EECS 501
CONDITIONAL PROBABILITY
Fall 2001
Three
There are 3 cards: red/red; red/black; black/black.
Card
The cards are shuffled and one chosen at random.
Monte
The top of the card is red. Pr[bottom is red]=?
3 possible lines of reasoning to solve this problem:
1. Bottom is red only if chose red/red card
→
Pr
= 1
/
3.
2. Not black/black, so either red/black or red/red
→
Pr
= 1
/
2.
3. 5 hidden sides: 2 red and 3 black
→
Pr
= 2
/
5.
Which is correct? They are ALL wrong! In fact,
Pr
= 2
/
3.
DEF:
Pr
[
A

B
] =
Pr
[event A occurs, GIVEN THAT event
B
occurrED].
•
Either
A
occurs or
A
doesn’t occur, even if
B
occurred.
•
Their
relative
probabilities (ratio) shouldn’t change,
after restriction to
B
occurring (
A
∩
B
and
A
0
∩
B
).
Want:
Pr
[
A

B
] +
Pr
[
A
0

B
] = 1 and
P r
[
A

B
]
P r
[
A
0

B
]
=
P r
[
A
∩
B
]
P r
[
A
0
∩
B
]
.
Know:
Pr
[
A
∩
B
] +
Pr
[
A
0
∩
B
] =
Pr
[
B
]. So just divide this by
Pr
[
B
].
THM:
Pr
[
A

B
] =
Pr
[
A
∩
B
]
/Pr
[
B
] =
Pr
[
A
∩
B
]
/
(
Pr
[
A
∩
B
] +
Pr
[
A
0
∩
B
]).
NOTE:
Forms:
Pr
[
A

B
] =
x
x
+
y
and
Pr
[
A
0

B
] =
y
x
+
y
. Ratio
x/y
, add to one.
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 Spring '10
 Lehman
 Topology, lim, Limit of a sequence, A priori and a posteriori, subsequence

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