Law Of Total Probability, Bayes' Formula And Binary Hypothesis Testing Worksheet With Answers - University Of Illinois, 2012 Page 2
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2
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6
7and also pass along the blue-eyed gene to their daughter. Then, there are three possible
gene pairs Dilbert can have because he has brown eyes: BN, N B, N N , where the first
gene comes from his father and the second from his mother. Let D denote the event
2
that Dilbert has one blue-eyed gene, then P (D) =
.
3
(b) Suppose that Dilbert’s wife has blue eyes. Find the probability that their first child will
have blue eyes.
Solution: Let C
be the event that their first child has blue eyes. Then, using the law
1
of total probability,
1
2
2
1
c
c
P (C
) = P (C
D)P (D) + P (C
D
)P (D
) =
+ 0 1
=
1
1
1
2
3
3
3
because Dilbert’s wife will surely pass the B gene (she has the gene pair BB because
she has blue eyes), so the eye color of the child depends only on the gene that Dilbert
passes.
(c) If their first child has brown eyes, what is the probability that their second child will
also have brown eyes?
Solution: Let C
be the event that their second child has blue eyes. Then, using the
2
definition of conditional probability and also the law of total probability,
c
c
c
c
c
c
c
c
P (C
C
)
P (C
C
D)P (D) + P (C
C
D
)P (D
)
c
c
2
1
2
1
2
1
P (C
C
) =
=
2
1
c
P (C
)
1
P (C
)
1
1
c
c
c
c
c
c
c
P (C
D)P (C
D)P (D) + P (C
D
)P (C
D
)P (D
)
2
1
2
1
=
1
P (C
)
1
1
1
2
2
3
+ (1)(1) 1
3
2
2
3
3
6
=
=
=
1
2
4
1
3
3
c
because given the gene pair of Dilbert (D or D
), the eye color of each child depends
only on the random choice of the gene from Dilbert, which is independent from one child
to the next.
3. [Bayes’ rule and total probability]
Suppose you have two fair coins and you flip them both. Let X be the number of heads that
show. Then, you grab a standard deck of 52 cards and draw X cards with replacement. Let
Y be the number of aces you draw.
(a) Find P Y = 1 X = 2 .
Solution: Each card drawn has a probability 1/13 of being an ace because there are
13 possible ranks A, 2, 3, . . . , 10, J, Q, K, and each draw is independent of the others
because of the replacement. Therefore, Y
Binomial(X, 1/13), and P Y = 1 X =
1
2 1
2
1
1
24
2 =
1
=
0.1420.
1
13
13
169
(b) Find the pmf of Y .
Solution: Each coin has a probability 1/2 of being heads, independently of each other,
so X
Binomial(2, 1/2). Using this fact, along with the law of total probability and
the fact that Y
Binomial(X, 1/13), we obtain for i = 0, 1, 2 :
p
(i) = P Y = i X = 0 P X = 0
+ P Y = i X = 1 P X = 1
Y
+P Y = i X = 2 P X = 2
2
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