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In this video, we're going to quickly review basic probability now probability is a mathematical way

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of describing the likelihood of an event happening for any event, A, the probability can be written

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like say.

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So the probability of event a copy of A is a number between zero and one with zero, meaning that the

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event is impossible to occur and one meaning that the event will always occur.

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Now, given the set of definitions for probability, we can say is if set s is a set of possible events,

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then the sum of all the probabilities in the set must be equal to one.

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For given all the possible events, one event has to happen.

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So the probability of something happening has to equal to one.

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Also, if we know the probability of event occurring, then the probability of event A not occurring

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is simply one minus the probability.

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So the probability of event A not occurring is one minus the probability of event occurring.

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Now, one way to describe events is to describe them as being mutually exclusive, so mutually exclusive

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events are just events that cannot occur at the same time.

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So we'll have a look at this Venn diagram here.

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We can say that this is the area for event A. And this is all the area for event B, but event A and

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Event B cannot occur.

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At the same time, there's no crossover between the two events.

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So mathematically, this can be written as a probability of event A and event B occurring at the same

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time is equal to zero.

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And then going back to the definition that the probability of all the different possible outcomes has

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to occur.

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One, we can say the probability of event A occurring or event B occurring is just a some of the probabilities.

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So the probability of A plus probability a B has to equal one according to the definition.

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Now, looking at the opposite type for non mutually exclusive events, this is saying that Event A can

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occur in this space and Event B can occur in this space.

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So it's possible to have event A and event B occurring at the same time.

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So that's this crossover amount here.

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So a non mutually exclusive event means that Event A and Event B can occur at the same time.

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So mathematically, this can be written as a probability of event A and Event B is equal to a non-zero

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value.

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So it's no longer impossible for them to happen at once.

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We can also say that the probability of Event A or the probability of Event B is now risen as the probability

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of A plus the probability of B minus the probability of A and B..

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So since the probability still has to equal to one, we want to make sure that the area of these two

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events here is equal to one.

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So we add up the area of A plus the area of B, and we would have already counted this this amount twice.

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So then we have to subtract this amount once of it.

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So that's this correction term over here.

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The next concept in probability is conditional probability, so events can be considered independent

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if the likelihood of one event does not affect the likelihood of another occurring.

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So this is like if you toss a coin multiple times or you roll the dice every time, you roll a certain

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number and then you roll it again, it doesn't mean that you're going to get a different probability

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of getting a different or the same number.

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Again, just because you flip two heads in a row does not mean you have a high chance of getting heads

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again.

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So this is independent events.

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Dependent events are the opposite.

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This is when one event occurs, it changes the probability of other events occurring.

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An example of this might be if you drawing cards out of a deck, once you draw one card out of the deck

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and you don't put it back, the probability of a drawing, another card of the same type is reduced.

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So the probability of event A and Event B occurring, if they are dependent, is the probability of

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A and B is just going to be equal.

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The probability of A multiplied by the probability of event B, given that event A happened, which

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means that we can write conditional probability in this form here.

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So the probability of event A happening, given that event B has occurred, it's just a..

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The probability of event A and B happening at the same time divided by the probability of B occurring.

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So this is called the conditional probability, this term up here is called the joint probability,

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and this term of here is commonly called the marginal probability.

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Now for independent events, we know for Event A and B occurring at the same time, it's just equal

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the probability of A multiplied by the probability of B.

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So then we can say that the conditional probability, the probability of a occurring given that they

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occurred, is just the probability of a..

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There's no dependence on B and likewise going the other way.

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The probability of occurring given that occurs is just probability or B, so using these definitions,

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here is the definition of independent events.

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Now, one powerful concept of this conditional probability is Bayes Theorem, so base theorems or Bayes

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Rules is one of the most important concepts used in Bayesian estimation, which is probabilistic estimation.

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Bayes Theorem allows you to calculate the likelihood or balance of an unknown parameter or event based

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on prior information related to that event.

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And this is called Bayesian inference.

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And Bayes Rule can be written like so.

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So this is a rule based on the conditional probability.

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So the first time here is the conditional probability.

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This is the probability of event A happening given the event B occurs over here, we have another conditional

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probability.

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This is the likelihood of event B happening, given the event A has occurred.

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So that's just the other way around of here.

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The margin probability is the likelihood of event occurring.

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And same thing down here.

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This is a marginal probability, the probability of Event B occurring.

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So we're only using Bayes Theorem.

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We often call the first term over here the posterior probability.

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So this is the updated probability, given all this information over here, over here is called the

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likelihood.

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This is the prior probability and this is the evidence.

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So we can get a better understanding of the actual probability of event A happening given this extra

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information over here.

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So this allows us to get a better estimate of whether A is going to occur by using information about

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Event B and knowing how B relates to A, we can get a better probability or a better idea of a weather

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event.

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A is actually going to happen.

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This was a very quick summary and review of basic probability, which is, I assume, require knowledge

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for this course.

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So please take the following quiz on probability and see if you have the concepts handled.

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If you are struggling with these concepts, then please read the notes included in this lecture on probability

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or investigate other learning sources for this background knowledge.