All language subtitles for 02 - Analyze a sample problem Kahneman’s Cabs

- If you haven't worked with Bayesian analysis before,

dealing with prior probabilities and base rates,

then the concept can be a little unclear.

I'd like to work through a classic example

from Tversky and Kahneman called at The Taxicab Problem

that will demonstrate how Bayesian analysis works

and give you better intuition.

Let's start with the facts as proposed within the problem.

A cab was involved in a hit-and-run accident at night.

There are two cab companies working in the city,

Green and Blue.

Furthermore, we know that a witness

identified the cab as blue.

However, 85% of the cabs in the city are green

and only 15% are blue.

Next, we need to know about witness reliability.

The court tested the reliability of the witness

under the circumstances

that existed on the night of the accident.

And concluded that the witness correctly identified

each one of the two colors 80% of the time

and failed 20% of the time.

If there are any attorneys watching this video, don't worry.

This is merely an example and it's not meant to be taken

as a reflection of witness reliability in the real world.

So your question is, what is the probability

that the cab involved in the accident

was actually blue rather than green?

So we're looking at the probabilities for blue versus green.

The answer might surprise you.

And Neel Ocean, in an internet posting,

provides an intuitive explanation

of how to arrive at the correct answer,

which is 41%.

And you can find the answer at the website listed below.

Let's walk through Neel Ocean's explanation

of why the answer is 41%.

We need to visualize the answer as a decision tree.

At the top, we know that our base rate of cabs in the city

are 85% green and 15% blue.

The witness is 80% accurate.

So if a cab is green,

then the witness will say green 80% of the time

and be correct and will be wrong 20% of the time,

meaning that the cab is actually blue.

And we have the same for the blue side.

If the cab is actually blue,

then it will be guessed 80% of the time.

If it's green,

then 20% of the time the witness will be wrong.

Now we take compound probabilities to see how likely it is

we end up in each branch of this decision tree.

So 85% of the time the cab is green

and it is guessed as being green 80% of the time.

.85 times .80, is .68.

Same thing for if the cab is actually blue,

but it's guessed green.

We have .85 times .20, which leads to .17.

And then we have similar calculations for blue

leading to probabilities of .12 and .03.

Now remember, that the witness said the cab is blue.

So we need to calculate the probability

that the witness guessed blue

when the cab was actually blue.

So we used the two values in the middle

where the cab was actually blue.

And we get this calculation,

.12 when the cab is actually blue,

divided by .12 plus .17,

which is when the cab is blue and was guessed blue.

But also when the cab was guessed green,

but was actually blue.

So we have .12, divided by .12, plus .17.

And that leads to a result of .41.

In the next movie, I will show you

how to create what is called a classification matrix

so we can get the values that we need

to perform this calculation in Excel.