Now here's another sort of model were we incorporate uncertainty and this sort of model is called a probability tree. And they're very natural when you have a process that moves through various stages. So the example that I've got going here relates to an activity that some people do and they probably shouldn't. And that's known as file sharing. And if you download or share content that you don't actually own, then typically you're breaking the law and people don't like that. And so one of the things that the industry does to try and deter people from sharing illegal content is to implement what is termed the notice program and this happens in many countries around the world. They're called graduated notice programs. So if an individual is found infringing, they are notified by their Internet service provider, by their ISP, and they're told not to do it. If they are seen infringing again, then they typically have to acknowledge that they're breaking the law. And if they are found infringing a subsequent time, then there might be some remedial action taken against them. Imagine we've implemented one of these notice programs and we're trying to now measure the efficacy of the program. A natural way of doing it is in a probabilistic sense and ask the question, well, if an individual has the potential to go all the way through this program, what's the probability that they actually stop infringing? And we can collect data to help us estimate those probabilities and ultimately measure the efficacy of the program. And so this tree, this probability tree is a natural way of representing that. Let's say you receive your first notice, so that is the first red ellipse in this graph. The top of the tree, sometimes called the root of the tree. So you've got your first notice. Now whether or not you stop infringing is going to be a random variable. And in this example, we've got a 10% or 0.1 probability that you stop infringing. So that's the first green circle. But a lot of people ignore those notices. And according to this model, you've got a 90% chance of continuing to infringe. Now if you're caught infringing again, the notice is a little bit more severe. You have to acknowledge the fact that you're breaking the law. And so on the second notice, given that you've got that second notice, then we have 15% of people stop infringing on the second notice and 85% continue. And finally, there's a third notice. And on the third notice, because there's some typically remediation action, we have in this particular model 20% of the people stopping infringing and 80% continuing. And so we've represented these probabilities, all stopping infringing at each stage in the tree. So one of the natural things, once you've got this tree established, is to ask a question like, well, given this program has been implemented, what's the probability that an individual stops infringing? And so the way that we can calculate that is really just adding up the probabilities in the three green ellipses, because they correspond to stopping infringing. And we can get those probabilities by propagating the probabilities down through the tree. So after having received the first notice as one of two options, you either go to the left, you stop infringing, or you go to the right, you continue infringing. So that's 0.1 probability that you stop after the first notice. Now for the second opportunity to stop infringing, the second green eclipse, the probability that you end up there is the probability that you go down to the right from the root note of the tree, that's 0.9. And then you stop infringing, so multiply that by 0.15. So that's the second term. And finally, the third ellipse in this tree, the third green one, you would get to by not stopping after the first, which is a 0.9 probability, and then not stopping after the second, which is a 0.85 probability, and then stopping after the third, which is a 0.2 probability. And so that's an example of propagating the probabilities down through the tree, and if we take those three probabilities and add them up, it comes to 0.388. So there's about a 40% chance that an individual faced with such a notice program stops infringing. Another way of actually getting that number is 1 minus the probability that you don't stop infringing. And the probability that you don't stop infringing is just that you go from the red, to the red, to the red. And that probability would be 0.9 x 0.85 x 0.8. And if you do 1 minus that, you're going to get the same number, the 0.388. So that's an example of a probability tree, a useful device when you have a sequence of events.