So let's take another example and get deeper into the per calculations. And how to use this method of this technique in finding the probabilities for the following example. And instead of starting by giving you the entire project and all the activities in the project. What you need to be aware of when you do the per calculations is to focus only on the critical activities. So for this example, I'm just covering the critical activities or providing for you just the critical activities. And here are the three duration estimates provided for you on each of the critical activities from A, G, F, and H. And I would love for you to answer all these four questions. Starting from what is the probability of completing the project before 24 days? The same question but instead 24, let's say 26 days. What is the probability for number three here that the project will be completed beyond 30 days. The last one is the probability find it in completing the project between 24 and 30 days. Now before you move forward with this example, I will ask you to solve all the expected durations and variance of each of the critical activities, A, G, F, and H. Then find the standard deviation of your project for the project in this example here. Then we will solve these four questions. So, go ahead, work on that expected durations for each of the activities and the variances, and find the standard deviation, these three points before we move forward. So let's compare the answers between the calculations. Again, the expected duration, I highlighted the equation here for you, and the variance as well. This highlight the answers. The expected duration for a 4.17 which is the 2 plus 4 times 4 plus 7 divided by 6 to give you 4.17, and the variance again is 7pb the pessimistic, 7 minus 2 the optimistic, the ta divided by 6, as we can see in the last column there and all to power of 2. And we do calculations also for the rest of the activities here as well as the variance. So from here, what we found is that total or that mean or the expected duration for the entire project is the summation of the expected values or the expected duration for all the critical activities you have A, G, F, and H in the project which will give you 26 dates. In addition, the variants on that path, for the critical path or that critical activities, the summation of all the numbers in the last column is 5.166. And the square root of that to give you the standard deviation or sigma would be 2.27. So let's go through the answers of the four questions because now we build the foundation which is the more important to highlight for further per calculation is that 26 the mean, 2.27 the sigma or the standard deviation. So, the first question we highlighted was, find the probability of completing the project before 24 day, as I explained to you just in a minute was for such question for per calculations you have 2 points. One to find the z value from the equation. Second is to find the probability from the z tables to highlight what's the probability before 24 days. Before I move forward, let me draw the normal distribution, the curve, that we have for our project. From what we just found in this table here. So, that will be, This is 26 days and we have, what? The standard deviation or sigma of 2.27. So that question was to find the 24 days, which will be here somewhere. So it's asking us to find the area under this curve here. So what's that probability? So that been said as I mentioned the first step, the z which is equal to 24. The time that we're trying to find the probability 4 minus the 26, the mean, the expected duration for the project divided by the standard deviation or sigma, which give you minus 0.88. Whenever you get a z in the negative value, that's mean that you are to the left side of the curve which is less than the 50%. Because the curve in the mean here, it split the curve to 50-50%. So having a minus 0.88 we go next to find that z value in the table which will be between these two numbers. So you can do a linear interpolation to highlight what's the probability. From the z tables and the interpolation you can find that the probability will be around 0.189 or 18.9% which is this is the case we have for this area here. 18.9% for the area. So let's move forward for the second question, the second question here we have probability, or computing the probability of completing the project before 26 days. So if you notice asking about the 26 days, that's mean you are asking about the probability of finishing the project at exactly the mean value, which is 26, which would be the same as here. So we are asking to find the area underneath this curve here, which in this aspect this area, of course, would be around 50%. If we want to highlight this from a link in to the table, if you have a t that you want to find which is 26 and you have a mean of 26. So, 26 minus 26 = 0. So the z value will give you here to equal to 0. Because 0 divided by the standard deviation would give you still 0. In the table, that's mean 0.5 for the probability. So, 50% it will give you for the second question. So let's move forward to the third question. For the third part of our example, we want to find the probability of completing the project beyond 30 days, so in this case, we do have, let's say 30 days here and we're trying to find the probability and the area under this curve. So in this aspect, again, we'll move forward with the two steps for the third calculation. The first one, to find the z. Finding the z, 30 minus 26 divided by 2.27 the simple, the more straightforward, and you'll find 1.76 as z value from the tables whenever you refer to a table from the z value, it usually gives you what? It gives you the area that choose different color, the area here. That is on the left side of the number that you're trying to calculate the value for. So from this equation here, or this example here, the 1.76. It gives you the rough estimate from the z tables around 96%. So the area here that on the left side of day 30, that's telling you that it will finish the project in 96% confidence. But probability, the question asking you of completing the project beyond the 30 days, that means the entire curve is 100% minus the 96% on the left to find the one on the right which will give you then 4%. So the answer would be the probability of completing than there is 4% probability of completing the project beyond 30 days in our project in this example. So let's go to the following example. Example number four. In example number 4, which we are asking to find the probability of completing your project between 24 and 30 days. So if you remember, if we have here the 24 days, if we have here the 30 days. We're asking to find the probability or in this case the area under the curve that covered this area Here. So, that's also very straightforward example. You'll find the z value for 30 and from the z value from the probability, from the tables, and then you go to the z value for 24 and you'll find the probability of 24 from the tables also after you finding the z value. And you subtract the area under the 30 minus the area under the 24 to give you the area in between. So from part three in our example, we covered the probability that below 30 that there will be 96% probability to finish the project in before 30 days and there is from the first part of our example. Probability of 18.9% to finish the product in less than or before 24 days. So the area between both of them is the subtraction so that will give us a 77%, in this area here, 77% probability of completing the project between 24 and 30 days in this specific project. So moving forward for the example I just covered, I want to highlight for you here the PERT technique limitations. And I highlight a couple of them which summarize them in three points. The first one is the three duration estimates that we get at the third calculations. It requires three durations. The optimistic, the most likely, the pessimistic duration estimates of each activity in your project or at least for the critical path, which is sometimes difficult for planners to estimate. So it's a cumbersome process. The second point is the values are all estimates. That estimated expected duration and variance of approximations involving some errors. The last point is I want to highlight the point of critical path versus the near-critical path. So, the PERT considers only the critical path in computing project completion time, and the project completion time probabilities and does not consider near critical path. Because the duration of the activities on the path are random. All the pathways, they are random. It is possible that some other path, in this case, close and near-critical could have an activity with a random duration longer than its expected duration. This potential longer path is not taken into consideration, or account, in the PERT calculations, which leads to an underestimation of the project duration.