CS 70 Fall 2006 — Solutions to Homework #9
November 30, 2006
1.
Marbles.
One approach is to explicitly construct the probability space with
•
The sample space Ω =
{
(
A, b
1
)
,
(
A, b
2
)
,
(
A, w
1
)
, . . . ,
(
A, w
5
)
,
(
B, b
1
)
,
(
B, w
1
)
, . . . ,
(
B, w
4
)
}
; and
•
The correct probability on each point—probabilities of
1
14
and
1
10
on each point with box
A
and
B
respectively.
(the events corresponding to box
A
and
B
must have total probability 0.5 each, and
each sample point within each event must have equal probability summing to total probability of
the event).
Then we can calculate the probabilities directly from Frst principles as follows.
•
The probability that the marble is black:
Involves deFning the event
B
=
{
(
A, b
1
)
,
(
A, b
2
)
,
(
B, b
1
)
}
that the marble is black and noticing that
Pr (
B
)
=
X
ω
∈
B
Pr (
ω
)
=
Pr ((
A, b
1
)) + Pr ((
A, b
2
)) + Pr ((
B, b
1
))
=
1
14
+
1
14
+
1
10
=
17
70
.
•
The probability that the marble came from A, given that it is white:
Involves the deFnition of condi
tional probability: let
A
=
{
(
A, b
1
)
,
(
A, b
2
)
,
(
A, w
1
)
, . . . ,
(
A, w
5
)
}
be the event that the marble came
from box A and
W
=
{
(
A, w
1
)
, . . . ,
(
A, w
5
)
,
(
B, w
1
)
, . . . ,
(
B, w
4
)
}
be the event that the marble is
white; then
Pr (
A

W
)
=
Pr (
A
∩
W
)
Pr (
W
)
=
∑
ω
∈
A
∩
W
Pr (
ω
)
∑
ω
∈
W
Pr (
ω
)
=
Pr ((
A, w
1
)) +
. . .
+ Pr ((
A, w
5
))
Pr ((
A, w
1
)) +
. . .
+ Pr ((
A, w
5
)) + Pr ((
B, w
1
)) +
. . .
+ Pr ((
B, w
4
))
=
5
/
14
(5
/
14) + (4
/
10)
=
25
53
.
1