Math 4653: Elementary Probability: Spring 2007
Homework #4. Problems and Solutions
1. Sec. 3.1: #8a):
A hand of ﬁve cards contains two aces and three kings. The ﬁve cards
are shuﬄed and dealt one by one, until an ace appears. Display in a table the distribution of the
number of cards dealt.
Solution.
Let
X
be the number of cards dealt until an ace appears. We compute that
P
(
X
= 1) =
2
5
,
P
(
X
= 2) =
3
5
·
2
4
,
P
(
X
= 3) =
3
5
·
2
4
·
2
3
,
P
(
X
= 4) =
3
5
·
2
4
·
1
2
·
2
2
.
These computations are summarized in the following table
x
1
2
3
4
P
(
X
=
x
)
2
/
5
3
/
10
1
/
5
1
/
10
2. Sec. 3.2: #14:
A building has 10 ﬂoors above the basement. If 12 people get into an
elevator at the basement, and each chooses a ﬂoor at random to get out, independently of the
others, at how many ﬂoors do you expect the elevator to make a stop to let out one or more of
these 12 people?
Solution.
Let
X
j
=
±
1 if the elevator stops at ﬂoor
j,
0 if the elevator does not stop at ﬂoor
j.
Then
10
∑
j
=1
X
j
is the number of ﬂoors at which the elevator makes a stop. Since the probability
that the elevator fails to stop at a particular ﬂoor is (0
.
9)
12
, the probability that the elevators
stops at a particular ﬂoor is 1

(0
.
9)
12
, and
E
(
X
j
) = 1

(0
.
9)
12
. Therefore,
E
(
X
) =
E
²
10
X
j
=1
X
j
³
=
10
X
j
=1
E
(
X
j
) = 10
·
[1

(0
.
9)
12
]
≈
7
.
1757
.
3. Sec. 3.3: #2:
Let
Y
be the number of heads obtained if a fair coin is tossed three times.
Find the mean and variance of
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