Okay, so what do we need to be able to formulate an analysis and a decision tree form? At our decision nodes, as I said before, we're going to need mutually exclusive decisions. And at our Event Nodes okay. We're going to need mutually-exclusive scenarios, either we awarded the contract or not, that type of thing. We've got to consider all possible scenarios. And of course of who are considering all possible scenarios, the sum of the probabilities for each scenario has to add to a 100% or one, okay? And across our stages along the time. Access here, Time or Stage access here. The probability is need to be independent. So, let's see how that looks here. Okay with here is another in this. It's a different simple analysis here, but what I want to show you here is that for any decision tree as we move along the time or stage access, the probabilities need to be independent. So what that means in this case, we're looking in this case at whether or not, we should buy a high-yield bond. We've got a 73.5% chance that there's going to be a market crash while we own this bond and a 26.52% chance that, sorry that we have seven days 3.5% chance that we wont have a market crash. While were holding responding we have 26.5% crash left. Percent chance that might be a market crash. Then when we go on, in time, we're saying, if the market is okay, we have about a 79% chance that the company which issued the bond will default, but that means a 21% chance, roughly, that the company won't default. And so, what we mean here by probabilities independent Is that these are independent okay. Okay and what we mean by our scenario's being mutually exclusive on the Y-axis here is any time that we have an event known. Only mutually exclusive things can happen. Either the market is going to be okay, or it's going to crash. We're going to consider both, and the probabilities for all of them have to add up to 100%. Same thing here, either the company is showing the bond, is going to default, or they're not. Here's no default, and here's default. So, they have to be mutually exclusive scenarios and they have to of course we have to consider everything so the probabilities have to add up to 100%. Few more definitions we're going to need to know to make this happen okay. The project expected value we're going to call EVP. I've been throwing that around without defining it carefully. This is the expected value of the net cash flows when moving through the tree considering only each optimal choice, each recommended choice through the tree, okay. And what's an optimal choice at a branch node? That's going to be, as I said, the choice or the branch with the highest expected value for the project. [COUGH] Okay. So how do we go ahead and solve a decision tree problem? It's always good to, whenever you're doing any real world financial project, define a risk reward policy for this project. And I'll talk about that a little more later when we get into our example. Then you want to build, one way or another a futures table. I've got a certain way of doing it, but all it really is just thinking about what could happen during the project. What the probabilities are, what all the decisions are. Or that we have to make because that's what you're going to have to put in those input to your Decision Tree, okay. Then we're going to build a Decision Tree using that Futures Table data and then we're going to solve. For all of our unknowns and we're going to start doing this without consideration of time value. And that's the equivalent to in our Maddox of money and the chapter, my review chapter for you on this. That would be considering that opportunity costs of capital is equal to zero. That's the way it works out. That's the same way it's equivalent to and gives us the same answers. Just saying we're not considering time value. At all, okay. Next, we're going to go back, so we'll solve that whole thing with, we're going to solve Of this whole thing, without considering the time value. And then next, that's going to help us debug technically the decision tree, make sure everything. Works good while we're keeping it simple, right. And once we feel like it's debugged it's working well in that stuff we're going to go back and consider the tree with time power view,okay.And to do that what are we going to have Have to do, we have cash flows in the trees that are going to happen a year from now, two years from now. We're going to have to take the present value of all of those, and once we take the present value of all of our cash flows, discount them back to their equivalent at t equals 0 or right now, that's going to give us, instead of what we call the expected value of the project That's going to give us the expected net present value of the project, which is awesome, that's what we want in a time value world. And of course to do this, to discount back, we're going to need to know our opportunity cost of capital. Next we're going to apply our risk reward policy and typically that's going to have a target that we're shooting for and a downside limit. So we say we want to maximize the expected value of this project But we want to make sure that no matter what happens we don't loose more than $ 50 million, that would be her downside criteria that type of thing so we apply both of those, and then we can make our decision. Do we want to go ahead with this yes or no? And then finally always good with something like this to consider the limitations of the model, the fancier we get with financial models the more simplifying assumptions we have to make and it's very important that we're aware of what those assumptions are. Because that tells us where the model may not apply, where we may not use this model or where if we're not careful, we may erroneously try to use this model. So we'll look at that too in the context of an example.Okay, and here again I have the last Instead of my self-serving, shameless, and gratuitous plugs. Helpful material for all of my modules. Including this module.