Probability I
Probability I: 8 Questions – 12 Minutes
1. A bowl contains blue and yellow candies
only. The ratio of blue candies to yellow
candies is 3/5. If a candy is chosen at
random from the bowl, what is the probability of choosing a yellow candy?
(A) 2/5
(B) 4/5
(C) 3/2
(D) 2/3
(E) 5/8
3. There are 50 candies of various colors in
a bowl. Which of the following could be
the probability of picking a red candy?
(A) 5/6
(B) 3/4
(C) 2/3
(D) 3/5
(E) 3/8
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2. A bowl contains red and green candies
only. If there are a total of 45 candies in
the bowl, which of the following could be
the ratio of the probability of picking a
red candy at random to the probability of
picking a green candy at random?
(A) 5/6
(B) 3/4
(C) 3/5
(D) 3/8
4. A bag has blue, green, and yellow marbles. There are 3 times as many green
marbles as yellow marbles and 4 times as
many blue marbles as green marbles.
What is the probability of randomly selecting a yellow marble?
(A) 1/16
(B) 1/15
(C) 1/12
(D) 1/8
(E) 1/7
(E) 2/7
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Probability I
5. A dartboard consists of 3 concentric circles of radii 5, 3, and 1 respectively. A
dart thrown at random at the board will
land somewhere inside the largest circle.
What is the probability that the dart will
land in the annular region between the
radius 3 circle and the radius 1 circle?
7. If a prime number between 1 and 30 is
chosen at random, what is the probability
that it will have a remainder of 2 when
divided by 3?
(A) 20%
(B) 30%
(A) 25%
(C) 40%
(B) 27%
(D) 50%
(C) 30%
(E) 60%
(D) 32%
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(E) 36%
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6. All of the numbers from 1 to 100 inclusive are written in a single line. If you
chose a digit at random from the line of
digits, what is the probability that you
would choose a 1?
(A) 1/8
(B) 3/20
(C) 4/35
(D) 5/48
(E) 7/64
8. A randomly-selected group of people
were asked the following question: “Do
you think Martians are currently mingling
with humans in cities all over the world?”
The results were as follows: 240 people
said NO; 100 people said MAYBE; 20
people said YES. Based on this survey, if
you asked another 90 randomly-selected
people the same question, how many
people would be expected to say NO?
(A) 45
(B) 60
(C) 72
(D) 75
(E) 80
End of Section
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Probability II
Probability II: 8 Questions – 40 Minutes
1. If three numbers from a list of the five
smallest positive integers are chosen at
random (the same number may not be
chosen twice), what is the probability that
the average of the three integers will be
one of the chosen integers?
2. If one letter is chosen at random from the
set {A, B, C, D} and one letter is chosen
at random from the set {A, B, E, F}, what
is the probability that the two chosen letters will be the same?
(A) 1/2
(A) 12.5%
(B) 1/4
(B) 20%
(C) 1/6
(C) 33.33%
(D) 1/8
(D) 40%
(E) 1/9
(E) 50%
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Probability II
3. According to one study, the probability
that an American adult ate too much
sugar as a child is 80%. According to a
companion study, 30% of people who ate
too much sugar as children have trouble
controlling their weight as adults while
only 10% of people who did not eat too
much sugar as children have a weight
control problem. According to both studies, how many American adults out of
100 would be expected to have trouble
controlling their weight?
(A) 20
(B) 22
(C) 24
4. If the probability of picking a yellow
marble out of a bag is 1/n and n is an integer greater than 1, what is the ratio of
the number of yellow marbles in the bag
to the number of other marbles?
(A)
1
n
(B) n
(C)
n
n−1
1
n−1
1
(E)
n+1
(D)
(D) 26
(E) 28
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Probability II
5. A bag contains N marbles. A marble selected at random from the bag could be
one of m different colors. Suppose there
are x (x > 1) marbles of each color in the
bag. What is the minimum number of
marbles you must select at random from
the bag in order to be 100% certain you
will have (outside the bag) a pair of marbles of the same color?
