STAT 100A MIDTERM EXAM Solution
Problem 1:
Suppose we generate two independent random variables
X
and
Y
uniformly over [0
,
1].
(1) (4 points) Calculate
P
(
X
2
+
Y
2
≤
1).
A: Let Ω be the unit square [0
,
1]
2
, and let
A
be the event that
X
2
+
Y
2
≤
1, then
P
(
A
) =

A

/

Ω

=
π/
4.
(2) (6 points) Calculate
P
(
X >
1
/
2

X
+
Y <
1).
A: Let
A
be the event that
X >
1
/
2, and let
B
be the event that
X
+
Y
<
1, then
P
(
A

B
) =
P
(
A
∩
B
)
/P
(
B
) = (1
/
8)
/
(1
/
2) = 1
/
4.
Problem 2:
Suppose at any moment, the probability that there is fire in a particular building is
α
.
Given there is fire, the probability that the fire alarm is heard is
β
. Given there is no fire, the probability
that the fire alarm is heard is
γ
.
(1) (5 points) At any moment, what is the probability that we hear the fire alarm?
A:
P
(alarm) =
P
(fire)
P
(alarm

fire) +
P
(nofire)
P
(alarm

nofire) =
αβ
+ (1

α
)
γ
.
(2) (5 points) If we hear the fire alarm, then what is the probability that there is actually a fire?
A:
P
(fire

alarm) =
P
(fire
∩
alarm)
/P
(alarm) =
P
(fire)
P
(alarm

fire)
/P
(alarm) =
αβ/
(
αβ
+(1

α
)
γ
).
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 Wu
 Probability, Probability theory, X1, ﬁre alarm

Click to edit the document details