Welcome back to Intuitive Introduction to Probability. In the last lecture I gave you some intuitive definitions of how we can define probabilities. There was a classical definition, the empirical probability definition, and the subjective probability definition. In this lecture I now want to be a little more precise and give correct definitions of what a probability exactly is. And we will return to those 3 definitions. So, we need a little language to describe the probabilities setting. As first, Random Experiment. Sounds kind of strange, kind of sounds like we're in a lab, but in probability theory anything that has an uncertain outcome is called Random Experiment. That could be rolling a die, that could be pulling a card this is roulette table, but it's also in the real world. The weather next Monday, or if you think about a stock price of your favorite company, the stock price next Tuesday. You don't know those, those are uncertain outcomes. Therefore in our language those are random experiments. Now, what can happen in the random experiment? Those are called Basic Outcomes. In a die it's very easy. 1, 2, 3, 4, 5, 6 are the basic outcomes. If you think about the exact weather of the exact stock price, this is already a bit more tricky and we will see examples as we go along. So, collection of all basic outcomes are called Sample Space. And I apologize, we need a little notation, here's our first notation in this class, S for the Sample Space. Now, when we want to define probabilities, we need subsets some elements, perhaps not all of them offer Sample Space. And those are called Event. And again we need a little notation. Those are depicted and ordered by capital letters A, B, C and so on. Now, having said this, we can define probability. That's the chance or the likelihood that an uncertain event will occur. That's a number that's between 0 and 1. Some people prefer to use percentages 0% to 100%. Now here be careful. A probability cannot be larger than 100%, cannot be 120%. it can also not be negative, -10%. Sometimes people confuse growth rates which can be negative or larger than 100% with probabilities. So, please be careful. Probabilities are numbers between 0 and 1, 0% and 100%. And now we can give precise definitions for the 3 probability concepts. The classical probability. concept rests on an important assumption all basic outcomes are equally likely. So, for example, in the Fair Die that's given every number 1, 2, 3, 4, 5, 6 is equally likely. And then the probability of an event is the number of elements in your event divided by the total amount of basic outcomes. In the Sample Space we'll see examples soon. Now, as I say, in addition to Fair Die that is true in a lottery at a roulette table and so on. Now, this key assumption that all the basic outcomes have to be equally likely is often not satisfied when we talk about real world applications. And therefore, this beautiful definition while helpful when you play a card game with your buddies or a dice game with some kids is essentially useless for real world applications. That's why we need more definitions. If you have data and you can derive proportions from historical data that's when we get to the empirical probability definition. Also called the relative frequency probability. Here the ideas now that we have repeated trials of an experiment. For example, I may look at the weather in a particular day for the last 20 years. And say "Oh, can I use some average there to make prediction what's the probability it will rain?" What's the probability it will be sunny? Or, if I think of stock prices in the stock market. People in finance love to look at the last 250 trading days and say how many days did the stock go up. How many days did it go down. And based on that, get empirical definitions. So, it's the definition says how often did an event occur in a series of trials divided by the number of trials. That's now the empirical probability definition also used in medicine, in the pharmaceutical industry whenever we do drug testing. Sometimes things get even worse. You don't have data, you have no clue and at that point we go to the subjective probability definition. Very important, for example, if you have a new product development you bring out a brand new product, really disruptive innovation you have no idea whether your customers like it or not. Maybe they have some, what managers like to call it, gut feeling or experience. And based on that experience, you say: "Oh, I think there's 75% chance that will be a successful product". And, so in that case we talk about subjective probabilities. Very important in managerial decision making and everyday decision making. Now, basic probabilities have to satisfy some goods. Now, I have to give you 3 rules, looks rather mathematical there's nothing to be understood here. These are also called axioms. Or after Russian mathematician who was the first to write these down in 