35% of the students are freshmen and 30% are
sophomores. If a University X student is selected at random, the probability that he or
she is either a junior or a senior is
A) 30%. B) 35%. C) 65%. D) 70%. E) 89.5%.
17. Event A occurs with probability 0.2. Event B occurs with probability 0.8. If A and B are
disjoint (mutually exclusive), then
A) P(A and B) = 0.16.
D) P(A or B) = 0.16.
B) P(A and B) = 0.84.
E) P(A or B) = 1.
C) P(A and B) = 1.
18. I flip a coin twice and count the number of heads. Which of the following is a valid
probability model for the number of heads observed in two flips?
A) Number of heads
0
1
2
Probability
1/3
1/3
1/3
B) Number of heads
0
1
2
Probability
1/3
1/2
1/4
C) Number of heads
0
1
2
Probability
1/3
1/2
1/3
D) Number of heads
0
1
2
Probability
1/4
1/4
1/4
E) None of the above.
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Chapter 6: Probability and Simulation: The Study of Randomness
19. Suppose there are three cards in a deck, one marked with a 1, one marked with a 2, and
one marked with a 5. You draw two cards at random and without replacement from the
deck of three cards. The sample space S = {(1, 2), (1, 5), (2, 5)} consists of these three
equally likely outcomes. Let X be the sum of the numbers on the two cards drawn.
Which of the following is the correct set of probabilities for X?
A) X
1
2
5
P(X)
1/3
1/3
1/3
B) X
3
6
7
P(X)
1/3
1/3
1/3
C) X
3
6
7
P(X)
3/16
6/16
7/16
D) X
3
6
7
P(X)
1/4
1/2
1/4
E) X
1
2
5
P(X)
1/4
1/2
1/4
Use the following to answer questions 20 and 21:
A stack of four cards contains two red cards and two black cards. I select two cards, one at a
time, and do not replace the first card before selecting the second card.
20. Referring to the information above, consider the events
A = the first card selected is red
B = the second card selected is red
The events A and B are
A) dependent. B) disjoint. C) complements. D) independent.
E) none of the above.
21. Referring to the information above, consider the events
A = the first card selected is red
B = the second card selected is black
The events A and B are
A) independent. B) disjoint. C) complements. D) reciprocals.
E) none of the above.
Use the following to answer questions 22 through 24:
Ignoring twins and other multiple births, assume that babies’ births at a hospital represent
independent events, with the probability that a baby is a boy and the probability that a baby is a
girl both being equal to 0.5.
22. Referring to the information above, the probability that the next three babies are all of
the same sex is
A) 1. B) 0.125. C) 0.250. D) 0.333. E) 0.500.
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Chapter 6: Probability and Simulation: The Study of Randomness
23. Referring to the information above, the probability that at least one of the next three
babies is a boy is
A) 0.125. B) 0.333. C) 0.500. D) 0.75. E) 0.875.
24. Referring to the information above, the events
A = the next two babies are boys
B = at least one of the next two babies is a boy
are
A) disjoint. B) complements. C) independent.
E) none of the above.
D) dependent.
25. Suppose that A and B are two independent events with P(A) = 0.3 and P(B) = 0.3.
P(A  B) is
A) 0.09. B) 0.51. C) 0.52. D) 0.57. E) 0.60.
26. Suppose that A and B are two independent events with P(A) = 0.2 and P(B) = 0.4.
P(A  BC) is
A) 0.08. B) 0.12. C) 0.52. D) 0.60. E) 0.92.
27. In a certain town 60% of the households own a cellular phone, 40% own a pager, and
20% own both a cellular phone and a pager. The proportion of households that own a
cellular phone but not a pager is
A) 20%. B) 30%. C) 40%. D) 50%. E) 80%.
Use the following to answer questions 28 through 31:
A system has two components that operate in parallel, as shown in the diagram below. Since the
components operate in parallel, at least one of the components must function properly if the
system is to function properly. The probabilities of failures for the components 1 and 2 during
one period of operation are .20 and .03, respectively. Let F denote the event that component 1
fails during one period of operation and G denote the event that component 2 fails during one
period of operation. Component failures are independent.
28. The event corresponding to the system failing during one period of operation is
A) F  G. B) F  G. C) (F  G)c. D) Fc  Gc. E) Fc  Gc.
29. The event corresponding to the system functioning properly during one period of
operation is
A) F  G. B) (F  G)c. C) Fc  Gc. D) Fc  Gc. E) F  G.
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Chapter 6: Probability and Simulation: The Study of Randomness
30. The probability that the system functions properly during one period of operation is
A) 0.994. B) 0.970. C) 0.940. D) 0.776. E) 0.006.
31. The probability that the system fails during one period of operation is
A) 0.994. B) 0.230. C) 0.224. D) 0.060. E) 0.006.
Use the following to answer questions 32-33:
An event A will occur with probability 0.5. An event B will occur with probability 0.6. The
probability that both A and B will occur is 0.1.
32. Referring to the information above, the conditional probability of A given B
A) is 0.3.
B) is 0.2.
C) is 1/6.
D) is 0.1.
E) cannot be determined from the information given.
33. Referring to the information above, we may conclude
A) that events A and B are independent.
D) that events A and B are disjoint.
B) that events A and B are complements. E) none of the above.
C) that either A or B always occurs.
34. Event A occurs with probability 0.4. The conditional probability that A occurs given that
B occurs is 0.5, while the conditional probability that A occurs given that B does not
occur is 0.2. What is the conditional probability that B occurs given that A occurs?
A) 0.
B) 0.5.
C) 4/7.
D) 5/6.
E) This cannot be determined from the information given.
35. Event A occurs with probability 0.8. The conditional probability that event B occurs
given that A occurs is 0.5. The probability that both A and B occur
A) is 0.3.
B) is 0.4.
C) is 0.5.
D) is 0.8.
E) cannot be determined from the information given.
36. Event A occurs with probability 0.3 and event B occurs with probability 0.4. If A and B
are independent, we may conclude that
A) P(A and B) = 0.12.
D) P(A or B) = 0.58.
B) P(A|B) = 0.3.
E) all of the above.
C) P(B|A) = 0.4.
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Chapter 6: Probability and Simulation: The Study of Randomness
37. A plumbing contractor puts in bids in on two large jobs. Let A = the event that the
contractor wins the first contract and let B = the event that the contractor wins the
second contract. Which of the following Venn diagrams has correctly shaded the event
that the contractor wins exactly one of the contracts?
A)
B)
C)
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Chapter 6: Probability and Simulation: The Study of Randomness
D)
E)
38. The probability of a randomly selected adult having a rare disease for which a
diagnostic test has been developed is 0.001. The diagnostic test is not perfect. The
probability the test will be positive (indicating that the person has the disease) is 0.99 for
a person with the disease and 0.02 for a person without the disease. The proportion of
adults for which the test would be positive is
A) 0.00002. B) 0.00099. C) 0.01998. D) 0.02097. E) 0.02100.
39. In question 38, if a randomly selected person is tested and the result is positive, the
probability the individual has the disease is
A) 0.001. B) 0.019. C) 0.020. D) 0.021. E) 0.047.
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Chapter 6: Probability and Simulation: The Study of Randomness
Answer Key
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B
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