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in this video we're looking at basic
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probability now when I say basic I mean
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probability as it relates to categorical
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variables like yes no type variables
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numerical variables have their own thing
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going on and I'll deal with that in
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later videos but we're going to be
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learning through the use of an example
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here and here it is the HBO cable
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network took a survey of 500 subscribers
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to determine people's favorite show now
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in this case let's say there were two
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categorical variables one which is
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gender male female and the other
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people's favorite show so I've Got Game
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of Thrones in there Westworld and I've
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just combined all the others into this
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other
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category I'm thinking that those two are
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probably the biggest shows on the HBO
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roster so let's see how this
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distribution pans out now out of 500
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subscribers 80 of them are male that
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like Game of Thrones 120 of them are
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female that like Game of Thrones etc etc
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and each of these squares we can call
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joint events they called Joint events
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because they depend on classes from two
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different variables now the other thing
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to note is that of course they're all
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going to add up to this total figure in
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the bottom right but we can also
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calculate this total column and total
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row where we just sum up the total
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number of people that thought Game of
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Thrones was their favorite show and
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that's 200 the total number of people
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that thought Westworld was their
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favorite show and that's 125 and we can
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also total up the males and females it's
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230 males and 270 females
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but this is not quite a probability
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distribution just yet so let's see how
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we get there here's that distribution
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again and if we divide everything by 500
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which is the total number of
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observations we get something which is
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called a probability distribution so
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where before we had 120 females who
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preferred Game of Thrones we now have
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.24 in that cell because that tells us
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that 0.24 or 24% of the distribution is
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defined by that joint event so that's
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why we call that a joint probability so
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that 024 is called a joint probability
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and the way we can write that is to say
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the probability of female the event
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where the person is female and the event
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where the person likes Game of Thrones
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is
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0.24 now you might see it written with
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the word and in between or else you
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might see it with this sort of upside
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down U which we're going to call
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intersection so that's just a fancy
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statistical term to say the intersection
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of female and Game of Thrones which
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which is of course that joint
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probability now collectively those six
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cells here form what's called The Joint
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probability distribution so all of these
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six cells are going to add up to one
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because if you think about it everyone
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in this distribution has to be in one of
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these cells you have to either be male
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or female and you have to have selected
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one of these options so it's no surprise
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that this should sum up to one now we
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can also calculate What's called the
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marginal probability called as such
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because it's in the marginal
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another term for this is the simple
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probability so just be aware of that if
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you see the phrase simple probability
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but here this 0.4 refers to the raw
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probability of someone liking Game of
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Thrones and I'm sure you can tell it's
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just the sum of both the male and female
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joint probabilities now of course this
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actual column here is called the
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marginal probability distribution so
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just like we had a joint probability
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distribution this one I've highlighted
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here also sums to one now the thing to
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appreciate is that this marginal
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probability distribution completely
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ignores gender it's as if that variable
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doesn't exist here but be aware also
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there's another marginal probability
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distribution down here and in this case
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it would ignore the show of preference
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but again this is going to sum to one
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because there are 46% males in our
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sample and 54% females in our sample and
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we're going to have to sum them up to
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get 100% so our joint probability
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distribution adds up to one but so does
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both of these marginal probability
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distributions okay so here's a couple of
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questions for you just using what we've
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gleaned from the last minute or so you
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should be able to answer some of these
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questions what's the probability of an
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HBO subscriber being male now I think I
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might have even said this before but the
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probability of a subscriber being male
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is purely that marginal probability of
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the male column so 46 now just be aware
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of terminology when we say probability
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we're after a number that's between 0
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and one sometimes they'll ask about
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percentages and stuff so you can turn
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that into a percent but if it just says
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probability you can just leave it
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as46 or the next question what's the
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probability of an HBO subscriber
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preferring Westworld pretty simple it's
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just going to be the 0.25 so we can keep
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going appreciating that those first two
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questions were asking for marginal or
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simple probabilities this question asks
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what's the probability of an HBO
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subscriber being male and preferring
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Westworld so that's 0.2 and again you've
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got that intersection symbol
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here and here's a slightly different one
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it says what's the probability of an HBO
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subscriber being male or preferring
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Westworld so this is not an intersection
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it's not a joint probability it actually
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requires us to think just a little bit
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and appreciate that to find this we're
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going to have to sum up all of the joint
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probabilities where this condition's met
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so if the subscriber is male we're in
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this column so we can highlight these
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three cells green and we can also
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highlight the Westworld row green as
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well so in summing up those four cells
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we're going to be able to get this
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probability and you can see here I've
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just gone all of those numbers added
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together and I've got
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0.51 now you might have noticed I used
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this upwards U here and this is the
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symbol that we call Union so this is the
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union of two events of someone being
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male and the event someone liking
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Westworld now if you've watched a few of
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my other videos you might know that I
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tend not to like formulas too much much
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because formulas tend to stop us
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thinking and many of them have an
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intuitive basis to start with and this
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is true of this formula here which
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provides us a way of calculating the
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union between two events so you might
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have seen this somewhere where we say
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the union between events A and B is the
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probability of a plus the probability of
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B minus the intersection and again you
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can calculate this one that way if you'd
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like which is to say that we add up the
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two marginal probabilities the 0.25 and
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the
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46 and then we subtract the joint
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probability at 0.2 now why do we
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subtract the 0.2 well if you think about
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each of these joint probabilities this
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one here sums up the entire row for
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Westworld this one here sums up the
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entire column for male thus we've
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actually added this cell twice the male
