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Grinstead-Snell-Probability
· Problem 4.1.8
Conditional Probability
Mathematics · independence-counterexample
Problem
Let
Ω
=
{
a
,
b
,
c
,
d
,
e
,
f
}
\Omega = \{a, b, c, d, e, f\}
Ω
=
{
a
,
b
,
c
,
d
,
e
,
f
}
. Assume that
m
(
a
)
=
m
(
b
)
=
1
/
8
m(a) = m(b) = 1/8
m
(
a
)
=
m
(
b
)
=
1/8
and
m
(
c
)
=
m
(
d
)
=
m
(
e
)
=
m
(
f
)
=
3
/
16
m(c) = m(d) = m(e) = m(f) = 3/16
m
(
c
)
=
m
(
d
)
=
m
(
e
)
=
m
(
f
)
=
3/16
. Let
A
A
A
,
B
B
B
, and
C
C
C
be the events
A
=
{
d
,
e
,
a
}
A = \{d, e, a\}
A
=
{
d
,
e
,
a
}
,
B
=
{
c
,
e
,
a
}
B = \{c, e, a\}
B
=
{
c
,
e
,
a
}
,
C
=
{
c
,
d
,
a
}
C = \{c, d, a\}
C
=
{
c
,
d
,
a
}
. Show that
P
(
A
∩
B
∩
C
)
=
P
(
A
)
P
(
B
)
P
(
C
)
P(A \cap B \cap C) = P(A)P(B)P(C)
P
(
A
∩
B
∩
C
)
=
P
(
A
)
P
(
B
)
P
(
C
)
but no two of these events are independent.
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