Activities A, B, C and D form the critical path for a project with a PERT network. The means and variances of the activity duration for each activity are given below. All activity durations follow the Gaussian (normal) distribution, and are independent of each other. The probability that the project will be completed within 40 days is _________ (round off to two decimal places). (Note: Probability is a number between 0 and 1).

Activity A B C D Mean (days) 6 11 8 15 Variance (days2) 4 9 4 9

Activity	A	B	C	D
Mean (days)	6	11	8	15
Variance (days2)	4	9	4	9

The correct answer is : 0.5

Step-by-Step Explanation:

1. Identify the Critical Path and its Activities:
As stated in the problem statement and visible in the provided image, the activities A, B, C, and D form the critical path of the project. Since these activity durations are independent random variables, the overall project duration is the sum of the durations of these critical activities.

2. Calculate the Expected Project Duration (Mean, T_E):
The expected project completion time is the sum of the mean durations of the individual activities along the critical path:
Activity A: 6 days
Activity B: 11 days
Activity C: 8 days
Activity D: 15 days

$$T_E=6+11+8+15=40\text{days}$$

3. Calculate the Variance of the Project Duration (σ²):
Since the activity durations are independent of each other, the variance of the project duration is the sum of the individual variances of the activities along the critical path:
Activity A variance: 4 days²
Activity B variance: 9 days²
Activity C variance: 4 days²
Activity D variance: 9 days²

$${\sigma }^2=4+9+4+9=26{\text{days}}^2$$

Thus, the standard deviation (σ) is:

$$\sigma =\sqrt{26}\approx 5.10\text{days}$$

4. Calculate the Standard Normal Variable (Z-score):
We want to find the probability that the project will be completed within a scheduled time of T_S = 40 days. The Z-score formula is:

$$Z=\frac{T_S-T_E}{\sigma }$$

Substituting the given values:

$$Z=\frac{40-40}{\sqrt{26}}=0$$

5. Determine the Probability:
Because the project duration follows a normal distribution, the probability of completing the project in less than or equal to the mean duration (Z = 0) is exactly 0.5 (or 50%):

$$P(Z\le 0)=0.5$$