P
(
E

F
) =
P
(
EF
)
P
(
F
)
, if
P
(
F
)
>
0.
In example with dice
P
(
E

F
) =
1
/
36
1
/
6
=
1
6
.
Conditional Probability
EXAMPLE 2b:
A coin is flipped twice. What is the conditional probability that
both flips land on heads, given that at least one flip lands on
heads?
Conditional Probability
EXAMPLE 2b:
A coin is flipped twice. What is the conditional probability that
both flips land on heads, given that at least one flip lands on
heads?
We are assuming that all four points in the sample space
S
=
{
(
h
,
h
)
,
(
h
,
t
)
,
(
t
,
h
)
,
(
t
,
t
)
}
are equally likely.
Let
B
=
{
(
h
,
h
)
}
be the event that both flips land on heads, and
let
A
=
{
(
h
,
h
)
,
(
h
,
t
)
,
(
t
,
h
)
}
be the event that at least one flip
lands on heads.
Conditional Probability
EXAMPLE 2b:
A coin is flipped twice. What is the conditional probability that
both flips land on heads, given that at least one flip lands on
heads?
We are assuming that all four points in the sample space
S
=
{
(
h
,
h
)
,
(
h
,
t
)
,
(
t
,
h
)
,
(
t
,
t
)
}
are equally likely.
Let
B
=
{
(
h
,
h
)
}
be the event that both flips land on heads, and
let
A
=
{
(
h
,
h
)
,
(
h
,
t
)
,
(
t
,
h
)
}
be the event that at least one flip
lands on heads.
P
(
B

A
) =
P
(
BA
)
P
(
A
)
=
P
(
{
(
h
,
h
)
}
)
P
(
{
(
h
,
h
)
,
(
h
,
t
)
,
(
t
,
h
)
}
)
=
1
/
4
3
/
4
=
1
3
Conditional Probability
EXAMPLE 2a:
A student is taking a onehourtimelimit makeup examination. Sup
pose the probability that the student will finish the exam in less than
x
hours is
x
/
2, for all 0
≤
x
≤
1. Then, given that the student is
still working after
.
75 hour, what is the conditional probability that
the full hour is used?
Conditional Probability
EXAMPLE 2a:
A student is taking a onehourtimelimit makeup examination. Sup
pose the probability that the student will finish the exam in less than
x
hours is
x
/
2, for all 0
≤
x
≤
1. Then, given that the student is
still working after
.
75 hour, what is the conditional probability that
the full hour is used?
Let
L
x
denote the event that the student finishes the exam in less
than
x
hours, 0
≤
x
≤
1, and let
E
be the event that the student
uses the full hour. We need to find
P
(
E

L
c
.
75
).
Conditional Probability
EXAMPLE 2a:
A student is taking a onehourtimelimit makeup examination. Sup
pose the probability that the student will finish the exam in less than
x
hours is
x
/
2, for all 0
≤
x
≤
1. Then, given that the student is
still working after
.
75 hour, what is the conditional probability that
the full hour is used?
Let
L
x
denote the event that the student finishes the exam in less
than
x
hours, 0
≤
x
≤
1, and let
E
be the event that the student
uses the full hour. We need to find
P
(
E

L
c
.
75
).
P
(
E

L
c
.
75
) =
P
(
EL
c
.
75
)
P
(
L
c
.
75
)
=
P
(
E
)
1

P
(
L
.
75
)
=
.
5
1

.
75
/
2
=
.
8
Conditional Probability
EXAMPLE 2c:
In the card game bridge, the 52 cards are dealt out equally to 4
players  called East, West, North, and South. If North and South
have a total of 8 spades among them, what is the probability that
East has exactly 3 of the remaining 5 spades?
Conditional Probability
EXAMPLE 2c:
In the card game bridge, the 52 cards are dealt out equally to 4
players  called East, West, North, and South. If North and South
have a total of 8 spades among them, what is the probability that
East has exactly 3 of the remaining 5 spades?
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 Spring '10
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 Physics, Conditional Probability, Probability theory