---
title: "PERT expected time and completion probability"
subject: "Industrial Engineering and Management"
chapter: 3
key: "semester_6/industrial_engineering_and_management:pert-probability-of-completion"
source: https://www.bctengineer.com/q/semester_6/industrial_engineering_and_management/pert-probability-of-completion
site: BCT Engineering
---

# PERT expected time and completion probability

Industrial Engineering and Management · Chapter 3: Loading and Scheduling Technique

- Asked in: year not given, 8 marks
- Video: [What is the probability that a project will finish early or late (PERT)](https://www.youtube.com/watch?v=dOWQDAcYhPo) — Engineer4Free
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/industrial_engineering_and_management/pert-probability-of-completion
- Whole chapter: https://www.bctengineer.com/md/semester_6/industrial_engineering_and_management/chapter-3

## Question

A project has the activities below with optimistic (a), most likely (m) and pessimistic (b) times in weeks:

| Activity | a | m | b | Predecessors |
|---|---|---|---|---|
| A | 2 | 4 | 6 | - |
| B | 3 | 5 | 13 | - |
| C | 4 | 6 | 8 | A |
| D | 5 | 8 | 17 | A, B |
| E | 2 | 3 | 4 | C |
| F | 3 | 5 | 7 | D |
| G | 1 | 2 | 3 | E, F |

(a) Calculate the expected time and variance of each activity. (b) Find the critical path and expected project duration. (c) Find the probability that the project is completed within 24 weeks and within 20 weeks.

## Short answer

Critical path is B-D-F-G with expected project duration 22 weeks and standard deviation 2.71 weeks. Probability of completion within 24 weeks is 77% (Z=0.74). Probability within 20 weeks is 23% (Z=-0.74).

## Answer

#### (a) Expected time and variance
PERT uses the beta distribution:

$$
t_e = \frac{a + 4m + b}{6}, \qquad \sigma^2 = \left(\frac{b-a}{6}\right)^2
$$

| Act. | $t_e$ | $\sigma^2$ | ES | EF | LS | LF | Float |
|---|---|---|---|---|---|---|---|
| A | (2+16+6)/6 = 4 | 0.444 | 0 | 4 | 2 | 6 | 2 |
| B | (3+20+13)/6 = 6 | 2.778 | 0 | 6 | 0 | 6 | 0 |
| C | (4+24+8)/6 = 6 | 0.444 | 4 | 10 | 11 | 17 | 7 |
| D | (5+32+17)/6 = 9 | 4.000 | 6 | 15 | 6 | 15 | 0 |
| E | (2+12+4)/6 = 3 | 0.111 | 10 | 13 | 17 | 20 | 7 |
| F | (3+20+7)/6 = 5 | 0.444 | 15 | 20 | 15 | 20 | 0 |
| G | (1+8+3)/6 = 2 | 0.111 | 20 | 22 | 20 | 22 | 0 |

#### (b) Critical path
Zero-float activities: B, D, F, G. Expected project duration $T_E = 6 + 9 + 5 + 2 = 22$ weeks.

#### (c) Probability
Variance of the project = sum of variances on the critical path only:

$$
\sigma_T^2 = 2.778 + 4.000 + 0.444 + 0.111 = 7.333, \qquad \sigma_T = 2.708\ \text{weeks}
$$

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

- For 24 weeks: $Z = (24-22)/2.708 = 0.74$, so $P = 0.770$ (from the standard normal table).
- For 20 weeks: $Z = (20-22)/2.708 = -0.74$, so $P = 1 - 0.770 = 0.230$.

**Answer: Critical path B-D-F-G, expected duration 22 weeks, sigma = 2.71 weeks; P(finish within 24 weeks) = 77%; P(within 20 weeks) = 23%.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the PERT expected time formula?
   A. (a + m + b)/3
   B. (a + 4m + b)/6
   C. (b - a)/6
   D. (a + b)/2
   Answer: B. The beta distribution gives t_e = (a + 4m + b)/6. (Key point: Expected time)

2. How is the activity variance calculated?
   A. ((b - a)/6)^2
   B. (b - a)/6
   C. (a + b)/6
   D. (m - a)^2
   Answer: A. sigma^2 = ((b - a)/6)^2. (Key point: Variance)

3. What are the critical path and expected duration?
   A. A-C-E-G, 16 weeks
   B. B-D-F-G, 22 weeks
   C. A-D-F-G, 20 weeks
   D. B-D-F-G, 24 weeks
   Answer: B. 6 + 9 + 5 + 2 = 22 weeks through B, D, F, G, which have zero float. (Key point: Critical path)

4. Project variance is the sum of variances of which activities?
   A. All activities
   B. Only non-critical activities
   C. Only the activity with maximum variance
   D. Only critical path activities
   Answer: D. Variance = 2.778 + 4.000 + 0.444 + 0.111 = 7.333, sigma = 2.71 weeks. (Key point: Project variance)

## Related in Industrial Engineering and Management

- [CPM schedule, critical path and float](https://www.bctengineer.com/question.md?key=semester_6%2Findustrial_engineering_and_management%3Acpm-network-critical-path-float) — Industrial Engineering and Management, chapter 3
- [Difference between CPM and PERT](https://www.bctengineer.com/question.md?key=semester_6%2Findustrial_engineering_and_management%3Adifference-between-cpm-and-pert) — Industrial Engineering and Management, chapter 3

## Related in other subjects

- [Expected time of contractors (PERT)](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Apert-contractor-expected-time) — Project Engineering, chapter 3
- [Dummy activity; use of critical path in CPM](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Adummy-critical-path-use) — Project Engineering, chapter 3
- [Critical path and dummy activity significance](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Acritical-path-dummy-significance) — Project Engineering, chapter 3
- [Bar chart; expected time and variance](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Abar-chart-draw-adv-te-variance) — Project Engineering, chapter 3