**Both tools lead to the same end: a critial
path and critical activities with slack time equal to zero. The differences between these tools come from how they treat the
activity time. PERT treats activity time as a random variable whereas CPM requires a single deterministic time value for each
activity. Another difference is that PERT focuses exclusively on the time variable whereas CPM includes the analysis of the
Time/Cost Trade-off.**

**PERT**

**We have a high degree of uncertainty in
regard to the completion time of many activities. It makes sense in the real world that you do not really know how long a
particular activity will take, specially talking about certain activities such as research and development. In this case,
we can look at the project completion time in a probabilistic fashion and for each activity we can define:**

**Optimistic time estimate: an estimate of the minimum
time an activity will require. **
**Most likely time estimate: an estimate of the normal
time an activity will require. **
**Pessimistic time estimate: an estimate of the maximum
time an activity will require. **

**These three estimates are considered to
be related in the form of unimodal probability distribution: m. What we need in any case is a specific duration for each activity
taking into consideration these three estimates. This can be possible calculating the expected or mean activity time for each
activity as**

**With the expected time for each activity
we can determine which is the critical path. Using three assumptions, we can conclude that project completion time or critical
path completion time has a normal distribution. Using this, we can determine probabilities, using completion time as a normal
random variable, mean and standard deviation.**

**P(Completion Time <= 15.5 months) = P
( z <= 15.5 - mean/ standard deviation) =**