Original subtitles

Instructor: Welcome back everybody.

Many scientific papers rely on conducting experiments

or surveys.

They often provide summarized statistics

we use to analyze and interpret

how certain factors affect one another.

An example would illustrate this better.

Imagine you conducted a survey

where 100 men and women of all ages were asked

if they eat meat.

The results are summarized in the table on the screen.

We see 15 of the 47 women that participated are vegetarian

as are 29 out of the 53 men.

Now, if A represents being vegetarian

and B represents being a woman

then P of A,

given B

and P of B,

given A

express different events.

The former equals 15 over 47

and represents the likelihood of a woman being vegetarian.

While the latter equals 15 over 44

and expresses the likelihood of a vegetarian being a woman.

Since 15 over 44 is greater than 15 over 47,

it is more likely for a vegetarian to be female

than for of a woman not to eat meat.

That's quite interesting

and shows you that in probability theory

things are never straightforward.

Now that you understand the distinctions between

the different conditional probabilities of two events,

we can introduce an important concept,

the law of total probability.

Okay.

Imagine A is the union of some finitely many events,

B1, B2 and so on.

This law dictates that the probability of A is

the sum of all the conditional probabilities of A

given some B multiplied

by the probability of the associated B.

To clear any confusion.

Let us go back to the survey example.

The probability of being vegetarian

equals the sum of the probability of being a vegetarian

given the person is male

times the probability of being male

and the probability of being a vegetarian

given the person is female

multiplied by the probability of being female.

If we plug in values from the table,

we get the probability of being vegetarian

equals 29 over 53,

times 53 over 100,

plus 15 over 47,

times 47 over 100.

That's 44 over 100

or 0.44.

Therefore, according to the survey

there is a 44% chance of someone being vegetarian.

We can refer back to the table and confirm

the probability of being vegetarian is indeed 0.44.

So our answer checks out.

Great.

In the next video

we are going to introduce the Additive Law

and the multiplication rule.

Thanks for watching.