© Scarborough, 2012
Math 141 Week In Review 9
1. Urn A has 3 green (G), 4 navy (N), and 5 red (R) marbles. Vase B has 2 green, 1 navy, 5 pink (P) and 3
red marbles. An experiment consists of randomly drawing one marble from Urn A and putting it into
Urn B. A marble is then randomly drawn from Vase B. Draw a probability tree diagram. Then answer
the following questions.
a. What is the probability of drawing a green marble from Vase B if you drew a red marble from
Urn A?
b. What is the probability of drawing a navy marble on both draws?
c. What is the probability of drawing a green marble from Urn A or drawing a pink marble from
Vase B?
d. What is the probability of drawing a navy marble from Urn A if you drew a green marble from
Vase B?
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MATH 141 Week In Review 9
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2. True or False?
a. P (E)  0 for any event E.
b. P (E  F) = P (E) + P (F) for any events E and F of sample space S.
c. E  F and EC  FC are mutually exclusive for any events E and F of sample space S.
d. P(S) = 1 for sample space S.
e. P (EC) = 1  P (E).
3. Let A and B be events in a sample space S such that P  A  0.42 , P  B   0.39 and P  AC  B   0.37 .
Evaluate P  A  B  .
4. If 10 people are present, what is the probability, to five decimal places, that at least two of them have the
same birthday?
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5. An experiment consists of observing the sum of the uppermost numbers when a pair of fair dice is
rolled.
a. TRUE or FALSE
The sample space associated with this experiment is uniform.
b. What is the exact probability that the sum is 6?
c. What is the exact probability that the sum is less than 5?
d. What is the exact probability that the sum is greater than 18?
e. What is the exact probability that one of the die is a 2 and the sum is 7?
f. What is the exact probability that one of the die is a 3 if it is known that the sum is 5?
g. If X is the sum of the uppermost numbers when a pair of dice is rolled, what are the mean, mode,
range, 1st quartile, 2nd quartile, 3rd quartile, inter-quartile range standard deviation, and variance?
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6. There is a 23% chance of a cyclone in Typhoon Alley tomorrow. What are the odds against a cyclone in
Typhoon Alley tomorrow?
7. A hurricane relief organization is hosting a fund-raising raffle. Two thousand tickets have been sold for
$20 each. There will be one first-place prize of $4000, 5 second-place prizes of $500 and 20 third-place
prizes for $50. Give the probability distribution. What are the expected net winnings of a person who
buys one ticket?
8. A new test for a virus will detect the virus 85% of the time in a person who has the virus. The test will
give a false positive 10% of the time. Thirty percent of the population has the virus. If the test is given
to a person selected at random, what is the probability, as an exact fraction, that the person has the virus
if the test detects the virus?
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9. Bevo is not playing with a full deck of cards. This short deck of cards contains only 2 spades S, 4
hearts H, 8 diamonds D and 6 clubs C. A playing card is drawn at random without replacement from
this short deck, and its suit is noted. Let R be for red and B be for black. A second card is then drawn
and its color is noted. Give all probabilities as fractions.
a. Draw a probability tree representing this experiment.
b. P  S  R  
c. P  H  B  
d. P  B | C  
e. P  D | R  
f. Are events R and S independent events?
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10. Let A and B be events in sample space S such that P  A  0.5, P  B   0.7 and P  A  BC   0.2 .
a.
P  A  B 
b. P  B | A 
c. P  A  B  
d. P  A | B  
11. Exactly evaluate P  A  B  if events A and B are independent such that P  A  0.35 and P  B   0.59 .
12. Exactly evaluate P  A  B  if events A and B are mutually exclusive such that P  A  0.35 and
P  B   0.59 .
13. The odds that I have a smile on my face are 7 to 3. What is the probability I am smiling?
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14. Classify the following random variables by type (non-binomial finite discrete, infinite discrete, binomial
finite discrete, continuous) and describe all possible values of the random variable.
a. Three fair dice are rolled until all three dice show the same number uppermost. Let X be the
number rolls.
b. A card is drawn, without replacement, until a face card (J, Q, and K) is drawn. Let X be the
number of cards drawn.
c. Let X be the number liters of water in the Fish Pond.
15. A gardener examined 60 peach trees and counted the number of ripe peaches on each tree. The results
are summarized in the table below.
Number of peach trees
12
18
20
10
Number of peaches per tree
25
30
35
40
a. What is the average number, to 2 decimal places, of peaches per tree? Remember your units!
b. What is the median to 2 decimal places?
c. What is the mode?
d. What is the variance, to 2 decimal places?
e. What is the range?
f. What is the inter-quartile range?
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16. The probability that a 10-gallon chinquapin oak tree survives for at least one year after transplanting it in
the ground is 0.82. If 1000 of these oaks are transplanted, what is the probability, to 4 decimal places,
that at least 800 but no more than 900 survive for at least one year?
