EE 464
Mitra, Spring 2009
Midterm Solutions
1.
Problem 1
(a) Suppose you are told that
P
[
A

B
]

P
[
A
] = 1
, what can you conclude about
P
[
A
]
and
P
[
B
]
?
What in turn does this tell you about the events
A
and
B
?
P
[
A

B
]

P
[
A
] = 1
→
P
[
A

B
] =
P
[
A
] + 1
→
P
[
A
] = 0
since
P
[
C
]
≤
1
∀
C
→
P
[
A

B
] =
P
[
A
∩
B
]
P
[
B
]
= 1
P
[
A
∩
B
]
≤
P
[
A
] = 0
→
P
[
A
∩
B
] = 0
since
P
[
A

B
] = 1
→
P
[
B
] = 0
since
0
0
= 1
We observe that since
P
[
A
∩
B
] =
P
[
A
]
P
[
B
] = 0
,
A
and
B
are independent.
(b) An urn contains eight balls. The letters
a
and
b
are used to label the balls. Two balls are labeled
a
; two balls are labeled
b
; and the remainder of the balls are labeled
ab
. Except for the labels,
the balls are identical. A single ball is drawn from the urn. Let
A
be the event that an
a
is
observed (note that if the ball is labeled
a
or
ab
, these both correspond to cases where an
a
is
observed). Similarly
B
is the event that a
b
is observed. Are the events
A
and
B
independent?
P
(
A
∩
B
) =
P
(
{
a, ab
} ∩ {
ab, b
}
) =
P
(
ab
) = 4
/
8 = 1
/
2
.
Also,
P
(
A
) =
P
(
a
) +
P
(
ab
) =
1
/
4 + 1
/
2 = 3
/
4
and
P
(
B
) =
P
(
ab
) +
P
(
b
) = 1
/
2 + 1
/
4 = 3
/
4
. Therefore,
P
(
A
)
P
(
B
) =
9
/
16
6
=
P
(
A
∩
B
)
, and so we conclude that the events are NOT independent.
(c) We are presented with three doors to choose  red , green , and blue  one of which has a prize
hidden behind it. We choose the red door. The presenter, who knows where the prize is, opens
the blue door and reveals that there is no prize behind it. She then asks if we wish to change
our mind about our initial selection of red. Should we change our mind?
Let
A
r
,
A
g
,
A
b
represent the events that the prize is behind a given door (red, green, blue).
Assume that
P
[
A
r
] =
P
[
A
g
] =
P
[
A
b
] =
1
3
. Let
B
correspond to the event that the presenter
selects the blue door. Without any prior information, we let
P
[
B
] =
1
2
.
i. Given that the presenter knows behind which door there is the prize and wants to re
duce your chances of winning through her selection, determine the following probabilities:
P
[
B

A
r
]
, P
[
B

A
g
]
, P
[
B

A
b
]
. Justify your calculations.
P
[
B

A
r
]
=
1
2
the host can open either blue or green as neither has the prize
P
[
B

A
b
]
=
0
the host would not open a door to reveal the prize
P
[
B

A
g
]
=
1
the host would not open a door to reveal the prize
ii. Use the probabilities above to determine whether you stay with the red door or switch to
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 Spring '06
 Caire
 Probability, Probability theory, probability density function, Cumulative distribution function

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