Of the 60 spam emails, 35 contain the word free.

Of the rest,

only three contain the word free.

If an email contains the word free, what is the probability that it is spam?

So what we want to do first is to organize this information into a probability tree.

We're going to start by dividing our population, our inbox in this case

is our population, into two, based on whether the email is spam or not spam.

So we have 60 emails that are spam, and 40 emails

that are not spam.

Now that we've done this branching, we can actually further

branch out from these and list how many of the spam

emails have the word free in them and how many of

them do not, and likewise for the no spam, non-spam emails.

Of the 60 spam emails, 35 have the word free in it, and of,

and the remainder 25 do not. And of the not spam emails, only three

of them have the word free in it, and 37 do not.

Now that we have organized the information that we're given

into a probability tree, what we want to do next

is to go back to the question and try to

figure out what it is exactly that we're being asked for.

The question is, if an email contains the word

free, what is the probability that it is spam?

So we know that the email contains the word free, so that's

going to be our given, and we're asked for the probability that it's spam.

So we can denote this as probability of spam

given that the word free is in the email.

Since we're saying that we know the word free is in the

email, we're basically saying we can in, ignore the rest of the email.

So first what we want to do is figure out how

many emails in total have the word free in them.

35 of them come from the spam folder and

three of them come from the not spam folder for a total of 38 and of these,

only 35 of them are of interest to us because those are the spam emails.

So 35 out of 38 gives us roughly 92%.

Here we've implicitly made use of the Bayes theorem.

What we have in the numerator is our joint probabilities, spam

and free, and what we have in the denominator is the marginal

probability of what we're conditioning on, the free.

Except instead of working with probabilities in this

case, to make things simple we've worked with counts.

So what we're going to do next is actually

move onto a situation where we're working with probabilities

from the get go, and we don't know the

sample size of the population size that we're dealing with.