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Westworld joint probability we've added
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that twice if we've just simply added
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these two blue cells together so we have
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to subtract one of those away again to
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be left with 0.51 which is no surprise
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the same answer we got using the other
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method this brings us to the next and
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more tricky type of probability which is
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called a conditional probability so here
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I've given you a question again as an
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example no just got an HBO subscription
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what's the chance that her favorite show
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will be Game of Thrones now again you
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might have seen a formula that looks a
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little bit like this where this says the
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probability of event a given event b
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equals the intersection of the two
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events divided by the probability of the
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condition so in this case what we do is
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we basically focus on the part of this
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distribution which is of interest and
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because non is female we can ignore the
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rest of the table completely we're only
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really considering this column here we
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take the joint probability which is .24
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and we divide by the probability of the
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condition which is 0 54 now the
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condition was that she was email that
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was given in the question so we can
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calculate this as24
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/54 which is 44444 which is about 49ths
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but that's okay we can leave it as a
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four decimal place probability here so
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what we can do now is create for
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ourselves a new column which is the
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probability of preferring each of these
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shows given someone is female so we just
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got 4444 for Game of Thrones here now we
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can do the same for Westworld and other
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we find here what's called the
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conditional probability distribution so
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this is the probability distribution of
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preferring various shows given someone
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as female and again because it's a
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complete probability distribution it's
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going to add up to one and indeed this
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does so what what we can do now is we
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can compare this conditional probability
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distribution to the marginal probability
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distribution that's this one here the
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original Total
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distribution and we can see that if we
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don't take sex into account 40% of
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people like Game of Thrones 25% like
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Westworld 35% like other shows but when
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we take gender into account these change
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a little bit so for females you can see
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that they're more likely to like Game of
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Thrones than the general population
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they're less likely to like Westworld
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than the general population and they're
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a little bit more likely to like other
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shows than the general population as
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well and we can actually use this to
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assess whether the two variables gender
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and show Choice are independent but
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before we do just appreciate that we can
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also go the other way with our
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conditions so for example if I asked you
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given that a subscriber's favorite show
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is Westworld what's the probability that
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they are male in this case the condition
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is that they like Westworld so our
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condition is actually an entire row so
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much like last time we can block out the
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rest of the table that doesn't coincide
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with Westworld and we can just focus on
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this row and the probability of being
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male is that 0.2 which is the joint
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probability divided by the probability
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of the condition
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0.25 so if you do that you get 0.8 and
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that's the probability of being male if
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you prefer Westworld in other words 80%
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of people that watch Westworld are
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males so this leads us to the topic of
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Independence between the variables
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now we've already touched on it just
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briefly to show that females have a
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little bit of a different preference of
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their HBO shows to the general
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population but the strict definition of
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independence actually has two different
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approaches the first of which says that
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if the two variables are independent
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then the probability of a given B is
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just equal to the probability of a
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another way to think of this is that
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imposing the condition B doesn't
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actually affect
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the probability of a at all so in our
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case we found the probability of liking
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Westworld if you're female is 0.093 we
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found that from our conditional
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probability distribution earlier but the
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probability of just preferring Westworld
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straight up is 0.25 it's in that final
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value here it's in that marginal value
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here so if these two variables gender
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and choice of HBO shows were independent
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these two values should be equal
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therefore the variables are not
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independent as 0.093 does not equal 0.25
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so clearly gender does influence the HBO
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show that they prefer now it's probably
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worth mentioning that because this is a
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sample you would never expect for these
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values to be exactly equal because we
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know in a sample there's going to be
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some random variation and even if the
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variables were completely independent
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they these two probabilities wouldn't be
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perfectly equal but in this case I think
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it's quite clear that they're very
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different probabilities one's 25% and
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one's 9% so I think we're quite safe to
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say these variables are not
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independent now as I said there were two
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approaches for assessing Independence
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and this one says the probability of a
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union B is equal to the two marginal
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probabilities multiplied together now
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I've got a feeling you might understand
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this equation in l so if I can take you
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away from this example just for a second
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say I flipped a coin in one hand and I
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rolled a dice in the other hand and I
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said I'm going to give you a hundred
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bucks if I roll a six and I flip aead if
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both of those events occur I'll give you
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a 100 bucks and I asked you what is the
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probability of me giving you a 100 bucks
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you probably say well that's one in 12
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right because you've got half a chance
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of getting ahead and a sixth of a chance
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of rolling a six
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so you multiply them together you're
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going to do the probability of a getting
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ahead times the probability of B and
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you're going to get that joint
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probability but the only reason you can
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do that is because rolling a dice and
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flipping a coin are completely
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independent variables it's not as if
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rolling a six influences the chance of
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getting ahead right so as I said you can
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kind of intuitively understand this
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formula but what that implies for our
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distribution here is that if we were to
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look at the intersection between
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Westworld and female we get
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0.05 and if you multiply the two
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marginal values together the
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0.54 and the
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0.25 you get
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0.14 so again these two values are not
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the same so we can say they're not
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independent
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variables which is no surprise because
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we just showed this using the other
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approach so either of these approaches
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would be fine to show that people's
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preference of HBO shows does depend on
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their gender all right so that's a wrap
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we've dealt with joint probabilities
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marginal probabilities conditional
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probabilities and we've also looked at
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how to test for Independence between two
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categorical variables now if you like
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the video I got plenty more you can
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check them out on the YouTube channel or
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heading to my website Zed statistics.com
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and if you've got any ideas feel free to
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get in touch adios
