Hello, my name is Jose, and I'm going to be your teaching assistant during this online course. What we are going to do here together is to solve a series of five exercises in which you are going to be able to apply the theory that you learned with Carl in the theoretical models, to a very specific question. Every module starts with a short story, a short introduction, from which we will take the information to answer very specific statistical question. So let's start with module one. In the movie Star Wars, there is a very famous scene where some of the characters of the movie are trying to escape with an aircraft, the Millennium Falcon, through an asteroid field. So we have Chewbacca, Han Solo, Princess Leia. And then, the droids, C-3PO or R2-D2, calculates the probability that they have to survive if they escape through the asteroid field. So for you to understand it, there is an asteroid field, which has some parts filled with air, through which the aircraft can easily fly through, and some parts full of asteroids. The density of this asteroid field is 90% asteroids, 10% air. There are two type of asteroids. There are asteroids of type A, which are small asteroid which we can find with a probability of 30% within the whole set of asteroids. There are also asteroids of type B, which are very big asteroids, which we can obviously find with a probability of 70%. So in the asteroid field we have 10% of air, 90% of asteroids. And then within the asteroids, we have 30% of small asteroids and 70% of big asteroids. If the aircraft crashes, again, one of the small asteroids, it will get damaged, but it will still be able to fly through the asteroid field. However, if the asteroid is of type B, is a big asteroid, and the Millennium Falcon crashes against it, the aircraft will be so damaged that the characters of the movie won't be able to make it. So let's start answering some questions regarding this assignment, this exercise. So we can scroll down a little bit and start answering the questions. First question is, can you identify the random experiment in this story? Well, it is actually a very simple question in which we can answer with a clear of course. The process leading to an uncertain outcome, which is what defines a random experiment, is flying through the asteroid field trying to avoid the asteroids. Then, we can move to the second question. What is the state space of this random experiment? Well, the state space of the random experiment are either we hit on air when we go through the asteroid field, or we hit on an asteroid of type A, or we hit on an asteroid of type B. So our state space in this random experiment is either air, asteroid of type A, or asteroid of type B. Then we can move to the next, also introductory question. What's the probability of hitting only air when crossing the asteroid field? Well, since we know that density of the asteroid field is 90% asteroids the remaining part is only 0.1 meaning 10% of probability for the state air. So we can answer this question with a clear 10%. Now we go to a slightly more complicated question, which is what's the probability of hitting an asteroid of type A when crossing the asteroid field? Well, we know that the probability of hitting an asteroid of any kind is 90%. Additionally, we know that 30% of the asteroids that we can hit are of type A. So if we multiply 90%, which is the probability of hitting one asteroid of any kind, by the probability of hitting an asteroid of type A, which is 30%, better said with the probability that we find among the asteroids one asteroid of type A, which is 30%. If we multiply these two values, 0.9 times 0.3, we get 0.27, meaning 27% of hitting an asteroid of type A when we just randomly cross the asteroid field. This leads us to a similar question, which is what is the probability of hitting an asteroid of type B, when crossing the asteroid field? Well, we have to do a very similar calculation. Probability of hitting an asteroid of any type is 90%. Probability that the asteroid, if we hit an asteroid, is of type B, is 70%, so if we multiply 0.7 by 0.9, we get 0.63, which is 63% of probability of hitting an asteroid of type B when randomly crossing the asteroid field. Now, we can scroll down a little bit further and go to the next question. What's the survival probability of the Millennium Falcon? Well, now we know that the asteroids of type A are not big enough to damage the Millennium Falcon such that it doesn't survive to a hit. So the Millennium Falcon will survive if it either hits on air or if it either hits on an asteroid of type A. Hence, we can multiply the probabilities of hitting on air which we know, had previously calculated is 10% and the probability of hitting on an asteroid of type A, which is 27%. If we add up these two probabilities, we get to 37, which is the probability that the Millennium Falcon, which is the name of the aircraft in which the characters of the movie are flying, actually survives. We can now move to the next question which is what's the complement of air? As you have probably noticed, this is also a very simple theoretical question. Well, the complement of air is hitting an asteroid of any kind, is just all the other events that are just not air. Last question, to wrap up a little bit, is if the laws of probability are fulfilled in these examples. Well, we can count on the three laws of probability again. First, the sum of the probabilities equals 1, which in this case, is true, probability of hitting on air is 10%. Probability of hitting on an asteroid of type A is 27%. And probability of hitting an asteroid of type B is 63% which add up equals to 1 meaning 100%. All the probabilities are between 0 and 1. Well, this is as you know defined in the exercise setting. And the third and the last law, is the probability of any two disjoint events is the sum of their probabilities, which by definition of this setting is obviously fulfilled. I hope that you have enjoyed the first exercise, and that you are now familiar when working with some data and some probabilities. Enjoy and see you in the next module. Thank you very much.