Welcome to our next module of intuitive introduction to probability. This module concerns "Conditional Probabilities". In my experience with students that's usually the topic where people are really struggling to wrap their head around the formulas so before we look at any formulas let's look at some data some empirical probabilities and develop some intuition. I put together a data set here with some population data from Switzerland. Before we look at the data let's think of the following. You come for vacation with it to Switzerland and you meet your first Swiss person. And here are some questions. What is the probability that the first person you meet is younger than twenty years old? So, is he or she somewhere between zero and nineteen? And then you may ask what is the probability that this person's main or first mother tongue language is French? So, we have data on this which I can tell you and show you in a moment. You talk to the person and the person tells you: "Actually, I am residing in the canton of Zurich." Question: Does this information reveal anything about the answers to the previous questions based on this assumption should you change your idea if this person, and the probability of him or her being younger than twenty or having the mother tongue French? Quick, for those of you who don't know much about Switzerland Switzerland is divided essentially in four regions when it comes to languages. The largest part in Switzerland people there, or the majority of the people residing there have German as their first language. The second largest region is the French speaking region predominantly in the south-west of Switzerland. Then there's a canton of Ticino where the mother tongue for most people is Italian. Many people are aware of these three languages but then in the canton of Graubünden there are some regions where there's a forth language as the main language Romansh. And so, we are now talking about the canton of Zurich. Here's now the Swiss population data. The person residing in Switzerland 20.2% of them are between 0 and 19 years old 22.4% of them have French as the first language. So, this is the data, now if you use the empirical probability definition we would say, you pick a random person out of Switzerland the first langue, French 22.4% is the probability or 0.224. Now the person tells you that he or she's residing in Zurich. Now the data tells us: "Wait a minute!" The probability that French is the first language now is suddenly 3.2%. What just happened? You received information from the person: "I am residing in the canton of Zurich." And now you need to adapt or you need to change the probability. Before, your probability of French was 22.4%, now you lower it to 3.2%. So, this probability changed. The occurrence of the event the person is residing in the canton of Zurich. made you change the probability you now have a different assessment of the likelihood than of the event, French. Here's now the general idea. Leaving the example behind us, thinking of the general concept of the conditional probability. You have an initial probability of an event P(A) that can be, again a classical probability, an empirical probability as we just saw in our data example or subjective probability. Then an even occurs, and even B occurs and as a result of B occurring, you now say: "Oh, I need to change my probability." I update it under the condition B, the probability's now different. Here now, we need to gain a little bit of terminology at the bottom of the slide here, the probability of A, then this vertical line B, we state the conditional probability of A given B, and that's the new probability here. So, back now to our example, your original probability P of French was 22.4%, now the event person from canton of Zurich occurs, and now the Probability of French under the condition that the person is from Zurich or as said more in the language of conditional probability is the probability of French given Zurich is 3.2%. And that's now a conditional probability. Now, here I've shown you where the canton of Zurich is. It's in the area of Switzerland where German is a predominant language but within this canton 3.2% of the people speak French as a first language, while in the entire country thanks to the large French part in the south-west it's 22.4%. Now, let me try again to get the intuition across. In the general setup of probability, here you see once again a venn diagram, we have entire sample space of possible outcomes S we have an event A, and and even B and then the intersection, A intersect B. Now the event B happens. How should you think of this? What should you now think that S is out of the window. We no longer think of the sample space of S. Now I know B occurred, so essentially this event B is like my new sample space. Back to our example. Originally we were talking about all of Switzerland. Now, after the person tells you: "I am from Zurich." Our new sample space adjusts to people residing in the canton of Zurich. So, from Switzerland we went to Zurich. Here now in general from the sample space S we go to just the even B and that's now our new sample space. And so the outcomes outside B are no longer possible and we can thing of the B as our new state space and then the question is: Now if this is our new state space B, how likely is the event A within B? That's the conditional probability. Probability of A given B is exactly the answer to that question. How likely is the event A within the new state space B. And that's it! That's the key intuition of conditional probability. To sum up. The occurrence of some events or some information that you're given may update, or may lead you to change the probability that you have of some events and how we do this, the concept of this is conditional probability. This sums up our first intuitive look at what a conditional probability is. In the next lecture we will formalize this intuition a little bit more with some precise definitions. So, please come back to the next lecture. Thank you very much.
