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

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

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

What does it mean subjective probabilities?

How do I develop it?

Can I just pick any numbers for subjective probabilities?

Of course, we can't.

The subjective probability cannot be a 120% or

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

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

your own subjective probability.

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

She's 31 years old, 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 going to give you eight 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, you've got feeling.

You need to develop some subjective probability.

So, what do you think?

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

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

active in the feminist movement?

Is she 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 one through eight.

Most likely a one, least likely an eight.

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.

It may not be exactly your subjective probability, but

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

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

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

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 highly.

Perhaps a one, a two or a three.

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

They ranked f, Linda is a bank teller.

Very, very low.

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

a bank based on the description.

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

the feminist movement, usually gets sort of 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 session to two events, A and B.

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

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

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

A and B may have an intersection.

We saw those, 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 it's also small line 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 the simple example of a Fair Die.

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

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

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

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

The five is neither 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, two and four.

And here look at this, the intersection.

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

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

This is a general rule.

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What this means is that Linda is a bank teller who is 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 is very likely that Linda is a bankteller.

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

You can rank bank teller ahead of feminist.

But no matter what you think,

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

So whatever your favorite ranking is,

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

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

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

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

This actually is an example of a famous fallacy,

a famous way how we, humans think incorrectly.

It's called the conjunction fallacy.

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

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 got 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 science.

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 science 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 calls, 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.

It cannot be true that anything is possible.

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

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

The conjunction fallacy.

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

Once again, thanks for your attention.