Law Of Total Probability, Bayes' Formula And Binary Hypothesis Testing Worksheet With Answers - University Of Illinois, 2012
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7University of Illinois
Fall 2012
ECE 313: Problem Set 5: Problems and Solutions
Law of total probability, Bayes’ formula and binary hypothesis testing
Due:
Wednesday, October 3 at 4 p.m.
Reading:
ECE 313 Course Notes, Sections 2.10 and 2.11
1. [Grabbing chocolates]
Suppose that you have two bags with white and dark chocolates. Bag 1 has two white
chocolates and six dark chocolates. Bag 2 has four white chocolates and two dark chocolates.
You choose one bag at random, both being equally likely, and you grab five chocolates with
replacement (you grab one, look at it, put it back, repeat) from the chosen bag. Let A be the
event that you grab three white chocolates and two dark chocolates.
(a) Find P (A).
Solution: Let B
be the event that bag i
1, 2 is chosen, then P (B
) = P (B
) = 1/2.
i
1
2
2
1
If bag 1 is chosen, the probability of grabbing a white chocolate is
=
. If bag 2 is
2+6
4
4
2
chosen, the probability of grabbing a white chocolate is
=
. Then, using the law
4+2
3
of total probability,
1
P (A) = P (A B
)P (B
) + P (A B
)P (B
) =
(P (A B
) + P (A B
))
1
1
2
2
1
2
2
3
5 3
3
5 3
1
5
1
1
5
2
2
103, 790
=
1
+
1
=
0.2086
2
3
4
4
3
3
3
497, 664
(b) Find the probability that you chose bag 1 given that event A occurred.
Solution: Using Bayes’ formula,
3
5 3 1
5
1
1
1
P (A B
)P (B
)
22, 394, 880
1
1
3
4
4
2
P (B
A) =
=
=
0.2107
1
103,790
P (A)
106, 280, 960
497,664
2. [Matching genes]
The color of a person’s eyes is determined by a single pair of genes. If they are both blue-eyed
genes, then the person will have blue eyes; if they are both brown-eyed genes, then the person
will have brown eyes; and if one is a brown-eyed gene and the other is a blue-eyed gene, then
the person will have brown eyes (the brown-eyed gene is dominant over the blue-eyed gene).
A newborn child independently receives one eye gene from each of its parents, and the gene it
receives from a parent is equally likely to be either one of the genes of that parent. Suppose
Dilbert and both of his parents have brown eyes, but Dilbert’s sister has blue eyes.
(a) Find the probability that Dilbert has a blue-eyed gene.
Solution:
Let B denote the blue-eyed gene and let N denote the brown-eyed gene.
Dilbert’s sister must have the gene pair BB because she has blue eyes, while Dilbert’s
parents must both have one gene of each color in order for them to have brown eyes
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