The PERT (Program Evaluation and Review Technique) method is commonly used in project management to estimate the completion time of a project. It considers three time estimates for each activity: optimistic time (O), most likely time (M), and pessimistic time (P). These estimates are used to calculate the project completion time and the probability distribution of completion time.

The probability distribution of project completion in PERT follows a normal distribution. Here's an explanation of why this is the case:

1. PERT formula:
The PERT formula is used to calculate the expected completion time (TE) of a project. It is given by:
TE = (O + 4M + P) / 6

2. Central Limit Theorem:
The Central Limit Theorem states that the sum (or average) of a large number of independent and identically distributed random variables will be approximately normally distributed, regardless of the shape of the original distribution. In the case of PERT, the completion time of a project is the sum of completion times of individual activities, which are assumed to be independent and identically distributed.

3. Weighted average:
In the PERT formula, the most likely time (M) is given the highest weight (4), while the optimistic time (O) and pessimistic time (P) are given lower weights (1 each). This weighting scheme ensures that the most likely time has a greater influence on the expected completion time.

4. Symmetry of distribution:
The PERT distribution is symmetric around the most likely time (M), which means that the probabilities of completing the project before or after the expected completion time are equal. This symmetry is a characteristic of the normal distribution.

5. Bell-shaped curve:
The normal distribution is characterized by a bell-shaped curve, with the mean, median, and mode all coinciding at the most likely time (M). This shape reflects the fact that most projects are likely to be completed close to the expected completion time.

In conclusion, the probability distribution of project completion in PERT follows a normal distribution because of the application of the Central Limit Theorem, the weighted average of time estimates, the symmetry of the distribution, and the bell-shaped curve.