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Open Probability
FAMAT State Convention 2016
For all questions, “NOTA” stands for “None of these answers.” A z-table is attached to the last page of
this test and may be required to answer several questions. Unless specified in a given question, you
may assume that dice, coins, or decks of cards are standard and fair.
1) A die is tossed, while a card is drawn from a standard deck at the same time. What is the
probability of obtaining at least one 6?
A) 19/78
B) 17/78
C) 4/13
D) 3/13
E) NOTA
2) The horse “Linder Macfarlane the Great” is predicted to win the Mu Alpha Theta Derby with
odds of 4 to 3. What is the probability that “Linder Macfarlane the Great” loses the race?
A) 1/4
B) 3/7
C) 4/7
D) 3/4
E) NOTA
3) A square grid is composed of 64 squares with each side having length 16. A circular disk of
diameter 4 is thrown at the grid, and the disk’s center lands on the grid. What is the probability
that the disk is not touching the side of any square within the grid?
A) 15/64
B) 7/16
C) 15/32
D) 49/64
E) NOTA
4) Walking across campus, a student interviewed a group of students. 25% of the students took a
finance class last semester, 50% took a marketing class last semester, and 40% took neither a
finance nor a marketing class last semester. What percent of the students in the group took
both a finance and a marketing class?
A) 15%
B) 20%
C) 25%
D) 30%
E) NOTA
Problems 5 to 8 are related. Information from the preceding questions may be used to answer the later
questions
5) Let 𝑊 be a random variable that is both continuous and distributed uniformly on the interval
1
1
[𝑎, 𝑏]. In probability theory, 𝐸[𝑊] = 2 (𝑎 + 𝑏) and 𝑉𝑎𝑟[𝑊] = 12 (𝑏 − 𝑎)2 .
Consider another distribution 𝐻 that is a continuous uniform distribution on the interval [0,1].
What is 𝐸[𝐻] − 𝑉𝑎𝑟[𝐻]?
A) 7/12
B) 1/2
C) 5/12
D) -5/12
E) NOTA
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Open Probability
FAMAT State Convention 2016
6) Julie needs help with her homework. The first question states: “Suppose 𝐻1 , 𝐻2 , … , 𝐻48 are 48
independent random variables where each variable is uniformly distributed on the interval
[0,1]. What is the probability that the sum 𝑆 = 𝐻1 + 𝐻2 + ⋯ + 𝐻48, when rounded to the
nearest integer, is 24?”
Which of the following probability statements correctly re-states the question from Julie’s
homework?
1
1
A) 𝑃 (𝑆 − 24 < 2)
C) 𝑃 (|𝑆 − 24| ≤ 2)
1
2
B) 𝑃 (𝑆 − 24 > − )
E) NOTA
1
2
D) 𝑃 (|24 − 𝑆| < )
7) Now that Julie knows the question to answer, she needs help actually answering it! Julie knows
that first step to solving this question is to calculate the mean 𝐸[𝑆] and the standard deviation
𝑆𝐷[𝑆] of 𝑆. What is 𝐸[𝑆]/𝑆𝐷[𝑆]?
A) 48
B) 4√3
C) 12
D) 1/6
E) NOTA
8) Julie almost has the answer! She recalls from her Statistics textbook that the Central Limit
Theorem has an application to a sample sum. She opens her book and reads the following:
“Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 be 𝑛 random variables that are independent and identically distributed with
mean 𝜇 and standard deviation 𝜎. Let 𝑆𝑛 = 𝑋1 + 𝑋2 + ⋯ + 𝑋𝑛 . The Central Limit Theorem
states that for sufficiently large 𝑛, the sample sum 𝑆𝑛 can be approximated using a normal a
normal distribution with mean 𝐸(𝑆𝑛 ) and standard deviation 𝑆𝐷(𝑆𝑛 ). Equivalently, the
standardized version of 𝑆𝑛 , 𝑆 ∗ =
𝑆𝑛 −𝐸(𝑆𝑛 )
,
𝑆𝐷(𝑆𝑛 )
is approximately standard normal.”
Using this fact from Julie’s book (assume we have sufficiently large n) and the z-table at the end
of this exam, which of the following is the closest approximation for the probability that the sum
𝑆 = 𝐻1 + 𝐻2 + ⋯ + 𝐻48, when rounded to the nearest integer, is 24?
