CONDITIONING AND EVIDENCE Fred is working on a major project. In planning the project, two milestones are set up, with dates by which they should be accomplished. This serves as a way to track Fred’s...

CONDITIONING AND EVIDENCE
Fred is working on a major project. In planning the project, two milestones are set up, with dates by which they should be accomplished. This serves as a way to track Fred’s progress. Let A1 be the event that Fred completes the ?rst milestone on time, A2 be the event that he completes the second milestone on time, and A3 be the event that he completes the project on time.
Suppose that P(Aj+1|Aj) = 0.8 but P(Aj+1|Ac j) = 0.3 for j = 1,2, since if Fred falls behind on his schedule it will be hard for him to get caught up. Also, assume that the second milestone supersedes the ?rst, in the sense that once we know whether he is on time in completing the second milestone, it no longer matters what happened with the ?rst milestone. We can express this by saying that A1 and A3 are conditionally independent given A2 and they’re also conditionally independent given Ac 2. (a) Find the probability that Fred will ?nish the project on time, given that he completes the ?rst milestone on time. Also ?nd the probability that Fred will ?nish the project on time, given that he is late for the ?rst milestone. (b) Suppose that P(A1) = 0.75. Find the probability that Fred will ?nish the project on time.