Okay, so, what I've got here is just a very nicely formatted version of the decision tree that we just created. I like to do this just to help ensure that we don't make any mistakes. So, I actually literally highlight all my input cells with this, sickly pink color. So that's going to be cash. We have to put into a project, or we'll get out of a project. Or, it's going to be a probability, at an event node, a set of probabilities, always adding up to one and in blue I have a solution quantity is the software gives us the expected value of the project. Assuming we take the decisions that It recommends through through the project, which which here one means take this branch try for zoning variants two means take this branch develop the property. So, if we pursued this project over and over and over again in the exact the same way in the long run on average if we make these recommended decisions, we'll come up with 9,000, also the program gives us the terminal value cash at the end point of each branch, okay. But, what we have to compute manually is what's the probability of getting to that branch, okay? So the things to remember with that probability of getting to that branch once again are that if we've got a decision it tells us don't follow that branch the probability is zero, okay. If we've got a live branch, okay, that arises from a recommended decision, how do we get this probability again. We got 56% is equal to our 80% here times our 70% here and that's because we can multiply like that because the probabilities are independent across stages. And then, down here if we take the sum of the probabilities. Where better get to a hundred and our expected value is the sum product of what's in this column with what's on this column as we discuss. Okay, so I'm going to keep moving on here and think a little bit now about what the decision tree is telling us, okay. So, as I said, at every branch it's telling us at the end on the right hand side. The net cash we would wind up with by pursuing this project if we get to that node, okay. So for example, if we try for the variants and it gets rejected, we're out 40,000. Cohesive costs us 40,000. Now, I'm using thousands units. Requests us 40,000 thousands of dollars to make that happen. So that's our terminal cash here. In this case we get our 60,000 from -40,000, -200,000 plus 300,000 that gives us our net 60,000. So, it tells us that It tells us the decisions that we should make to maximize the expected value of that treaty, okay. And assuming that we make those decisions it also tells us, as I just showed, the probability of ending at any particular branch. And I've gone over how to compute, for example, things like our probability of getting 60,000, is 56%. But let's just jump back up the there and look at that. What that means is the probability of no crash given approval. Okay, and we got 56% that's the terminal node where we got a net of 60,000 and how'd we get that 56 56% probability of winding up there, are 80% times, or 70%, okay. And it gives us, of course, the expected value of the project, as I said, which is both computed by the program and checked by us using some product over here. Okay, so here's a little bit more on just to remind us all, make sure we don't make a mistake, how to compute our terminal path probabilities. So, first thing we want to do is exclude any branches where the program telling us don't take a decision that leads you to that branch. So we set their probabilities to 0, and then we just compute the combined probabilities at every branch. So again, that probability of getting to our $60,000 terminal cash was the probability of no cash, given a approval because of our independence of probabilities across events. That's the same thing as the probability of no crash and approval and that of course means that we just multiple the probabilities and we get to 56%. Also I always wanted to check that all of our probabilities when we had them up get to 100%, okay? And then to compute the expected value probability, you never ever want to just believe this software, we want to compute it ourselves over here. Up here, and the way to do that is to take this terminal cash times this probability, plus this terminal cash times this probability, plus this terminal cash times this probability, plus this terminal cash times this probability, plus this terminal cash times this probability which in Excel, is done with the sum product function, that I've showed you. And, which in algebraic terms,just looks like this summation, okay. It's really a dot product summation. For the less mathematically inclined, among us