(A)
N
+1
m
6. In a bag containing purple, magenta, and
cyan colored marbles, the probability of
picking a purple marble is x times the
probability of picking a magenta marble
and the probability of picking a magenta
marble is y times the probability of picking a cyan marble. Which of the expressions below represents the probability of
picking a magenta marble in terms of x
and y?
(A) 1−
y
x
(B) 1−
x
y
(B) m + 1
(C) mx + 1
(D) x + 1
N
+1
(E)
mx
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(C)
y
xy + 1
(D)
y
x + y +1
(E)
y
xy + y + 1
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Probability II
7. A bag contains blue, green, and yellow
marbles only. The probability of picking a
blue marble is 1/2. The probability of
picking a green marble is 1/3. There are
twenty yellow marbles in the bag. Which
of the following is true about this bag of
marbles?
I.
There are 40 blue marbles.
8. Another bag. Same colors. There are
twice as many green marbles as yellow
marbles. The probability of picking a blue
marble is 4/5. The number of green marbles is x. What is the number of blue
marbles?
(A) 15x
(B) 12x
II. There are 30 green marbles.
(C) 7.5x
III. The probability of picking a yellow
marble is 1/5.
(D) 6x
(A) none
(B) I, only
(E) 4x
End of Section
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(C) II, only
(D) III, only
(E) I, II, and III
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Probability III
Probability III: 2 Questions – Unlimited
1. The evil blue, green, and yellow marble
bag has an equal number of blue and
green marbles. There are four times as
many yellow marbles as blue marbles.
Some marbles are solid and some are hollow. For each color, there are five times
as many solid marbles as hollow marbles.
What is the probability of randomly
choosing a solid green marble?
(A) 7/20
(B) 5/36
(C) 4/27
(D) 4/25
(E) 2/15
2. A slightly more interesting bag with blue,
green, and yellow marbles has some marbles that explode when removed from the
bag. Only blue marbles explode. There
are 162 blue marbles in the bag, not all of
which explode. Two-thirds of the marbles
in the bag are blue. If one marble is chosen at random from the full bag, the probability that it will explode is 1/9. What is
the ratio of exploding blue marbles to all
blue marbles?
(A) 2/27
(B) 2/9
(C) 1/9
(D) 1/6
(E) 1/3
End of Section
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ETS vs Bowdoin
ETS vs Bowdoin
Bowdoin is a highly selective college in Maine. They do not require the SAT.
They dropped the requirement because they discovered that the majority of
people who did really well at Bowdoin actually did NOT have stellar SAT
scores. Here’s the quote from Owen’s book.
ETS did not take kindly to its [Bowdoin’s] optional SAT policy. ‘Bowdoin can
make whatever decisions it wants,’ Arthur Kroll [an ETS executive] told me
somewhat snippily when I asked him in 1983 if he thought Bowdoin was irresponsible not to require the SAT. ‘I certainly wouldn’t classify them as irresponsible.’
Kroll went on to say that the SAT helped colleges weed out in advance students
who probably wouldn’t be able to hack it and that ‘there’s not that much benefit in admitting a lot of people who are going to end up flunking out.’ But if the
SAT helps colleges do that, I said, isn’t Bowdoin being just the teeniest bit irresponsible in not requiring it?
‘It depends on what their goals are,’ he said. ‘They may not have a goal of
maximizing every student’s educational performance at Bowdoin. Bowdoin may
care less as to whether all students in fact benefit from that experience. But if
that was their goal, then I would say that they were being irresponsible.’
There, I’m glad he got that off his chest. But it’s been nearly 30 years since the
plan went into effect, and Bowdoin hasn’t had to auction off its dormitories
yet. Its academic reputation is still as high as it used to be.*
You have to admire the ETS guy. He could not only spout nonsense like a pro
— he had apparently reached the point where he believed it himself!
Answers to Probability I: EEDADEEB (solutions on page A91)
Answers to Probability II: DDDDBEAD (solutions on page A93)
Answers to Probability III: BD (solutions on page A97)
*Owen, David None of the Above: The Truth Behind the SATs, Rowan and Littlefield
Publishers Inc., Revised Updated Edition (1999) 0-8476-9507-7 p. 238.
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