Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 778 — #786
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Chapter 18
Conditional Probability
will be released”, then his own probability of release will drop to 1=2 because he
will then know that either he or Voldemort will also be released, and these two
events are equally likely.
Dark Lord Sauron has made a typical mistake when reasoning about conditional
probability. Using a tree diagram and the four-step method, explain his mistake.
What is the probability that Sauron is released given that the guard says Foo-Foo is
released?
Hint: Define the events S, F and “F ” as follows:
“F ” D Guard says Foo-Foo is released
F D Foo-Foo is released
S D Sauron is released
Problem 18.8.
Every Skywalker serves either the light side or the dark side.
The first Skywalker serves the dark side.
For n 2, the n-th Skywalker serves the same side as the .n 1/-st Skywalker
with probability 1=4, and the opposite side with probability 3=4.
Let d n be the probability that the n-th Skywalker serves the dark side.
(a) Express d n with a recurrence equation and sufficient base cases.
(b) Derive a simple expression for the generating function D.x/ WWD P 1
1 d nx n .
(c) Give a simple closed formula for d n .
Problem 18.9. (a) For the directed acyclic graph (DAG) G 0 in Figure 18.3, a
minimum-edge DAG with the same walk relation can be obtained by removing
some edges. List these edges (use notation hu!vi for an edge from u to v):
(b) List the vertices in a maximal chain in G 0 .