Chapter 16 Events and Probability Spaces560
P.x;y/WWD8z: E.x;z/ORE.y;z/P.x;y/WWD8z: x¤yIMPLIES ŒE.x;z/ORE.y;z/çFor the following parts (c)–(e), letVbe a fixed set ofn > 3vertices, and letGbe a
graph with these vertices constructed randomly as follows: for all distinct vertices
x;y 2 V, independently include edgehx—yias an edge ofGwith probabilityp.
In particular, PrŒE.x;y/çDpfor allx¤y.
(c)For distinct verticesw,x,yandzinV, circle the event pairs that are indepen-
dent.
1.E.w;x/versusE.x;y/2..E.w;x/AND E.w;y//versus.E.z;x/AND E.z;y//3.E.x;y/versusP.x;y/4.P.w;x/versusP.x;y/5.P.w;x/versusP.y;z/(d)Write a simple formula in terms ofnandpfor PrŒnotP.x;y/ç, for distinct
verticesxandyinV.
Hint:Use part (c), item 2.
(e)What is the probability that two distinct verticesxandy lie on a three-
cycle inG? Answer with a simple expression in terms ofpandr, whererWWD
PrŒnotP.x;y/çis the correct answer to part (d).
Hint:Expressxandybeing on a three-cycle as a simple formula involvingE.x;y/
andP.x;y/.
Class Problems
Problem 16.4.
[A Baseball Series]
The New York Yankees and the Boston Red Sox are playing a two-out-of-three
series. (In other words, they play until one team has won two games. Then that
team is declared the overall winner and the series ends.) Assume that the Red Sox
win each game with probability3=5, regardless of the outcomes of previous games.