Welcome back to intuitive introduction to probability. Currently we are talking about conditional probabilities. In the last lecture I tried to give you a first intuitive look at what conditional probabilities are in this lecture I now want to be a little more precise and define the exact, or give you the exact definition of the conditional probability. So, as we have seen last time, once event B has occurred we can thing of the state space S shrinking down to B. All the outcomes outside B no longer play a role. They didn't occur, we know B has occurred so now, we want to focus on B and thing what can we now say about his other even A? Before we had probability of A under entire state space S. What is left of that now? Just B. And this gives us now the definition of the conditional probability. The probability of A given B is what is left of A in B it's the intersection of A and B. In this case the probability of A intersected B and we're dividing this intersection probability by the probability of the event B. In some sense you can think of this, we are normalizing our probabilities on the new states space B, the event that occurred. If that sounds abstract to you let's look at our stand up basic example: A Fair Die. We first have the event A, an even number, 2, 4, 6 ex ante before we receive more information A, an even number, is an even number probability of a half, 50%. The even B are the smaller half of the numbers so, 1, 2, 3, also as a probability of a half. Now, I roll a die, and the even B occurs. What happens now to the probability of A? That's now a question for you. Do you think with this information that B happened 1, 2, 3, does the probability of A happening go up stays the same or decrease? Here we do a brief in class quiz, please answer this and then it's back to me, thank you. Okay! Thanks for answering the quiz question now let's see what happened. So, the die shows 1, 2, or 3. What does that mean? If it's a 1, A did not occur. If it's a 3, A did not occur. But if it's a 2, 2 is an element of A than A indeed occurs. So, let's look at the diagram. Instead of the large sample space S, we can forget about this now we are down to just B. B now has three elements, 1, 2, 3, so we see a 2 out of those three, probability should be a third. In two cases, 1 and 3, it doesn't occur in the last case, 2, A occurs, one out of three. So the probability of A now is decreasing from a half, 50% to a third, 33.33%. Let's look what the definition of the conditional probability tells us. It tells us to do the probability of A intersect B divided by the probability of B. These are the original probabilities if the section probability is 1/6 that's just the element of 2 probability of B was a half. 1/6 divided by 1/2 is 2/6 or 1/3. And indeed, this definition gives us the correct probability that we intuitively would calculate looking at the picture of just B. To summarize, your original probability ex ante was a half an even number has a probability of a half then you're given the information the event B occurred I roll the die and tell you B occurred, it's a 1, 2 or 3. Now you update the probability using the conditioned probability definition and you realize the probability decreased and the likelihood is now down to 1/3. Now let me remind you of the concept of the previous module so the concept of independence and dependence. Notice, the information of B now changed the probability of A so the event A, the likelihood of the event depends of B occurring or not. That (INAUDIBLE) means, A and B are dependent. Now, let's look at a slight change of the problem. Here now, again as the set A, the even numbers but instead of B now we're looking at a set B' which are the smallest four numbers. So we change B from B to B' by adding the 4 to that event. Again, I roll a die, you can't see it, and I tell you: "Listen! B' occurred." Now, we go from the said S down to the event B'. What happens now? In B' we have four numbers, 1 and 3 means A does not occur 2 and 4 means A also occurs. So now the probability of A given B' you do the math, it's again 1/2. So what happened? Before you said the even number had the probability of a half B' occurs, and the probability hasn't changed. Nothing changed. You're not updating the probability, if you're updating, nothing changed it's still a half. Does this remind you of the concept of independence from the previous module? Now, B occurring does not make you change the probability of A. In that sense, A and B' are independent. Let's summarize what we have seen here. I've given you a proper definition of conditional probability. Probability of A given B equals to the probability of the intersection of A and B divided by the probability of B. We have seen when A and B are dependent then the conditional probability is different from the original probability. In the case that A and B are independent then the probabilities are the same. So the probability of A equals to the probability of A given B. We will take another look at these concepts in the next few lectures. Thank you and please come back.