The correct answer is option D: 50%. Let's understand why.

1. PERT Analysis:
PERT (Program Evaluation and Review Technique) is a project management tool used to analyze and estimate the time required to complete a project. It uses three time estimates for each activity: the optimistic time (TO), the most likely time (TM), and the pessimistic time (TP).

2. Expected Time (TE):
The expected time (TE) for each activity is calculated using the formula:
TE = (TO + 4TM + TP) / 6

3. Critical Path:
In PERT analysis, the critical path is the longest path of activities that determines the minimum time required to complete the project. It consists of activities with zero slack or float time, meaning any delay in these activities will delay the entire project.

4. Probability of Completion:
To calculate the probability of completing the project within a certain time, we need to consider the critical path and the sum of the expected times for all the activities on that path.

In this case, the earliest expected time (TE) of the last event is given as 40 days. This means that the sum of the expected times for all the activities on the critical path is 40 days.

5. Probability Calculation:
To calculate the probability of completing the project in 40 days or less, we need to consider the distribution of the critical path.

Since PERT uses a beta distribution, the expected time (TE) represents the mean of the distribution. In a symmetrical beta distribution, the standard deviation (SD) is calculated as:
SD = (TP - TO) / 6

Using this information, we can calculate the Z-score (standard score) for the desired completion time. The Z-score represents the number of standard deviations the desired completion time is away from the mean.

Z-score = (Desired Completion Time - TE) / SD

6. Z-score Calculation:
In this case, the desired completion time is 40 days and the TE is also 40 days. Since the SD is positive (assuming TP is greater than TO), the Z-score will be zero.

Z-score = (40 - 40) / SD = 0

7. Probability Calculation:
The probability of completing the project in 40 days or less can be calculated using the Z-table or a statistical calculator. A Z-score of zero corresponds to a probability of 50%.

Therefore, the probability of completion of the project in 40 days is 50%.