Intro to Conditional Probability - YouTube

In this video, we're gonna talk about something called "conditional probability".

The idea with a "conditional" probability is that we want to figure out the probability of something,…

…GIVEN...

…that we have some OTHER piece of information,…

…other piece of information that we can bring to the table, that will…

…influence the probability of whatever it is that we're investigating.

Now, the way we write this down, is with the following notation.

We're gonna refer to the probability of…

…"A, given B".

So, this vertical bar that we have here is…

…is referred to as, or read off as, "GIVEN",…

…and what it means is, that the probability of A, the thing I want,…

…the thing I'm investigating, the probability of some event A,…

…GIVEN that I have this other piece of information,…

…GIVEN that I KNOW that this event B has occurred.

So, we're gonna have some formula for how we're gonna compute this.

It is a quotient of two things. It is the probability…

…that we have BOTH the event A and the event B.

So I write this as an intersection. It means the probability of A AND B.

All divided out by the probability just of B.

Now, the way I like to think about this formula, is that…

…we're looking, NOT at all possible events,…

…we're looking at this…

…this sort of narrowed sample space,…

…of knowing that this event B has occurred, and we're asking,…

Well, what's the probability of A occurring, GIVEN B?

So, the sort of…

…sum total of all the possibilities that are going to be…

…is whatever the possibility is that B is going to be,…

…because we KNOW that B HAS occurred, but…

…but then we're asking, well…

…well, how often is it likely the case that you have…

…A…

…AND B occurring, amongst all the possibilities where B occurs?

So, that's why we have this formula. This... this is...

What is the probability that A AND B occurs? That's the numerator, divided by…

…all the possible ways that B can occur.

That's the denominator.

Note that, in most cases,…

...if we're just talking about non-conditional probability,...

…what the probability of A is,…

…without any other information,…

…we actually are typically dividing by something, in the same way, just that we…

…divide by the probability of ALL possible events,…

…the entire sample space, the entire universe of possibilities, which is just 1.

So, we are in fact…

…always dividing out by something.

In the case of conditional probability, is that we're dividing out…

…by the probability of B, the thing that we're TOLD has to exist.

Now, we can sort of visualize how this is going to work, by using a Venn diagram.

So, a Venn diagram: I'm gonna list my entire universe of possibilities.

And then I can say that...

Look, I have within my entire universe, I can have some…

…probability of an event A occurring. And...

…I can also have…

…a probability of some event B occurring.

Well,...

If I'm looking at this sort of conditional probability, well,...

…conditional probability is saying: Look, I… I'm in the scenario…

…where this event B is occurring, I'm… I'm ASSUMING that event B occurs.

And then, what I'm really interested in, is…

…what is the probability of this event A occurring,…

…GIVEN that I'm in this event B, that this event B HAS occurred?

In which case, I'm really interested in this intersection here.

In other words, I care about this ratio of the probability of the intersection…

…to the probability of B occurring.

The "red" divided up by the "yellow" area.

Now, in this example,…

…I've given you something, I've given you an intersection.

I've told you that the percent of adults who are both: 1) male,...

…and 2) are alcoholics…

…is going to be… it looks like 2.25%. I looked this up.

And then the question is: …

…what is the probability that you're an alcoholic if you KNOW that you're a male?

Well… we can look at what the probability of being an alcoholic is…

… in general, but I'm interested in this problem.

If… if you KNOW that your patient is a male, what is the probability…

…that they're an alcoholic?

So, we're gonna use conditional probability to figure this out…

…because we know something about the intersection, the probability of being…

…an alcoholic and a male, that's 2.25 %

We know what the probability of being a male is, or we approximate it at 50%.

And we can use those 2 facts together to get the conditional probability,…

…the probability that you're an alcoholic, GIVEN…

...that you're an adult male.

Now, I need to define a little bit of notation before I plug into my formula.

I'm gonna let A…

This is going to be equal to this particular event of being an alcoholic.

And then I'm going to let B, be the event…

…that you're gonna be a man.

So, there's 2 different things we could wish to get: 1) probability of being an alcoholic,...

2) probability of being a man…

The probability of being an alcoholic AND a man, that's the thing that we were given.

We were given in particular that the probability of being alcoholic…

…AND being a man, that this was equal to…

…0.0225, or, in other words, 2.25%.

And so, what we're really asking when I ask, what is the probability that you're an alcoholic…

…given that you're a man, is…

…we're asking, what is the probability of being A — being an alcoholic —…

…given that you're a man, and… and we know that, from our formula, this is going to be the…

…probability of A intersect B,…

…so, the probability of being both, all divided by the probability…

…of just B, in other words, the probability of being a man,...

…and we know these 2 different answers.

The top is going to be 0.0225,…

…and then I need to be dividing this by the probability of being a man,…

…and I'm going to say that that's 0.5, I assume it's exactly 50%.

And therefore, what I'm gonna get is 0.045.

We're about 4.5% for adult males.

By the way, it's a little bit smaller for adult females. It's about 2.5% for adult females.

So, this is the way that we've been able to compute this result of…

…probability of… of being an alcoholic given that you're a male,…

…via this technique of conditional probability.