17. The random variable X only assumes values 4, 5, 6, 7 and 8. If the tick marks on the vertical axis have a
1
scale of , complete the probability distribution histogram for this random variable.
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a. Shade the part of the histogram associated with P(6  X  8) .
b. What is the value of P  X  5  P  X  7  ?
c. What is the expected value?
d. What is the median?
e. What is the mode?
f. What is the standard deviation?
g. What is the variance?
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18. An exam has exam has 20 multiple choice questions with five possible answers to each. A student
wildly guesses the answers.
a. What is the probability that he/she will get exactly 12 questions correct?
b. What is the probability that he/she will get more than 70% correct?
c. What is the probability that he/she will get no more than 8 correct?
d. What is the probability that he/she will get at least 13 correct?
e. What is the probability that he/she will get between 11 and 17, inclusive correct?
f. What is the mean?
g. What is the variance?
h. What is the standard deviation?
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19. The probability that my dog Bentley wags his tail when I get home is 0.95, the probability that
Elizabeth’s dog in Indiana wags her tail when she gets home is 0.98, and the probability that Michele’s
dog in North Carolina wags his tail when she gets home is 0.93. What is the probability that exactly 2
dogs will wag their tail when their keepers get home today?
20. Let S  s1 , s2 , s3 , s4 , s5  be the sample space for an experiment with the probability distribution below.
If P s4   s2 , s3  
Outcome
Probability
13
, fill in the missing values in the probability distribution table.
25
s1
s2
s3
s4
s1
p1
6
25
1
5
p4
3
25
21. Out of 800 batteries, 12 are defective. If 6 batteries are picked at random, what is the probability at least
1 is defective? What are the odds against getting at least 1 defective?
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22. The table shows number of trees by type that were found in certain areas.
Rainforest
Desert
Grassland
Mountain
Total
Evergreen
180
5
15
80
280
Deciduous
75
5
30
60
170
Semi-evergreen
25
10
10
35
80
Find the probability
a. that a tree selected at random is evergreen
b. that a tree selected at random is a deciduous desert tree
c. that a semi-evergreen tree selected at random lives in the mountains
d. that a rainforest tree is evergreen
e. that a tree is deciduous if it is in the grassland
Total
280
20
55
175
530
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MATH 141 Week In Review 9
23. Out of 630 food aficionados
 200 liked only Mediterranean (M) dishes
 82 liked South American (S) dishes but not Mediterranean dishes
 95 liked Mediterranean dishes and French dishes
 77 liked South American dishes and French dishes
 118 like Mediterranean dishes and South American dishes but not French dishes
Use the above information along with the given Venn diagram to find the probability, as an exact
fraction in lowest terms that a food aficionado
a. who likes French food likes Mediterranean food
b. who does not like Mediterranean food likes South American food
24. A letter is selected at random from the word knickknack.
a. Describe a non-uniform sample space, S, for this experiment.
b. Describe a uniform sample space, T, for this experiment.
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25. There are 5 blue colored eggs, 8 pink colored eggs, and 9 green colored eggs.
a. If all of the eggs are lined up at random, what is the probability that all of the eggs of the same
color would be next to each other?
b. If 5 eggs are selected at random what is the probability of getting exactly 1 pink egg or at least 3
blue eggs?
26. If six cards are picked at random from a standard deck of cards, what is the probability that six of the
same suit is drawn?
27. The probability of the Brazos River Bridge failing during the year is 0.0035, the probability of the
Golden Gate Bridge failing during the year is 0.00057, and the probability of the Brooklyn Bridge
failing during the year is 0.0039. What is the probability that all bridges will fail in the same year?
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28. Let X be the sum of the faces of two rolled fair dice, where one of the die is a 3-sided die and the other is
a 4-sided die. Find the probability distribution and then represent it graphically with a histogram.
29. Out of 500 flowers, 250 are red, 230 are fragrant, and 400 are red or fragrant.
a. If a flower is selected at random, what is the probability it is red and not fragrant?
b. If a red flower is selected at random, what is the probability it is fragrant?
c. If a non-fragrant flower is selected at random, what is the probability it is not red?
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30. Determine whether the experiment is a binomial experiment. Justify your answer. If it is binomial gives
its mean, variance, and standard deviation. If it is binomial, calculate P  X  1 and P  X  2  .
a. Rolling a fair die twelve times and observing the number of times a 5 appears uppermost
b. Rolling a fair 8-sided die until a 1 appears uppermost
c. Rolling a pair of 10-sided dice and observing the sum of the numbers uppermost
d. Recording the mass in kilograms of students on campus on a given day
e. Recording the number of free throws made by a basketball player who usually makes 3 out of 5
free throws who attempts 800 free throws in a row with no break
f. Five cards are selected from a deck of cards and their rank is recorded
g. Five cards are selected, one at a time with replacement, from a deck of cards and it is noted
whether the card is a heart or not