Now we switch to another experiment which is rolling dice. So actually, unfortunately I don't have a dice now, so I can only show you a picture. But probably you have seen it, so you can also toss it and roll like this, or whatever. And you can get a number which is between 1 and 6, 1, 2, 3, 4, 5, 6. So it's useful if six people go in a restaurant and can decide who of them pays. And natural science part is that these six outcomes appear equally often, it's observation. And mathematics can make some conclusions but of course what we will say this is our trivial conclusion. So for example, we can conclude that even number 2, 4, 6 appears in 50% of cases and a multiple of 3 or 3 or 6 appears just one-third of cases. Why, because it's very easy. We have just 6 outcomes, 1, 2, 3, 4, 5, 6. And if we want to get an even number among this six outcomes, there are three favorable outcomes. 2, 4 and 6. So we have 3 over 6 which is 50%. And for multiple of 3, we have 2 over 6. We have two favorable outcome, 3 and 6. So we get one-third. Trivial, what is not so trivial is what happens if we roll two dice? So imagine if we have a red and blue dice. And then if we roll them or if we do something other way to make them produce random numbers, then we get an outcome which is a red number and blue number. Each of them, for example, x is red and y is blue. And each of them is between 1 and 6. How many of them do we have? If you remember something from combinatorics part of the course, you will tell immediately, it's 36. And even if you don't remember, you can just write all 36 outcomes and believe that they are equiprobable. So this is a table. So red, let me see just for example. 4 on the red dice and 1 on the blue dice on the table. There all possible combinations. If you don't believe me, you can count all the numbers here and you get 36. And the assumption is that they all appear equally often the yield world. So if we assume this, we can compute the probability. So let me explain the language of probability theory. So there is something which is called probability space. It's just the set of all outcomes, so in our case we have 36 outcomes and they form probability space. Also there is some event, something, this is property of outcome, it may happen or not. And it's a set of outcomes and these outcomes are called favorable. We want something somehow. Of course we can consider probability of undesirable events, so it will be unfavorable count. But just this is a kind of language used so if we have an event we call the outcomes from this event favorable. So for example, imagine you have an event that red number is bigger than blue number. And we want to compute the probability of this event. And probability's just the fraction of this outcomes in the long series. So if we believe that all 36 will appear with the same frequency, we just want, what we should do we should compute the number of favorable outcomes and divide it by 36. So where the red is bigger, red number is bigger. So here, here, here, here they are equal so this is all this. No, I think I prepared a nice big chart. Yeah, with boxes, so these boxes show favorable outcomes. And then you can count them 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. And so we got 15 of our 36. We know that favorable outcome will in in a long run form 15 of a 36 fraction. This is called the probability of the event. What is very important, it's called independence. We assume that red and blue dice are independent. Let me explain what does it mean. So we have 36 outcomes and we believe that there appear equally often. So red dice can give 1, 2, 3, so there are, let's see. Rows, we can have 1, 2, 3, 4, 5, 6, it's equally often. And also blue dice appears with 1, 2, 3, 4, 5, 6. And these cases are equally often. But this is not only, it's very important things that they are also independent. For example, imagine some magic dice which always for some strange reason, red and blue show the same thing. And then of course, on almost all outcomes, it appear what we have is something like this. This is the only six possibilities. And these six possibilities are equiprobable and then indeed the red dice is okay in dilation, and the blue is okay in dilation. But of course all our theory doesn't work. Because they are somehow dependent. That's the language of probability theory. We will return to this dependence question later. So independence is somehow more than it's equiprobability for both dice. We assume that all 36 outcomes have the same frequency. And it actually happens in the real world, for many settings. For example, you can roll the two dice at the same time. Or another thing you can do is you can take, first you can roll the red one and then the blue one. You can do even more. You can take one dice and roll it twice. And first you get the red number and then you get the blue number. And then all three settings actually infect in the real world all combinations will appear equally often. This is observation about the real world but then we can compute the probabilities as we have done in our example. So this is again the division, equiprobable model happens in the real world and then mathematicians use it for computation of probabilities.