A) 20%
B) 15%
C) 10%
D) 5%
E) NOTA
9) What is the probability that the equations 𝑥 + 2𝑦 = 1 and 𝐴𝑥 + 𝐵𝑦 = 3 have no solution for 𝑥
and 𝑦 where 𝐴 and 𝐵 are equal to the values obtained when one rolls two fair, independent dice
numbered 1 to 6?
A) 1/36
B) 1/18
C ) 1/12
D) 1/9
E) NOTA
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Open Probability
FAMAT State Convention 2016
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10) If 𝑃(𝐻) = 0.2 and 𝑃(𝐺|𝐻) = 3, what is 𝑃(𝐺 ∩ 𝐻)?
A) 3/5
B) 8/15
C) 7/15
D) 1/15
E) NOTA
11) There are 4 red socks, 4 white socks, and 4 blue socks in a box. Three are drawn without
replacement. What is the probability that at least two of the socks match?
A) 39/55
B) 17/32
C) 43/72
D) 11/24
E) NOTA
12) Suppose a committee of 8 people is selected in a random manner from 15 people. Determine
the probability that two particular people, Weezy and Sue, will both be selected .
A) 1/15
B)
7!
15!
C)
13! 8!
6!7! 15!
D) 4/15
E) NOTA
13) The digits 2, 4, 6, and 7 are each used once to form a 4-digit number. What is the probability
that the number is divisible by 4?
A) 1/4
B) 1/3
C) 7/24
D) 4/9
E) NOTA
14) Three fair 6-sided dice are rolled. What is the probability that the total is 10?
A) 1/16
B) 1/10
C) 1/9
D) 1/8
E) NOTA
15) It has been determined that at a certain intersection, cars arriving from the west go straight 10%
of the time, turn left 70% of the time, and turn right 20% of the time. It is also known that 80%
of drivers use their signals regularly (you can assume always) while 20% use them rarely (you
can assume never). You, who are heading into the intersection from the west, are sitting behind
a driver who does not have his signal on. What is the probability that he is turning left?
A) 1/2
B) 5/9
C) 3/5
D) 7/9
E) NOTA
16) Suppose the population for the baldness factor, which is a numerical measure on the interval (0,
100) with the high numbers representing increased baldness, is normally distributed. Now the
baldness factor population has a mean of 64 and a standard deviation of 6. What is the chance
that a randomly selected man has mild balding, which is a score ranging from 76 to 82,
exclusively? Round to the nearest percent.
A) 2%
B) 3%
C) 8%
D) 9%
E) NOTA
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Open Probability
FAMAT State Convention 2016
17) What is the probability that a random point on the interior of a circle of radius 3 is more than 2
units from the center?
A) 1/3
B) 4/9
C) 1/2
D) 5/9
E) NOTA
18) A cube measuring 3 inches on a side is painted and then cut into 27 cubes measuring 1inch on a
side. One of the cubes is randomly selected and tossed. What is the probability all 5 of the
faces showing are unpainted?
A) 1/27
B) 2/27
C) 1/9
D) 7/27
E) NOTA
D) 60
E) NOTA
19) How many ways can 6 objects be placed on a key chain?
A) 720
B) 360
C) 120
20) 10 world leaders are randomly seated at a round table for dinner. Each seat comes with a
unique entrée, one of which is sweet breads. What is the chance that Frank, the President of
Casa de Cards, sits at the spot where sweat breads are served?
A) 1/10
B) 1/45
C) 1/9
D) 1/5
E) NOTA
21) A gambler plays a certain game. For each play, the gambler will get $10 if he wins and lose $10
otherwise. Suppose that the probability to win is 𝑝, where 0 ≤ 𝑝 ≤ 1, and that the gambler
begins playing with a fortune of $100. What is the expected value of her wealth after 𝑛
independent game plays?
A) 100 + 10𝑛𝑝(1 − 𝑝)
B)
100 + 10𝑛𝑝(1 − 2𝑝)
C) 100 + 10𝑛𝑝(2𝑝 − 1)
D) 100 +
E) NOTA
𝑝
10𝑛 1−𝑝
22) Akash loves math! He loves math so much that he will spend hours after school on a computer
program to study. The program randomly generates multiple-choice questions (each question
has 4 possible responses) that Akash answers until he picks the incorrect choice for one of the
questions. At this point, the computer will explain to him why he missed the question.
