averaging time it takes to complete a project

Question

Consider a project consisting of four activities A, B, C, and D.
a) A and B, the first activities of the project, can be started simultaneously.
b) C can be started only after A is completed.
c) D can be started only after B is completed
Suppose the activity times for the activities are A = 2 weeks, B = 2 weeks, C = 2 weeks, D = 2 weeks.

a) How long does the project take to complete?
b) We ask the engineer working on activity A to factor in uncertainty in activity times and to provide better estimates of the activity times. The engineers tell us that activity A takes 1 week 50% of the time, and 3 weeks 50% of the time. How long does the project now take to complete (on average)?
c) We now also ask the engineer working on activity B to factor in uncertainty in activity times and to provide better estimates of the activity times. The engineer now tell us that activity B takes 1 week 50% of the time, and 3 weeks 50% of the time. How long does the project now take to complete (on average)? [ASSUME THAT BOTH ACTIVITY A AND ACTIVITY B ARE UNCERTAIN]

(a) is 4 weeks
however i am confused about b and c wouldn't the time average out to 2 weeks so both for a and b the time to complete a project would again be 4 weeks, this answers is sketchy though because it seems too easy to get
my logic .5(1)+ .5(3) = 2 average, 2 + 2 = 4

Answer 1

One way in which part (c) can be evaluated is to use a "tree diagram" of the possible outcomes (which I'm not going to attempt to render right now using TeX...):

Task A could take either 1 week (0.5 probability) or 3 weeks (also 0.5 prob.). Independently of that (we assume), task B has the same probabilities for such outcomes. So a "tree" looks something like:

                          1 week (0.5) --- 0.25 prob.
  1 week (0.5) ---  B --
                          3 weeks (0.5) --- 0.25 prob.

A --
1 week (0.5) --- 0.25 prob. 3 weeks (0.5) --- B -- 3 weeks (0.5) --- 0.25 prob.

Since tasks A and B are performed concurrently, the probability that both are done after just one week is 0.25. Otherwise one or the other or both take three weeks, which occurs with a probability of 0.25 + 0.25 + 0.25 = 0.75 .

Now tasks C and D each take two weeks, but cannot start until A and/or B are complete. So if A and B are both done in a week, C and D can both get started and everything will finish in 3 weeks. There's a 0.25 probability of that happening. Otherwise, at least one of A and B will not be done for 3 weeks, which means at least one of C and D will need another 2 weeks after that. In those cases, the project will need (3 + 2) = 5 weeks; there's a 0.75 probability of that.

Hence, the expected time to complete the project with the uncertainities for tasks A and B considered is $ \ 0.25 \cdot 3 \ + \ 0.75 \cdot 5 \ = \ 0.75 \ + \ 3.75 \ = \ 4.5 $ weeks.