Cross Product Sample Spaces 497Table 8.1 Link Status Sample SpacesSample Space Link Probability Density Function021 AB pl(^0 ) = 0.1, pi(I) = 0.9
022 BC P2(0) = 0.^2 , P2() = 0.^8
023 AC p3(0) = 0.5, p3() = 0.^5
the outcome (0, 1, 1) means that AB is down, BC is up, and AC is up. This outcome is
assigned the probability
p(A, 1, 1) = Pl(0) • P2() • p3(1)
= (0.1)(0.8)(0.5)
= 0.04Figure 8.3 shows an enumeration of the entire sample space QŽ1 x ý22 x Q3 and gives
the values of the cross product probability density function p.
0.5 up (1, 1, 1) (.9)(.8)(.5) = .36
0.80.5 down (1, 1, 0) (.9)(.8)(.5) =.360.5 up (1, 0, 1) (.9)(.2)(.5) =.090.5 down (1, 0, 0) (.9)(.2)(.5) =.090.5 up (0, 1, 1) (.1)(.8)(.05) = .04
0.5 down (0, 1, 0) (.1)(.8)(.5)
=.040.5 up (0,0, 1) (.1)(.2)(.5) = .01
0.2 down
0.5 d (0, 0, 0) (.1)(.2)(.5) = .01AB BC AC 1.00Figure 8.3 Tree diagram for the communication network.
The event "A is connected to C" occurs when AB and BC are both up; it also occurs
when AC is up. The set E representing this event is