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In today's lecture, I want to talk about an example of subjective probabilities.

Â In my experience, whenever I introduce the concept of subjective probabilities,

Â students early on feel a little bit insecure about the idea.

Â What does it mean subjective probabilities?

Â How do I develop it?

Â Can I just pick any numbers for subjective probabilities?

Â Of course, we can't.

Â The subjective probability cannot be a 120% or

Â minus 5%, that would violate the exhumes of probability.

Â Now, let's look at an example where I would like you to develop

Â your own subjective probability.

Â Here's a description of a woman by the name of Linda.

Â She's 31 years old, single, outspoken, very smart, majored in philosophy.

Â When she was a student, she was deeply concerned with discrimination,

Â social justice.

Â She participated in antinuclear demonstrations.

Â Now I'm going to give you eight options that describe Linda and

Â I now would like you to think about what do you think is most likely,

Â second most likely, lowest probability?

Â Of course, based on this description, your idea of Linda that you have in

Â your head may be very different than the idea I have of Linda after reading this.

Â And so now based on this and your experience, you've got feeling.

Â You need to develop some subjective probability.

Â So, what do you think?

Â Do you think that she is now a teacher in an elementary school?

Â Or does she work in a bookstore and take yoga classes,

Â active in the feminist movement?

Â Is she a psychiatric social worker?

Â And so on and so on.

Â When I give this question in class for my students to develop subjective

Â probabilities, I ask them to give me numbers one through eight.

Â Most likely a one, least likely an eight.

Â In this format, I can't quite do this.

Â So here now we do a little in-class, in-lecture quiz.

Â I'm now going to offer you a couple solutions and pick the one,

Â the ranking that's closest to your subjective probability.

Â It may not be exactly your subjective probability, but

Â pick the one that you think is closest to your feelings.

Â 2:33

I hope you now filled out the in-class quiz and picked your favorite description.

Â Here is now, I want to show you one answer that was among the options that you have,

Â that's very popular among my students and that's the following.

Â People usually rank many, many do, very high that she's active

Â in the feminist movement based on the description they got.

Â That's usually something people rank very, very highly.

Â Perhaps a one, a two or a three.

Â What I see from the vast majority of people will have to answer this question.

Â They ranked f, Linda is a bank teller.

Â Very, very low.

Â 3:20

So they say, it's very unlikely that Linda now works for

Â a bank based on the description.

Â And the last option, Linda is a bank teller who is active in

Â the feminist movement, usually gets sort of average rank.

Â Now, let's think about whether this is really possible.

Â Before we think about Linda,

Â allow me to go back in an abstract session to two events, A and B.

Â 3:51

And here, we have what's called a Venn diagram.

Â There's a sample space, S all the possible outcomes.

Â There's a set of some outcomes, B with some other outcomes and

Â A and B may have an intersection.

Â We saw those, an example of this sort in a previous lecture.

Â Now notice the intersection of A and B.

Â Remember from middle school math,

Â those are the elements that are both in A and in B.

Â This set A intersected B is smaller than A and it's also small line B.

Â It builds a subset of A and it builds a subset of B.

Â What this means is that it has at most as many,

Â usually fewer elements in it than A and B.

Â And therefore, the probability of the intersection must be smaller or

Â equal than the probability of the individual events.

Â P of A and P of B.

Â Let me illustrate this again with the simple example of a Fair Die.

Â Look at this picture here, a Fair Die has six possible.

Â Outcomes one, two, three, four, five, six.

Â A are the even numbers, two, four, six.

Â B are the first four numbers, one, two, three, four.

Â The five is neither A and B, but it's an S.

Â So that's outside the two circles representing the events, but

Â it's still within S.

Â Notice now the intersection.

Â The elements that are both in A and in B.

Â Those are the two numbers, two and four.

Â And here look at this, the intersection.

Â Probability of A intersection B is two out of six, that's smaller equal three and

Â six of A and it's also small equal four and six of B.

Â This is a general rule.

Â 6:21

What this means is that Linda is a bank teller who is active

Â in the feminist movement must be ranked below bank teller and

Â must be ranked below feminist movement.

Â Of course, your personal opinion may be it is very likely that Linda is a bankteller.

Â Maybe she sold out and she wanted to make a lot of money later in life.

Â You can rank bank teller ahead of feminist.

Â But no matter what you think,

Â the intersection probability must is going to be smaller or equal.

Â So whatever your favorite ranking is,

Â h must rank below f and it also must rank below c.

Â 7:15

I can tell you, I have given out hundreds and hundreds of times this questionnaire,

Â usually more than 80% of the students in the class get this wrong.

Â So if you picked the wrong option before, you are in good and large company.

Â This actually is an example of a famous fallacy,

Â a famous way how we, humans think incorrectly.

Â It's called the conjunction fallacy.

Â It was first documented in a series of experiments done by two

Â famous psychologists, Amos Tversky and Daniel Kahneman.

Â And here, I give you citation of a famous paper in psychology from 1983.

Â Sadly, Amos Tversky died of cancer before he could have got the Nobel Prize.

Â So Daniel Kahneman got the Nobel Prize in economics for

Â his work not on this fallacy and other fallacies, as well sometime later.

Â This work is extremely influential in areas, such as behavioral economics and

Â behavioral science.

Â You may have heard about these fields that are now very popular not only in academia,

Â also in industry and I encourage you to Google these terms.

Â Conjunction fallacy, behavioral economics,

Â behavioral science to learn more about these fallacies.

Â These mistakes that we make in decision making.

Â To wrap up, two events occurring simultaneously cannot

Â be more likely than the individual events by themselves.

Â But often in our judgement calls, we make that error.

Â It has been well-documented in many experiments.

Â It's called the conjunction fallacy.

Â So, be careful with your subjective probability.

Â It cannot be true that anything is possible.

Â Obviously, the probabilities are between zero and one not 120%, not minus 5%.

Â But in addition, there's also this intersection rule.

Â The conjunction fallacy.

Â So, be careful when you develop your gut feeling and your subjective probabilities.

Â Once again, thanks for your attention.

Â