Test 1
An Aggie does not lie, cheat, or steal, nor tolerate those who do.
February 11, 2015
Last name, First name (print):
Signature:
• There are 10 true/false and 6 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.
P (Ei |F ) =
P (F |Ei )P (Ei )
P (F |E1 )P (E1 ) + P (F |E2 )P (E2 ) + ... + P (F |En )P (En )
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1. (10 points) Indicate whether the following statements are true or false. U is a
universal set containing two subsets E and F .
a)
∅ ∈ U.
b)
∅ ⊆ U.
c)
(E ∩ F ) ⊆ E.
d)
E ⊆ (E ∪ F ).
e)
F ∈ (E ∪ F ).
f)
(E ∩ F ) ∈ E.
For the following questions also assume F ⊆ E.
g)
(E ∪ F ) = E.
h)
∅ ⊂ ∅.
i)
(E ∩ F ) = E.
j)
(E ∩ E) = (E ∪ E).
2. (15 points) Construct a truth table for the symbolic expression (p ∨ q) ∧ (∼ p∨ ∼ q).
Is this expression a tautology, a contradiction, or neither?
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3. (15 points) Consider a universal set U = {x|x is a positive integer and x < 20} and
consider the following sets:
A = {x|x is prime and x ∈ U }
B = {x|x is a multiple of 3 and x ∈ U }
C = {x|x is even and x ∈ U }
D = {x|x > 10.5 and x ∈ U }.
Use A, B, C and D to give the following subsets of U . Do NOT list the individual
elements of the sets.
a) E = {x|x is an odd prime no greater than 50}
b) F = {x|x is an even multiple of 3 (x is a multiple of 6) or x is prime}.
c) G = {x|(x > 10 and x is even) or (x < 10 and is prime)}.
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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.
every customer had at least one buttermilk pancake,
75 people had at least one chocolate chip pancake,
everyone who had no chocolate chip pancakes had at least one blueberry pancake,
and 30 people had blueberry pancakes,
how many had all 3?
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5. (8 points) In a recent pretend survey of 175 3-month old babies,
27 said they enjoy the flavor of peas, while the others vomited on me at the
mention.
79 said they enjoy carrots,
and 169 enjoy pears.
a) What is the empirical probability that a baby will not like peas?
b) What is the empirical probability that a baby will like pears?
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6. (18 points) A standard 52 card deck of playing cards is dealt into 4 stacks of 13
cards each.
Stack A contains 5 red cards and 8 black cards,
stack B contains 3 red cards and 10 black cards,
stack C contains 9 red cards and 4 black cards and
stack D contains 9 red cards and 4 black cards.
Suppose a stack is selected then a card is drawn from it at random. Assume all 4
stacks are equally likely to be selected.
a) What is the probability that stack B is selected and a black card is drawn from
it?
b) What is the probability that stack B is selected or a black card is drawn?
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c) If a red card is drawn, what is the probability that stack C was selected?
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7. (24 points) Suppose 5 six-sided dice are in a dark bag. In place of the normal 1-6
markings,
there are 2 dice of type M which have 4 sides showing one and 2 sides showing two,
and 3 dice of type N which have 3 ones and 3 twos.
A die is selected at random from the bag and is rolled. Assume the five die have
equal likelihood of being drawn from the bag.
a) What is the probability of rolling a one?
b) If a one is rolled, what is the probability that the die drawn was type M?
c) If 3 dice are drawn from the bag with replacement (one die is drawn, observed
and returned to the bag 3 times) what is the probability that all three are die
of type N?
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d) If 3 dice are drawn from the bag without replacement (one die is drawn, observed
and is not returned to the bag 3 times) what is the probability that exactly two
are type N? (Hint: construct a probability tree and add the probabilities of all
outcomes in this event).
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