1933, also called Kolmogorov's axioms. These are just assumptions that we put in place for probability theory. Everything else can be derived from this. Rule #1. The probability of any outcome in sample space P of S must be 1. Something must happen when we do our random experiment. P of S is 1. Second rule, very intuitive, any probability is a number between 0 and 1. Can't be larger than 1, cannot be negative. Please keep that in mind. And finally, the first little more complicated rule if you have 2 events that have no elements in common also called disjoint by mathematicians the end of probability that A or B happens or the probability that A union B happens equals the sum of the individual probabilities P of A + P of B. If that looks a little tricky already to you, let's look at an example and let's go back to Fair Die. What's random experiment? I'm rolling a fair die. I don't know what's going to happen. That's now my experiment. What are the possible outcomes? 1, 2, 3, 4, 5, 6. And they together build the sample space. Now, what I told you about events? Events are subsets of S, and here are defined I pick an event A, the even numbers 2, 4, 6 and the event B 1 and 5. And now let's look at A and B and their probabilities and see how this works. What's the appropriate definition here? I don't have to make up probabilities. There's no subjective probability here. I don't have to roll the die 1000 times to determine empirical probabilities. Since we believe it's a fair die, we can use a classical definition All 6 numbers are equally likely. And so, I'm allowed to just divide. So, clearly P of any number 1 through 6 is equal to 1. Seven cannot happen, zero cannot happen, pi cannot happen. Now, let's look at the probabilities of the 2 events. A has three elements - 2, 4, 6 3 out of 6 is a 1/2 = 0.5 = 50%. B only has 2 elements. So, probability of B = 2 divided by 6, 1/3. So, here we used the first definition - classical probability. Now A union B. Hopefully you remember from your middle school math class if I have 2 sets and I build the union I take all the elements together, so if I take A (2, 4, 6) with the union of 1 and 5 I get 1, 2, 4, 5, 6. Probability now of A union B of either A happening or B is 5 divided by 6, 5 elements divided by 6. Notice that those 2 elements have no elements in common. There's no number that's in A and in B. So, they are disjoint, the intersection is the empty set. And therefore now I can use my probability rule. The probability of A union B is a sum of P of A and P of B 3/6 plus 2/6 is 5/6 and, guess what? That's exactly the right answer that we saw before. Those are now the... I showed you now the 3 axioms of fundamental rules. From those we can derive further rules, some additional rules that are very very helpful. First, the complement rule. What's the compliment of a set? The compliment of an event A or the set A are all the elements in S that are not in A. And not surprisingly the complement rule says the probability that the opposite of A happens is just 1 minus the probability that A happens. And then we have addition rules. The general addition rule that always holds even when A and B are not disjoined when there's something in the intersection. And then the rule gets a little more complicated then the probability of A union B if P of A plus P of B but then now I need to subtract the probability of the intersection because otherwise there will be some double counting. Let's look again at our little example. What's the opposite of an even number? 2, 4, 6, you see. Odd numbers, so compliment of the odd numbers. 1, 3 and 5. So probability of an odd number is 1 minus the probability of the even numbers, 1 minus a half, "Bingo!" is a half again. Not let's look at an event C that has elements 1, 2, 3, 4. A union C, now is 1, 2, 3, 4, 6 5 numbers, 5 our of 6 so the probability should get 5, 6. Now if I use rule, probability of A union B equals P of A plus P of C and I add A is 3/6, C has a probability of 4/6 I get 7/6. Oh! That's not so correct probability because I'm double counting the number 2 and 4 which are both in A and in C. And therefore I need to subtract them out, that why we now have this general rule, and bingo, I get again the right answer 5 divided by 6. To summarize this lecture I gave you formal definitions of the 3 probability concepts that we have. Very important, familiarize yourself of them. It's not always a classical probability. That we learn as kids as soon as we play a dice game or we play a card game. There are more important definitions for real world decision making. The empirical probability definition and the subjective probability definition. I showed you the fundamental rules, also called the axioms of probability and finally 2 derived rules which are very helpful in application and we will see them in action in the next couple lectures. Thanks for your attention. I look forward to see you back in the next lecture.