Welcome back to an Intuitive Introduction to Probability. In today's lecture I want to talk about an example of Subjective Probabilities. In my experience, whenever I introduce a concept of Subjective Probabilities students early on feel a little insecure about what does it mean Subjective probabilities. How do I developed it? Can I just pick any numbers for subjective probabilities? Of course he can't. The subjective probability cannot be 120% or -5%. That would violate the axioms of probability. Now, let's look at an example where I would like you to develop you own subjective probability. here's the description of a woman by the name of Linda. She's 31 y.o., single, outspoken, very smart, majored in philosophy. When she was a student she was deeply concerned with discrimination, social justice, she participated in antinuclear demonstrations. Now I'm gonna give you 8 options that describe Linda and I now would like you to think about what do you think is most likely, second most likely, lowest probability. Of course, based on this description your idea of Linda that you have in your head may be very different than the idea I have of Linda after reading this. And so, now based on this and your experience, your gut feeling you need to develop some subjective probability. So what do you think? Do you think Linda is now a teacher in an elementary school, or does she work in a bookstore and takes yoga classes, active in the feminist movement, if she is a psychiatric social worker and so on, and so on. When I give this question in class for my students to develop subjective probabilities, I ask them to give me numbers 1 through 8. Most likely a 1, least likely an 8. In this format I can't quite do this, so, here now we do a little in class, in lecture quiz. I'm now going to offer you a couple solutions. And pick the one, the ranking, that's closest to your subjective probability. May not be exactly your subjective probability, but pick the one that you think is closest to your feelings. I hope you now filled a out the in class quiz and picked your favorite description. Here now I wanna show you one answer that was among the options that you had that's very popular among my students. And that's the following: people usually rank (many, many do) very high that she's active in the feminist movement based on the description they got. That's usually something people rank very very high. Perhaps a 1, 2 or 3. When I see of the vast majority of people who have to answer this question, they rank "f) Linda is a bank teller" very very low. So, they say it's very unlikely that Linda now works for bank based on the description. And the last option, "Linda is a bank teller who's active in the feminist movement" usually gets sort of an average rank. Now, let's think about whether this is really possible. Before we think about Linda, allow me to go back in an abstract fashion to two events. A and B. And here we have what's called a van diagram. There's a sample space S- or the possible outcomes there's a set on event A of some outcomes B of some other outcomes and A and B may have an intersection. We saw an example of this sort in a previous lecture. Now notice, the intersection of A and B remember from middle school Math, those are the elements that are both in A and in B. This set A intersected B is smaller than A and is also smaller than B. It builds a subset of A and it builds a subset of B. What this means is that it has at most as many usually fewer elements in it than A and B. And therefore the probability of the intersection must be smaller or equal than the probability of the individual events. P of A and P of B. Let me illustrate this again with a simple example of a Fair Die. Look at this picture here. A Fair Die has 6 possible outcomes. 1, 2, 3, 4, 5, 6. A are the even numbers. 2, 4, 6. B are the first four numbers. 1, 2, 3, 4. The 5 is neither an A and B, but it's an S, so that's outside the two circles representing the events but it's still within S. Notice now the intersection. The elements that are both in A and in B those are the two numbers 2 and 4. And here look at this, the intersection probability of A in the sector B is 2 out of 6 that's smaller equals 3 and 6 of A, and it's also small equal 4 and 6 of B. This is a general rule. So, if this is a general rule, your subjective probabilities must obey this rule. Now let's return to Linda. Look at C, F and H. "C" is "Linda's active in the feminist movement". "F" is "Linda is a bank teller". "H" says "Linda is a bank teller who is active in the feminist movement". Ah! Look at H! That's C and F simultaneously! That's intersection. What this means is that Linda is a bank teller who's active in the feminist movement. Must be ranked below bank teller And must be ranked below feminist movement. Of course, your personal opinion may be it's very likely that Linda is a bank teller, maybe she thought it out and she wanted to make a lot of money in later life? You can rank bank taller ahead of feminist. But no matter what you think the intersection probability is gonna be smaller or equal. So, whatever your favorite ranking is H must rank below F and it also must rank below C. So, subjective probability must satisfy these 2 inequalities. I can tell you I have give out hundreds and hundreds of times this questionnaire usually more than 80% of the students in a class get this wrong. So if you picked the wrong option before you're in good and large company. This actually is an example of the famous fallacy a famous way how we, humans, think incorrectly it's called a conjunction fallacy. It was first documented in a series of experiments done by 2 famous psychologists Amos Tversky and Daniel Kahneman. And here I give you citation of a famous paper in psychology from 1983. Sadly, Amos Tversky died of cancer before he could have gotten the Nobel Prize so Daniel Kahneman got the Nobel Prize in economics for his work, not on this fallacy and other fallacies as well sometime later. This work is extremely influential in areas such as behavioral economics and behavioral finance. You may have heard about these fields that are now very popular not only in academia, also in industry and I encourage you to google these terms: conjunction fallacy, behavioral economics, behavioral finance to learn more about these fallacies, these mistakes that we make in decision making. To wrap up. Two events occurring simultaneously cannot be more likely than the individual events by themselves. But often, in our judgement course we make that error it has been well documented in many experiments it's called the conjunction fallacy. So, be careful with your subjective probability. Ir cannot be true that anything is possible. Obviously, the probabilities are between 0 and 1. Not 120%, not -5%. But in addition, there's also this intersection rule to conjunction fallacy, so be careful when you develop your gut feeling into subjective probabilities. Once again, thanks for your attention. I see you in the next lecture.