Okay, so if I don't misunderstand things, it should be over now. Okay. So who of you

got an error at some point? And who of you got an error that was prohibitive enough that

you couldn't actually continue? So apparently no one. So every one of you managed to get

through the quiz ultimately, right? You didn't? Okay, who else didn't? Please hands up just

I get a good number of things. Okay, so those are a few. So who of you got an error that was

severe enough that you could not finish the quiz? One, two, three, four. Okay. Of any one of you

who got an error? Did you try the like test quiz at some point before? And did it work then? Okay,

so it worked earlier, but it didn't work this time. Okay. In that case, I will have to like

that to our front-end engineer and see what might have gone wrong there. For those of you again who

got an error, was it all a client-side error or did any one of you get a different error? Okay,

so who did not get a client-side error? You? Sorry, can you quiet down please a bit? Sorry?

Okay, I'm gonna make a stood on post later tonight regarding the quiz today. And any one of you who

had sufficiently severe issues, please add a comment there so that we can collect and see what

happened. And then ideally get things resolved until next week. Used to work fine last semester

after the first four or five trials. So I'm somewhat confident that it will be similar this

semester. Okay, before we continue with the actual content, one more admin note. We've changed the

tutorial sessions. Oh, if the thing is on, it actually works. We changed the tutorial sessions.

The most pressing thing is that the session on Thursday got scrapped. That's a bit disappointing

maybe to some of you. The problem there being that the tutors are also students and also have

schedules. So we didn't get around that. But keep in mind that those are now the updated ones. I've

updated them on stood on. So don't accidentally try to go to a session on Thursday that doesn't

exist. Okay. Now for the interesting things. There we go. Okay. For the math, we started talking a

bit about that last week. So the foundation of everything we're going to do this semester is

naturally probability theory. So we need to know what probabilities even are. And this is the

mathematical framework for what probabilities are. Namely, we have a pair consisting of a set of

outcomes. And we have a function P that assigns probabilities to sets of those outcomes. The

reason why we use sets is that if our outcomes are distributed continuously, i.e. if we have any

kind of events we're interested in where the result is a real number, then sets are the only

things that we can assign meaningful probabilities to in the first place. And the idea is that we

have some kind of experiment where omega is just the set of all possible outcomes of that

particular experiment. And then we need to make sure that the function P that we add to that

probability space satisfies these two probabilities. The first one basically just saying that the

probability that anything happens is precisely one. And the other one making sure that if I have

disjoint sets of outcomes, that the probabilities of the union of those disjoint sets is just the

sum of the probabilities of the individual sets. The simplest example for that is something like

throwing a coin or throwing a dice. In that case, the set of outcomes is just a finite set. And the

probabilities of the sets literally are just the sums of the probabilities of the individual

singleton sets of the outcomes. i.e. if I throw a coin, singleton set head is 0.5, singleton set tail

is 0.5. And this particular, the second axiom that makes sure that pairwise disjoints probabilities

I can sum up is exactly the one that guarantees that if I say the probability of heads is 0.5,

then the probability of tails also needs to be 0.5. Why? Because the singleton set head and the

singleton set tail are disjoint and we know that together they have to sum up to one. Therefore,

the same for any Boolean random variable, whatever probability I give the one value,

one minus that probability will have to be the probability of the other outcome. In the simple

case, we really just have two outcomes in the first place. For a dice, I can throw, for example,

two dice. In that case, the set of outcomes would just be the set of pairs i, j, where i and j are

between 1 and 6. One more thing to mention, which I also already mentioned last week,

if you Google this definition and you read the word sigma algebra, think power set every time you

hear the word sigma algebra, if you read anything about a measurable space, just think a set. That's

basically the gist of it. And the things that we are actually interested in, which we can now define