CS 440: Introduction to AI
Homework 4 Solution
Due: November 9 11:59PM, 2010
Your answers must be concise and clear.
Explain sufficiently that we can
easily determine what you understand.
We will give more points for a brief
interesting discussion with no answer than for a bluffing answer.
Please email your solution to the TA at [email protected]
1
Probability
1.
[10 pts.] Consider a joint probability distribution
P
(
X, Y, Z
) over vari
ables X, Y, and Z. Assume that the probability of each outcome is non
zero.
Is the following equation always true?
Prove its correctness or
incorrectness.
P
(
X

Y, Z
) =
P
(
Y

X, Z
)
P
(
X

Z
)
P
(
Y

Z
)
(Solution) The given equation is always true.
RHS of the equation has
the numerator P(Y,X,Z)/P(Z) and has the denominator P(Y,Z)/P(Z) by
Bayes rule.
The resulting fraction P(Y,X,Z)/P(Y,Z) = P(X

Y,Z) by
Bayes rule, and this is equivalent to the LHS of the equation.
2.
[10 pts.] After your yearly checkup, the doctor has bad news and good
news. The bad news is that you tested positive for a serious disease and
that the test is 95% accurate (i.e., the probability of testing positive when
you have the disease is 0.95 and the probability of testing negative when
you don’t have the disease is 0.95). The good news is that this is a rare
disease, striking only 1 in 100,000 individuals. Why is it good news that
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 Spring '08
 Levinson
 Artificial Intelligence, Conditional Probability, Probability theory, pts, Bayesian probability, Bayesian network

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