In this video, we're going to quickly review basic probability now probability is a mathematical way
of describing the likelihood of an event happening for any event, A, the probability can be written
like say.
So the probability of event a copy of A is a number between zero and one with zero, meaning that the
event is impossible to occur and one meaning that the event will always occur.
Now, given the set of definitions for probability, we can say is if set s is a set of possible events,
then the sum of all the probabilities in the set must be equal to one.
For given all the possible events, one event has to happen.
So the probability of something happening has to equal to one.
Also, if we know the probability of event occurring, then the probability of event A not occurring
is simply one minus the probability.
So the probability of event A not occurring is one minus the probability of event occurring.
Now, one way to describe events is to describe them as being mutually exclusive, so mutually exclusive
events are just events that cannot occur at the same time.
So we'll have a look at this Venn diagram here.
We can say that this is the area for event A. And this is all the area for event B, but event A and
Event B cannot occur.
At the same time, there's no crossover between the two events.
So mathematically, this can be written as a probability of event A and event B occurring at the same
time is equal to zero.
And then going back to the definition that the probability of all the different possible outcomes has
to occur.
One, we can say the probability of event A occurring or event B occurring is just a some of the probabilities.
So the probability of A plus probability a B has to equal one according to the definition.
Now, looking at the opposite type for non mutually exclusive events, this is saying that Event A can
occur in this space and Event B can occur in this space.
So it's possible to have event A and event B occurring at the same time.
So that's this crossover amount here.
So a non mutually exclusive event means that Event A and Event B can occur at the same time.
So mathematically, this can be written as a probability of event A and Event B is equal to a non-zero
value.
So it's no longer impossible for them to happen at once.
We can also say that the probability of Event A or the probability of Event B is now risen as the probability
of A plus the probability of B minus the probability of A and B..
So since the probability still has to equal to one, we want to make sure that the area of these two
events here is equal to one.
So we add up the area of A plus the area of B, and we would have already counted this this amount twice.
So then we have to subtract this amount once of it.
So that's this correction term over here.
The next concept in probability is conditional probability, so events can be considered independent
if the likelihood of one event does not affect the likelihood of another occurring.
So this is like if you toss a coin multiple times or you roll the dice every time, you roll a certain
number and then you roll it again, it doesn't mean that you're going to get a different probability
of getting a different or the same number.
Again, just because you flip two heads in a row does not mean you have a high chance of getting heads
again.
So this is independent events.
Dependent events are the opposite.
This is when one event occurs, it changes the probability of other events occurring.
An example of this might be if you drawing cards out of a deck, once you draw one card out of the deck
and you don't put it back, the probability of a drawing, another card of the same type is reduced.
So the probability of event A and Event B occurring, if they are dependent, is the probability of
A and B is just going to be equal.
The probability of A multiplied by the probability of event B, given that event A happened, which
means that we can write conditional probability in this form here.
So the probability of event A happening, given that event B has occurred, it's just a..
The probability of event A and B happening at the same time divided by the probability of B occurring.
So this is called the conditional probability, this term up here is called the joint probability,
and this term of here is commonly called the marginal probability.
Now for independent events, we know for Event A and B occurring at the same time, it's just equal
the probability of A multiplied by the probability of B.
So then we can say that the conditional probability, the probability of a occurring given that they
occurred, is just the probability of a..
There's no dependence on B and likewise going the other way.
The probability of occurring given that occurs is just probability or B, so using these definitions,
here is the definition of independent events.
Now, one powerful concept of this conditional probability is Bayes Theorem, so base theorems or Bayes
Rules is one of the most important concepts used in Bayesian estimation, which is probabilistic estimation.
Bayes Theorem allows you to calculate the likelihood or balance of an unknown parameter or event based
on prior information related to that event.
And this is called Bayesian inference.
And Bayes Rule can be written like so.
So this is a rule based on the conditional probability.
So the first time here is the conditional probability.
This is the probability of event A happening given the event B occurs over here, we have another conditional
probability.
This is the likelihood of event B happening, given the event A has occurred.
So that's just the other way around of here.
The margin probability is the likelihood of event occurring.
And same thing down here.
This is a marginal probability, the probability of Event B occurring.
So we're only using Bayes Theorem.
We often call the first term over here the posterior probability.
So this is the updated probability, given all this information over here, over here is called the
likelihood.
This is the prior probability and this is the evidence.
So we can get a better understanding of the actual probability of event A happening given this extra
information over here.
So this allows us to get a better estimate of whether A is going to occur by using information about
Event B and knowing how B relates to A, we can get a better probability or a better idea of a weather
event.
A is actually going to happen.
This was a very quick summary and review of basic probability, which is, I assume, require knowledge
for this course.
So please take the following quiz on probability and see if you have the concepts handled.
If you are struggling with these concepts, then please read the notes included in this lecture on probability
or investigate other learning sources for this background knowledge.