University of Illinois
Fall 2009
ECE 313:
Problem Set 8: Solutions
Independent Events; System Reliability; CDFs
1.
[“I before E except after C”]
Let
C
= (
A
∪
B
)
c
denote the event that
neither
A
nor
B
occurs on a trial of the experiment, and
notice that
one
of the three events
A
,
B
, and
C
always occurs on a trial. Then,
A
occurs before
B
does if for
n
= 1
,
2
,
3
,...
,
C
occurs on the 1st, 2nd, .
. . , (
n

1)th trials and
A
occurs on the
n
th
trial. By independence of the trials,
P
{
A
before
B
}
=
P
(
A
) +
P
(
C
)
P
(
A
) + [
P
(
C
)]
2
P
(
A
) +
···
=
P
(
A
)
·
1
1

P
(
C
)
=
P
(
A
)
P
(
A
) +
P
(
B
)
.
Obviously,
P
{
B
before
A
}
= 1

P
{
A
before
B
}
=
P
(
B
)
P
(
A
)+
P
(
B
)
. The way to think about this is we can
ignore all trials on which
C
occurs. On the very ﬁrst trial on which
one
of
A
and
B
occurs, what are
the chances that
A
occurs? Obviously
P
(
A

(
A
∪
B
)) =
P
(
A
)
P
(
A
∪
B
)
=
P
(
A
)
P
(
A
)+
P
(
B
)
. See also Example 4h
in Chapter 3 of Ross.
2.
[What happens in Vegas stays in Vegas]
(a)
P
(wins on ﬁrst roll) =
6
36
+
2
36
=
2
9
.
P
(loses on ﬁrst roll) =
1
36
+
2
36
+
1
36
=
1
9
.
P
(establishes point
i
on ﬁrst roll) =
6
 
7

i

36
,i
= 4
,
5
,
6
,
8
,
9
,
10. Check: probabilities sum to
1.
(b) If the shooter’s point is
i
, then
P
(makes point

point is
i
) =
6
 
7

i

36
while
P
(craps out) =
6
36
so that
P
(game ends) =
12

7

i

36
. Else, the shooter rolls the dice again. Since
the events “roll
i
” and “roll 7” are mutually exclusive, we can use the result of Problem 1 to get
P
(shooter wins

point is
i
) =
6
 
7

i

12
 
7

i

which equals
3
9
,
4
10
,
5
11
,
5
11
,
4
10