1. Prove the following identity using induction:
Pr (α1, . . . , αn | β) = Pr (α1 | α2, . . . , αn, β)Pr (α2 | α3, . . . , αn, β). . .Pr (αn | β).
2. A well is being drilled on a farm. Based on what has happened to similar farms, we judge
the probability of oil being present to be 0.5, the probability of natural gas being present to
be 0.2, and the probability of neither being present to be 0.3. Oil and natural gas cannot be
present at the same time. If oil is present, a geological test will give a positive result with
probability 0.9; if natural gas is present, it will give a positive result with probability 0.3; and
if neither are present, the test will be positive with probability 0.1. Suppose the test comes
back positive. What’s the probability that oil is present?
3. We have a bag of three biased coins a, b, and c with probabilities of coming up heads of 20%,
40%, and 80%, respectively. One coin is drawn randomly from the bag (with equal likelihood
of drawing each of the three coins), and then the coin is flipped three times to generate the
outcomes X1, X2, and X3. A bell will ring “on” if all coins flips come out the same. Draw the
Bayesian network corresponding to this setup and define the necessary CPTs (Conditional
4. Consider the DAG in Figure 1:
(a) List the Markovian assumptions asserted by the DAG.
(b) True or false? Why?
• d separated(A, F, E)
• d separated(G, B, E)
• d separated(AB, CDE, GH)
(c) Express Pr (a, b, c, d, e, f, g, h) in factored form using the chain rule for Bayesian networks.
(d) Compute Pr (A = 1, B = 1) and Pr (E = 0 | A = 0). Justify your answers.
Pr (A = 0) Pr (A = 1)
Pr (B = 0) Pr (B = 1)
C D E
Figure 1: The DAG of a Bayesian network.
Pr (E = 0 | B) Pr (E = 1 | B)
B = 0 .1 .9
B = 1 .9 .1
Pr (D = 0 | A, B) Pr (D = 1 | A, B)
A = 0, B = 0 .2 .8
A = 0, B = 1 .9 .1
A = 1, B = 0 .4 .6
A = 1, B = 1 .5 .5
5. Consider the joint probability distribution in Table 1 and the propositional sentence
α : A ⇒ B.
(a) List the models of α.
(b) Compute the probability Pr (α).
(c) Compute the conditional probability distribution P r(A, B | α) as in Table 1.
(d) Compute the probability P r(A ⇒ ¬B | α).
A B Pr (A, B)
w0 T T 0.3
w1 T F 0.2
w2 F T 0.1
w3 F F 0.4
Table 1: A joint probability distribution.