Afterwards, Akash continues to answer questions in a similar manner as described before.
Assuming that Akash has an equally likely chance of answering any randomly generated
question when he starts, what probability distribution best describes this situation?
A) Uniform
B) Binomial
C) Geometric
D) Poisson
E) NOTA
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Open Probability
FAMAT State Convention 2016
23) Mr. Mac owns a farm and will sometimes bring in fresh fruit for his students. On Monday, he
has a 75% of bringing in fruit, assuming that the Rays won during their weekend game. He will
normally bring in fruit on Monday about 50% of the time. Interestingly, the Rays have a 90%
chance of winning if Mrs. Mac brings in fresh fruit the week before. What is the probability that
the Rays win a random weekend game (assume this value is constant and it’s unknown when
Mr. Mac last brought in fruit)?
A) 13/20
B) 3/5
C) 2/3
D) 4/5
E) NOTA
24) Suppose the scores on this test are approximately normally distributed with an average of 50
points and a standard deviation is 10 points. In order to qualify for a trophy, a student’s score
must be in the 90th percentile. What’s the minimum score that a student must have in order to
win a trophy?
A) 76
B) 75
C) 65
D) 64
E) NOTA
25) If 𝑃(𝑆│𝐽) = 0, and 𝑃(𝑆) and 𝑃(𝐽) are positive, then events S and J could be:
A) Mutually exclusive
B) Independent
C) Both A and B
D) Neither A nor B
E) NOTA
26) The maintenance staff of a large office building regularly replaces florescent ceiling lights that
have gone out. During a visit to a typical floor, the staff may have to replace several lights. The
manager of this staff has given the following probabilities to the number of lights (identified by
the random variable 𝑌) that need to be replaced on a floor:
𝑌
𝑃(𝑌 = 𝑦)
0
20%
1
15%
2
20%
3
30%
4
15%
If a crew usually takes 6 lights to a floor, how many are left on average after replacing those that
are out (rounded to the nearest light bulb)?
A) 5
B) 4
C) 3
D) 2
E) NOTA
27) The probability that a widget will lose 10% or more of its sprockets in a given year is one in
twenty. This means:
A)
B)
C)
D)
E)
A widget will experience a 10% loss of its sprockets every twenty years.
In the next 200 years, a widget will experience a 10% loss of sprockets about 5 times.
In the next 200 years, a widget will experience a 10% loss of sprockets about 10 times
In the last 200 years, a widget experienced a 10% loss of sprockets about 5 times.
NOTA
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Open Probability
FAMAT State Convention 2016
28) Weezy is preparing for her National Nanny exam and needs to pass a multiple choice test before
she is certified. Each question on the test has 4 choices. In preparing, Weezy hired an expert
test taker to help her master this test. The test taker, who has developed a full proof way of
improving people’s score, will ask Weezy questions until she gets one right. Once she does,
Weezy will get a bite of coconut ice cream. Weezy just started studying and currently she only
answers a question correctly 20% of the time. Assume that Weezy’s chance of getting a
question right will remain the same no matter how many bites of coconut ice cream she eats. In
any given run of questions, on what question is Weezy most likely to receive a bite of coconut
ice cream?
A) 5
B) 4
C) 2
D) 1
E) NOTA
29) Region R is a square in the x-y plane with vertices 𝐽 = (– 1, – 2), 𝐾 = (– 1, 4), 𝐿 =
(5, 4), and 𝑀 = (5, – 2). What is the probability that a randomly selected point in region R lies
below the line 3𝑥 – 5𝑦 = 10?
A) 5/12
B) 5/18
C) 5/24
D) 5/36
E) NOTA
30) There are two sets of letters, and you are going to pick exactly one letter from each set.
Set #1 = {A, B, C, D, E}; Set #2 = {K, L, M, N, O, P}
What is the probability of picking a C or an M?
A) 1/30
B) 1/15
C) 1/6
D) 1/3
E) NOTA
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Open Probability
FAMAT State Convention 2016
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