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Hello, my name is Jose, and

Â I'm going to be your teaching assistant during this online course.

Â What we are going to do here together is to solve a series of five exercises

Â in which you are going to be able to apply the theory that you learned with Carl

Â in the theoretical models, to a very specific question.

Â Every module starts with a short story, a short introduction,

Â from which we will take the information to answer very specific statistical question.

Â So let's start with module one.

Â In the movie Star Wars, there is a very famous scene where

Â some of the characters of the movie are trying to escape with an aircraft,

Â the Millennium Falcon, through an asteroid field.

Â So we have Chewbacca, Han Solo, Princess Leia.

Â And then, the droids, C-3PO or R2-D2, calculates the probability

Â that they have to survive if they escape through the asteroid field.

Â So for you to understand it,

Â there is an asteroid field, which has some parts filled with air,

Â through which the aircraft can easily fly through, and some parts full of asteroids.

Â The density of this asteroid field is 90% asteroids, 10% air.

Â There are two type of asteroids.

Â There are asteroids of type A, which are small asteroid

Â which we can find with a probability of 30% within the whole set of asteroids.

Â There are also asteroids of type B, which are very big asteroids,

Â which we can obviously find with a probability of 70%.

Â So in the asteroid field we have 10% of air, 90% of asteroids.

Â And then within the asteroids, we have 30% of small asteroids and

Â 70% of big asteroids.

Â If the aircraft crashes, again, one of the small asteroids,

Â it will get damaged, but it will still be able to fly through the asteroid field.

Â However, if the asteroid is of type B, is a big asteroid, and

Â the Millennium Falcon crashes against it, the aircraft will be so

Â damaged that the characters of the movie won't be able to make it.

Â So let's start answering some questions regarding this assignment, this exercise.

Â 2:16

So we can scroll down a little bit and start answering the questions.

Â First question is, can you identify the random experiment in this story?

Â Well, it is actually a very simple question

Â in which we can answer with a clear of course.

Â The process leading to an uncertain outcome, which is what defines a random

Â experiment, is flying through the asteroid field trying to avoid the asteroids.

Â Then, we can move to the second question.

Â What is the state space of this random experiment?

Â Well, the state space of the random experiment are either we hit

Â on air when we go through the asteroid field, or

Â we hit on an asteroid of type A, or we hit on an asteroid of type B.

Â So our state space in this random experiment is either air,

Â asteroid of type A, or asteroid of type B.

Â 3:08

Then we can move to the next, also introductory question.

Â What's the probability of hitting only air when crossing the asteroid field?

Â Well, since we know that density of the asteroid field is 90% asteroids

Â the remaining part is only 0.1 meaning 10% of probability for the state air.

Â So we can answer this question with a clear 10%.

Â Now we go to a slightly more complicated question, which is what's

Â the probability of hitting an asteroid of type A when crossing the asteroid field?

Â Well, we know that the probability of hitting an asteroid of any kind is 90%.

Â Additionally, we know that 30% of the asteroids that we can hit are of type A.

Â So if we multiply 90%, which is the probability of hitting one asteroid

Â of any kind, by the probability of hitting an asteroid of type A,

Â which is 30%, better said with the probability that we find

Â among the asteroids one asteroid of type A, which is 30%.

Â If we multiply these two values, 0.9 times 0.3,

Â we get 0.27, meaning 27% of hitting an asteroid of type

Â A when we just randomly cross the asteroid field.

Â 4:27

This leads us to a similar question, which is what is the probability of

Â hitting an asteroid of type B, when crossing the asteroid field?

Â Well, we have to do a very similar calculation.

Â Probability of hitting an asteroid of any type is 90%.

Â Probability that the asteroid, if we hit an asteroid,

Â is of type B, is 70%, so if we multiply 0.7 by 0.9,

Â we get 0.63, which is 63% of probability of hitting

Â an asteroid of type B when randomly crossing the asteroid field.

Â Now, we can scroll down a little bit further and go to the next question.

Â What's the survival probability of the Millennium Falcon?

Â Well, now we know that the asteroids of type A are not big enough to damage

Â the Millennium Falcon such that it doesn't survive to a hit.

Â So the Millennium Falcon will survive if it either hits on air or

Â if it either hits on an asteroid of type A.

Â Hence, we can multiply the probabilities of hitting on air which we know,

Â had previously calculated is 10% and

Â the probability of hitting on an asteroid of type A, which is 27%.

Â If we add up these two probabilities, we get to 37,

Â which is the probability that the Millennium Falcon, which is the name of

Â the aircraft in which the characters of the movie are flying, actually survives.

Â 6:08

Last question, to wrap up a little bit,

Â is if the laws of probability are fulfilled in these examples.

Â Well, we can count on the three laws of probability again.

Â First, the sum of the probabilities equals 1,

Â which in this case, is true, probability of hitting on air is 10%.

Â Probability of hitting on an asteroid of type A is 27%.

Â And probability of hitting an asteroid of type B is 63% which

Â add up equals to 1 meaning 100%.

Â All the probabilities are between 0 and 1.

Â Well, this is as you know defined in the exercise setting.

Â And the third and the last law,

Â is the probability of any two disjoint events is the sum of their probabilities,

Â which by definition of this setting is obviously fulfilled.

Â I hope that you have enjoyed the first exercise, and

Â that you are now familiar when working with some data and some probabilities.

Â Enjoy and see you in the next module.

Â Thank you very much.

Â