HW 1-1 (Due Aug. 25, 2016)
Name:
Problem 1. Suppose that A and B are two events. Write expressions involving unions, intersections,
and complements that describe the following:
1. Both events occur
2. At least one occurs
ANB
AUB
And
3. Neither occurs
4. Exactly one occurs
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( 5)
An
U
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Problem 2. suppose a family contains two children of di↵erent ages, and we are interested in the
gender of these children. Let F denote that a child is female and M that the child is male and let
a pair such as F M denote that the older child is female and the younger is male. THere are four
points in the set S of possible observations:
S = {F F, F M, M F, M M }.
Let A denote the subset of possibilities containing no males; B, the subset containing two males;
and C, the subset containing at least one male. List the elements of A, B, C, A [ B, A \ B, A [ C,
A \ C, B [ C, B \ C, and C \ B̄.
A={
C=
FF
}
(
FM
{
AUB=
B={
,
MF
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MM
MM
,
}
}
}
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cnD=
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}
Problem 3. Define the sequence of sets Aj = (1
1/j, 2 + 1/j), for j = 1, 2, . . . . Then what are
1
[1
j=1 Aj and \j=1 Aj ?
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Aj
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,
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=
=
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,
,
1,2
,
3
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=
( 0,3
)
HW 1-2 (Due Aug. 25, 2016)
Name:
Problem 1. If A and B are two sets, draw Venn diagrams to verify the followings:
1. A = (A \ B) [ (A \ B̄)
2. if B ⇢ A then A = B [ (A \ B̄)
Use the identities A = A \ S and S = B [ B̄ and a distributive law to prove that (mathematically,
not graphically)
1. A = (A \ B) [ (A \ B̄)
2. if B ⇢ A then A = B [ (A \ B̄)
3. Further, show that (A \ B) and (A \ B̄) are mutually exclusive and therefore that A is the
union of two mutually exclusive sets, (A \ B) and (A \ B̄).
4. Also show that B and (A \ B̄) are mutually exclusive and if B ⇢ A, A is the union of two
mutually exclusive sets, B and (A \ B̄).
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Problem 2. Suppose two dice are tossed and the numbers on the upper faces are observed. Let S
denote the set of all possible pairs that can be observed. For example, let (2, 3) denote that a 2 was
observed on the first die and a 3 on the second.
1. Define the following subsets of S: A: The number on the second die is even. B: The sume of
the two numbers is even. C: At least one number in the pair is odd. Using equally likely rule
to calculate P (A), P (B) and P (C).
2. List the points in A, C̄, A \ B, A \ B̄, Ā [ B and Ā \ C.
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Problem 3. Suppose two balanced coins are tossed and the upper faces are observed.
1. List the sample points for this experiment
2. Assign a reasonable probability to each sample point. (Are the sample points equally likely?)
3. Let A denote the event that exactly one head is observed and B the event that at least one
head is observed. List the sample points in both A and B.
4. From your answer to part 3., find P (A), P (B), P (A \ B), P (A [ B) and P (Ā [ B).
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