STAT 100A HWI Solution
Problem 1:
Suppose we flip a fair coin 4 times independently.
(1) What is the sample space?
A: The sample space Ω consists of all the 2
4
= 16 sequences of heads and tails.
(2) What is the set that corresponds to the event that the number of heads is 2? What is its
probability?
A: The event is
{
HHTT, HTHT, HTTH, THHT, THTH, TTHH
}
. The probability is 6
/
16 =
3
/
8.
(3) Let
Z
i
= 1 if the
i
th flip is head, and
Z
i
= 0 otherwise, for
i
= 1
,
2
,
3
,
4. A: Let
X
be the
number of heads. Express
X
in terms of
Z
i
.
A:
X
=
X
1
+
X
2
+
X
3
+
X
4
.
(4) What is the probability distribution of
X
? That is, what is
P
(
X
=
k
) for
k
= 0
,
1
,
2
,
3
,
4?
A: Using the same method as in the answer to question (2),
P
(
X
= 0) = 1
/
16.
P
(
X
= 1) = 1
/
4.
P
(
X
= 2) = 3
/
8.
P
(
X
= 3) = 1
/
4.
P
(
X
= 4) = 1
/
16.
Problem 2:
Suppose we roll a fair die twice independently. Let
X
and
Y
be the two numbers we
get.
(1) What is the sample space? Let
A
be the event that
X >
4, and
B
be the event that
Y >
4.
What are
P
(
A
),
P
(
B
)?
A: The sample space Ω consists of the 36 pairs of numbers.
P
(
A
) =
P
(
B
) = 2
/
6 = 1
/
3.
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 Fall '07
 Wu
 Probability, Probability theory, Probability space, zi

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