So far we've looked at modeling the overall probability of a single event, or a single collection of events. For example, we've developed forecasts for the demand of our product next year using the probabilities we saw in historical sales data. These are sometimes called prior probabilities. It's also possible to design a model that includes a series of related events, each with its own probability of occurrence. For example, a company might test market a new product in a selected city or region before launching a full national campaign. The test market might go well or it might go badly, and historical data with similar products or other forms of product research can assign a probability to that test. If the test goes well, the probability of success on a national rollout is higher than if the test goes poorly. Models that capture these joint probabilities are called probability trees. To make the model more useful for decision making, you can add estimated expenses to the events in the model, as well as some estimated revenue returns. The result is a decision tree model. In this lecture and demonstration, we will look at the layout and design of a model for calculating joint probabilities of a series of events. Then we'll add projective revenues and expenses to develop the decision tree. Let's turn to the spreadsheet to look at an example. This is an example of a probability tree model. By the way, all the models I'm showing you are available on your course site. You may want to open them up while you're going through this material. In a probability tree, we're modeling a series of sequential events, each with its own probability, and then we will combine them to generate an overall probability for the series. So, let's take a practical example. It's a common practice for retailers to do market research on a new product, then later test it in a regional market, and then roll out a national sales campaign. This spreadsheet model captures that set of activities. First is a research phase that you see here in Column D. The blue circles indicate an event with its related probabilities. In this case, for example, this blue circle and the connected probabilities for success and failure are an indication that market research gives green light to new projects. Only about 40% of the time, 60% of the time, new projects fail. Following that branch of the tree, we come to the test market event, which is located here in section G9, and its related probabilities in Column H for success and failure of the test market. When market research gives a green light, on average, our test market results are good about 75% of the time. Now moving on to the national marketing campaign, located here in column L. We see that, when you get a green light from market research, and that's combined with good test market results, success happens in the national campaign about 90% of the time. Each of these gray boxes represents a decision to be made by company managers. Each decision creates a new branch on the tree. On this branch that I'm pointing to in Cell G13, market research gave a red light to the project, but the managers decided to push ahead anyway with a regional test market. Those results tended to be good in 20% of the cases and were good in the national campaign as well in 70% of the case where the test market was successful. Each branch of the tree has its own combined probability, so for example, look at this cell in 5. 27% of the time there are positive results from market research, which leads to a successful regional sales campaign which leads in turn to a successful national campaign. Compare that to the 8% of the time that bad research leads to a successful test market and a successful national campaign. If I show you that formula, you see the combination of those three probabilities, one multiplied against the other. In this case, in the bad market research results, and in the case in Cell B or in 5, in the successful market research results. This is a simplified tree focusing on combined probabilities of all of its branches. To make this into a true decision tree, we would need to add expenses into our forecasts, then we could use those forecasts to prune the tree. For example, if you look at Cell N10, the likelihood of success in the national campaign after a failed regional campaign is less than 1%. It's just not worth the risk.