STA 4321
I
ntroduction to
P
robability
T
heory
Assignment 1
Assigned Wednesday January 13
Due Wednesday January 20
Show your work to receive full credit.
1. Let
A
1
, A
2
, . . .
be a countable collection of sets. Give a formal proof (not a picture) of
the following statements.
(a)
T
∞
i
=1
A
i
=
S
∞
i
=1
A
i
(b)
T
n
i
=1
A
i
=
S
n
i
=1
A
i
(c)
A
1
∩
A
2
=
A
1
∪
A
2
2. Let
A
and
B
be arbitrary events. Use the axioms of probability to prove that
P
(
A
∪
B
) =
P
(
A
) +
P
(
B
)

P
(
A
∩
B
)
.
[Hint: Note that
A
∪
B
=(
A
∩
B
)
∪
(
A
∩
B
)
∪
(
B
∩
A
) and
A
∩
B, A
∩
B
and
B
∩
A
are disjoint events.] Be sure to specifiy where exactly the axioms of probability are used.
3. Let
A
be an arbitrary event. Use the axioms of probability to prove that
P
(
A
) = 1

P
(
A
)
.
STA 4321  Introduction to Probability Theory Spring 2010
1
Jorge Rom´an
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STA 4321
4. Roulette is a casino and gambling game named after a French diminutive for “little
wheel.” In the game, players may choose to place bets on either a number, a range of
numbers, the colors red or black, or whether the number is odd or even. There are 18
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 Spring '08
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 Probability, Probability theory, Parity, Evenness of zero

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