Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 793 — #801
18.9. Probability versus Confidence 793
An alternative example is
Class Problems
A WWD f1g
B WWD f1; 2g
E WWD f3; 4; 5g:
Problem 18.28.
Event E is evidence in favor of event H when Pr H j E > PrŒH , and it is
evidence against H when Pr H j E < PrŒH .
(a) Give an example of events A; B; H such that A and B are independent, both
are evidence for H , but A [ B is evidence against H .
Hint: Let S D Œ1::8
(b) Prove E is evidence in favor of H iff E is evidence against H .
Problem 18.29.
Let G be a simple graph with n vertices. Let “A.u; v/” mean that vertices u and v
are adjacent, and let “W.u; v/” mean that there is a length-two walk between u and
v.
(a) Explain why W.u; u/ holds iff 9v: A.u; v/.
(b) Write a predicate-logic formula defining W.u; v/ in terms of the predicate
A.:; :/ when u ¤ v.
There are e WWD n 2
possible edges between the n vertices of G. Suppose the
actual edges of E.G/ are chosen with randomly from this set of e possible edges.
Each edge is chosen with probability p, and the choices are mutually independent.
(c) Write a simple formula in terms of p; e, and k for PrŒjE.G/j D k.
(d) Write a simple formula in terms of p and n for PrŒW.u; u/.
Let w, x, y and z be four distinct vertices.
Because edges are chosen mutually independently, events that depend on disjoint
sets of edges will be mutually independent. For example, the events
A.w; y/ AND A.y; x/