So if you remember, we mention that part help us to answer the question, what is the probability that the project can be completed in x number of days given the variable durations of the activities? Moreover, we said under the normal distribution, 99.7% of the values of normally distributed variables will lie in a range defined by three standard deviation below the mean and three standard deviation above the mean or the expected value. In this case, the 17.5 for our example we just covered. So in this example, let's draw quickly how that normal distribution will look like. So, we have a normal distribution like this. And we said. The TE is 17.5. So in this case, we have, if we go 3 sigma here, 17.5+3 times, sigma was 2.33. The standard deviation and that will give you 24., 49. So in this aspect, there is a better than 99.7% a chance that the project can be completed in x or the mean, 17.5+3 sigma of the standard deviation, which will equal 24.4 or 24.49 days. In this case, if this is the 3 sigma here. The area underneath this curve here is 99.7%. So in other words, we can almost 100% sure that the project can be completed in 25 days. So what we did so far, let's sum it up. So what we did so far, let's sum it up. One we calculated the expected duration and variance on each of the activities in our project, this is one. Second, we found the critical path of the project. We looked from the expected durations. We blogged them in on all that a project or activity nodes that I highlighted for you. And from that, we looked at all the possible paths we have in the project to find what's the longest one. And that longest one, we found the critical path of the project. Third, what we did is we calculated the mean of the project, which is our example was 17.5, the expected duration to finish the project, 17.5 days. Then the number four, we calculated the standard deviation of the project by summing up all the variances we have in the critical activity. And because we have two critical path as expend look at the higher variance of any critical path available for you and take it more for the further calculations for the third, because that's more distributed you have under the curve or the normal distribution curve. So, the standard deviation then will be the square root of that variance that we found or the sigma. Now this four points that I just highlighted we went through, built the foundation and the basics for us to start work on answering the questions that per calculations can figure out such as find the probabilities on finishing the project in x number of days. So, let's take for the example here to build on our current exercise that we just covered. So, if I ask you what is the probability of completing the project in 19 days? So if you remember, we do have the normal distribution was as following, 17.5. And I ask you in the question here, what is the probability of completing the project in 19 days? The 19 days, that means is going to be roughly here. So, I'm asking in the example to find the area under this curve and this is what we're trying to understand or to calculate. The area behind or below the 19 to the left will be the probability of completing the project in 19 days. How to go through that is through to two steps. One is to calculate what we call the z value. The z value is equal to T minus mean or the TE, so the T here refers to the days that we're trying to calculate the number for. In our example here, it's the 19 and the mean would be the TE of the project, which is 70.5. And the standard deviation of the project. We have and we saw that 2.33. So, The z value would be 19-17.5 divided by 2.33, which give us 0.644. This is the first point on per calculations to answer the first question here. The second point is to find the probability for the z value, which here, the 0.644 from the z tables. The z tables, I summarize here the bigger picture kind of quick summary of the z table, but you will find the z table in much more details at many textbooks as well as you can just look it up online and you can find the more detailed ones. And I mean by the more details, the z values if you notice I'm giving in the first column goes from -2 all the way to 0, but you might find bigger z tables that can go from, let's say, -4 to 0 with have much more breakdown than just like 0.5 here and then 0.2 and then 0.3 and 0.1 and so on. You can find like 0.99, 0.98, 0.97 and so on. So, the probability you will find it in this table I just gave here in column number two and column number three and the z values are the first column and the last column. So once you find the z value, you look at the table and identify where's in the table it will be located. In our example here, the 0.644 is located between the plus 0.6 and the positive 0.7. So the probability will be between 0.73 and 0.76. Or in other words, it will be between 73% to 76%. So if you have such a situation, do a simple linear interpolation to find what's the probability if your z number will not be shown in the table. So perform a linear interpolation between all these four numbers, the interpolation showed that the number approximately to be around 74%. So in this case, the value here would be or the area underneath, this is around 74%. So if I want to rephrase it, the probability of completing the project in 19 days will be 74% probability to finish it in 19 days.
