 Hello, I am Ishtar Satish Tharnange, Assistant Professor, Department of Civil Engineering, Walsh and Institute of Technology, Sallabu. In today's session, we will see regarding the probability completion time of the project. At the end of the session, the learner will be able to determine the probability completion time of the project, which is very helpful to make a decision regarding the project completion, or you can say probability completion time of the particular project. Here, see what are the normal distribution curve? See it is a probability distribution curve, which is in symmetric at its apex, and which is called, when it is symmetric at that time, it is called as a normal distribution curve. And here, the mean of the normal distribution curve is generated by the mu. And in this normal distribution curve, in which we are observing that the particular it is divided into two symmetric parts on the left-hand side and the right-hand side. This is a discontinuous line, which is highlighting that is a mean of the particular curve. When we observe on the left-hand side and as well as on the right-hand side, the 50 percent area is covered on the both the side. Here, the particular range that is from the mu minus 3 sigma to up till the mu plus 3 sigma is a range from which covers almost 99.70 percent of overall area, which is coming under the normal distribution curve. And the second is the range from the mu minus 2 sigma till mu plus 2 sigma, which is the area, or you can say the portion of the curve, which covers 95 percent of the overall. And the next is mu minus sigma to mu plus sigma, which covers 68 percent area of overall curve. Here, this range are very most important, which is helpful to define the particular probability of the particular curve, or you can say the project. Here, there are some of the formulas for the normal distribution curve. The first one is the expected time, that is, T is equal to T O plus 4 T M plus T P by 6. And the standard deviation is denoted by sigma, that is, sigma is equal to T P minus T O by 6. That is, T P is a arithmetic time, and that is T O is a optimistic time. And the variation is a square of the standard deviation, which is denoted by the sigma square. Here, 6 is nothing else, that is, a particular area, or you can say the range that is minus from the mu minus 3 sigma plus mu plus 3 sigma, that is, 1, 2, 3, 4, 5, 6. That is range from minus 3 sigma to plus 3 sigma, that is why it is taken as a divided by 6. And the 4 is a coefficient of most likely time, because mostly all our activities present in the project complete at the most likely time, so the weightage is more given to the most likely time. And the 1 is a coefficient of optimistic time as well as prismatic time. This is overall vertical concept behind the expected time, standard deviations, and the variations in the particular probability distributions curves, that is, a normal distribution curve. Here, the beta distribution curve. The beta distribution curve is nothing else, skewed distribution curve. As we have seen the normal distribution curve, the particular curve was symmetric at its apex or at the mu or at the center, but here we observe that the curves are skewed at the left-hand side as well as right-hand side. That is nothing else, that is, a beta distribution curve. Now, let's see the probability completion time of the project. So, it is necessary to know the probability completion time of the project within the scheduled completion time, that is, T s, which is represented by or denoted by T s, for the decision making. Yeah, with the help of this, we are going to define the normal deviation as well as the particular probability completion of the particular project. Now, steps to be followed for finding out the probability of the particular project, probability completion time of the project, you can say. As the project is defined, we should go for the construction of the network diagram and determine the critical path, which is available with the particular project. Next step, let us to know what to be determined, whether we have to determine the scheduled completion time or the probability completion time or the probability of the particular project. Means, a clear curve, we have to know what we have to find or what is expected to be to find out. Now, step three, with the help of that, with the help of overall data of the activities, their duration, the formula for the variance is sigma square is equal to T p by T p minus T u by 6 bracket raised to square. Yeah, next step, after getting the variance, we are going to find out the standard deviation for the particular project with the help of the critical path. Yeah, the sigma is equal to square root of some of the variance along the critical path. You have the standard deviation, we have to find out for the project with the help of the critical path only. If there are more than two critical path, the longest critical path we have to consider for the finding the variance or you can say the standard deviation of the overall project. Now, finally, knowing the T s and the T, that is the scheduled completion time and the expected completion time of the particular project, we can find out the normal deviation that is Z and the formula for Z is equal to T s minus T by particular standard deviation. Finally, when we get the normal deviation with the help of normal deviation, we are going to find out the probability completion time of the particular project in the percentage. So, we have to refer these tables that is the standard normal distributions function table in which the particular Z values that is a normal deviation as well as the probability completion of the particular project in the percentage is been mentioned plus 3 to minus 3. Yeah, in this table when we observe that they are given the particular guideline that if the Z value is equal to 0, then the chance of completing the project are 50-50 percent and if the project is in positive, then the chance are more than 50 percent. If the Z value is negative, then the chance are less than 50 percent. Means what? There is very less chance less than 50 percent chance to complete the particular project as per the schedule. Now, let's see the slide. Select the correct answer for the following questions. Hope so you have selected the marked one or highlighted one. These are the references for the today's session. Thank you.