Test 2
An Aggie does not lie, cheat, or steal, nor tolerate those who do.
March 6, 2015
Last name, First name (print):
Signature:
• There are 9 work out problems on this test.
• You have 75 minutes.
• Show all your work and indicate your final answer clearly. You will be graded not merely on the
final answer, but also on the work leading up to it.
• Calculators are allowed on this test.
• Your calculator’s memory must be cleared before taking the test.
1
1. (10 points)
a) In how many seating arrangements can 10 people sit in a row of 10 seats?
b) In how many arrangements can 10 people sit in a row of 11 seats?
c) 12 seats? (hint: find the number of permutations of 10 people and 2 indistinguishable empty
seats)
d) n seats for n> 10?
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2. (15 points) A theater seats 50 people in 10 rows of 5 seats each. If 10 groups of friends, each with
5 people, attend a movie in this theater
a) in how many ways can they sit so that each group occupies 1 row?
b) If all 50 people are assigned seats randomly, what is the probability that each row will contain
1 of the 10 groups?
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3. (15 points) A bag contains 7 red cubes, 5 white cubes and 3 blue cubes. If 3 cubes are removed
from the bag without replacement, and all cubes equally likely to be removed,
a) what is the probability of getting exactly 0 blue cubes?
b) exactly 1 blue cube?
c) exactly 2 blue cubes?
d) exactly 3 blue cubes?
e) What is the expected number of blue cubes?
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4. (10 points) Suppose 100 people ate breakfast at the campus IHOP yesterday. Suppose further that
all of the pancakes ordered were buttermilk, blueberry or chocolate chip.
a) If each person ate exactly one pancake, and 35 buttermilk, 45 chocolate chip and 20 blueberry
pancakes were ordered, in how many distinguishable ways could the pancakes have been served?
b) If 50 customers were chosen at random, what is the probability that exactly 3 had blueberry
pancakes?
5. (8 points) A coin is weighted so that the probability of getting heads is 0.6. If this coin is flipped
50 times, what is the probability of getting tails at least 12 but fewer than 15 times?
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6. (12 points) A raffle offers 1 grand prize of $1000, 2 $250 prizes, 3 $50 prizes and 4 $10 prizes.
a) In how many ways can the prizes be distributed among 10 winners?
b) If 2000 tickets are sold for $5 each, what is the expected return on a ticket?
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7. (14 points) 100 imaginary chocolate chip cookies were analyzed. Let X be the random variable
giving the number of chocolate chips in a cookie.
a) Fill in the table with the probability distribution.
x
number of cookies
P (X = x)
10
28
11
6
12
30
13
13
14
23
b) Determine the expected number of chocolate chips, the variance and standard deviation.
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8. (8 points) A random variable X has mean 100 and standard deviation 40. Use Chebyshev’s inequality to approximate P (0 ≤ X ≤ 200).
9. (8 points) If a random variable X is normally distributed with mean 6 and standard deviation 2,
find P (4 ≤ X ≤ 5).
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Question
Points
1
10
2
15
3
15
4
10
5
8
6
12
7
14
8
8
9
8
Total:
100
Score
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