Stat 330
Solution to Homework 3
1
Independence, Baron p. 40
Events
A
and
B
are independent. Show, intuitively, and mathematically, that
(a) Their complements are also independent.
We know, that
P
(
A
∩
B
) =
P
(
A
)
P
(
B
)
.
We want to find out about the probability of the intersection of the complements.
Intuitively, the
complements should be independent, since the occurrence of the event or the negative of the events
should not matter in the case of independence.
Mathematically, we get:
P
(
A
∩
B
)
=
P
(
A
) +
P
(
B
)

P
(
A
∪
B
) =
=
P
(
A
) +
P
(
B
)

(1

P
(
A
∩
B
)) =
=
P
(
A
) +
P
(
B
)

(1

P
(
A
)
·
P
(
B
)) =
=
1

P
(
A
) +
P
(
B
)

1 +
P
(
A
)
·
P
(
B
)) =
=
P
(
A
)
(
1

P
(
B
)) +
P
(
B
) =
P
(
A
)
·
P
(
B
)
(b) If they are disjoint, then
P
(
A
) = 0 or
P
(
B
) = 0.
Since we know, that
P
(
A
∩
B
) =
P
(
A
)
P
(
B
)
the statement follows directly, if we also assume, that the
events are disjoint, i.e.
P
(
A
∩
B
) = 0
(c) If they are exhaustive (i.e.
A
∪
B
= Ω), then
P
(
A
) = 1 or
P
(
B
) = 1.
From
1 =
P
(
A
∪
B
)
we get,
1 =
P
(
A
∪
B
)
=
1

P
(
A
∩
B
) = 1

P
(
A
)
·
P
(
B
)
Therefore
P
(
A
)
·
P
(
B
) = 0
,
which implies that either
P
(
A
) = 0
or
P
(
B
) = 0
. Then either
P
(
A
) = 1
or
P
(
B
) = 1
.
(4 points)
2
Bayes’ Theorem
(a) Suppose that a barrel contains many small plastic eggs. Some eggs are painted red and some are painted
blue.
40% of the eggs in the bin contain pearls, and 60% contain nothing.
30% of eggs containing
pearls are painted blue, and 10% of eggs containing nothing are painted blue. What is the probability
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 Spring '08
 Staff
 Probability theory, Bayesian probability, 0.999..., test pos

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