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Math 3101
Exam 1
10.27.2009
Directions:
Write your name and your TA’s name on the cover of your exam
booklet. Show all work and clearly mark your ﬁnal answers. Answers must be sim
pliﬁed unless otherwise indicated (by a footnote). You may use one sheet of paper
with anything you want written on it, but you may not use a calculator.
1. (10 pts) Let
A, B
and
C
be events, each having probability 1
/
4. Suppose
A
and
B
are independent,
A
and
C
are disjoint, and
P
(
C

B
) = 1
/
3. Find
P
(
A
∪
B
∪
C
).
2. You have a fair die and a coin that ﬂips heads 2
/
3 of the time. Suppose you
roll the die and then ﬂip the coin the number of times that the die shows (i.e.
if you roll a 3, you ﬂip the coin three times).
(a) (10 pts) What is the probability that you ﬂip exactly four heads?
1
(b) (10 pts) Show that the probability that you rolled a ﬁve, given that you
ﬂipped exactly four heads, is 5
/
13.
3. Let
X
be exponential with parameter
λ
.
(a) (10 pts) What is the probability
X
is less than 2 or greater than 4?
(b) (10 pts) Let
Y
= 1
/
(
X
+ 1). Find the distribution function of
Y
.
(c) (10 pts) Let
Z
=
√
X
. Find a density function of
Z
.
4. Suppose
X
is geometric with parameter
p
.
(a) (10 pts) Let 0
≤
a
≤
b
. Find
P
(
X
≥
b

X
≥
a
).
(b) (10 pts) Let
Y
=
±
X
if
X
≤
3
4
if
X
≥
4
. Find the density function of
Y
.
5. Suppose you deal a sixcard hand from a standard 52card deck.
(a) (6 pts) What is the probability that your sixcard hand is a ﬂush (i.e. your
cards all belong to the same suit)?
1
(b) (6 pts) What is the probability your hand contains 3 of a kind and a pair
(i.e. your six cards have values
x
,
x
,
x
,
y
,
y
, and
z
, where
x, y, z
are
diﬀerent)?
1
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 Spring '11
 SARVER
 Math

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