Question

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).

Hint:

In PERT analysis, the total project duration is the sum of the means of the activities on the critical path, and the probability is calculated using the standard normal distribution.

Answer 1

Correct Answer: 0.5

The total duration of the critical path is the sum of the means of the activities A, B, C, and D. Therefore: \[ \text{Total mean} = 6 + 11 + 8 + 15 = 40 \text{ days}. \] The total variance is the sum of the variances of the activities: \[ \text{Total variance} = 4 + 9 + 4 + 9 = 26 \text{ days}^2. \] The total standard deviation is the square root of the total variance: \[ \text{Total standard deviation} = \sqrt{26} \approx 5.1 \text{ days}. \] Now, we need to find the probability that the total duration will be less than 40 days. This is given by the Z-score: \[ Z = \frac{40 - 40}{5.1} = 0. \] From standard normal tables, the probability corresponding to \(Z = 0\) is 0.5. Thus, the probability that the project will be completed within 40 days is: \[ \boxed{0.5}. \]