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5.3b - Conditional Probability & Independence | MrRzMath | YouTubeToText
YouTube Transcript: 5.3b - Conditional Probability & Independence
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Core Theme
This content explains the concept of statistical independence in probability, defining it as a lack of association where knowing the outcome of one event provides no information about another, and detailing how to mathematically verify independence using probability formulas and real-world examples.
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so we are in our final lesson of chapter
5 tackling the concept of Independence
in probability questions and
independence is a topic we've talked
about before um a little bit and I'm
gonna remind you guys of what that means
as we get things started here so first
of all before we answer this question
and I tell you how you tell if things
are independent I'll talk about how what
what even means for things to be
independent where would we have heard
this word before it was back in like
chapter 1 when we talked about the idea
of a conditional distribution
conditional distribution where we were
looking for if one variable had an
association with another so we said that
two variables have an association if one
variable knowing the value of one
variable gives us information about
another we like we're careful not to say
cause and effects well if two things are
independent it means that there is no
Association and if there's no
about the other in like a very simple
sense if I flip a coin and flip a second
coin knowing the outcome of my first
coin does nothing for me about my second
coin earlier in the chapter I talked
about that roulette wheel where they
show what the last spin was well the
spins are independent of one another so
knowing one doesn't help you at all with
the other now when we talk about
independence in AP stats you very very
rarely will be talking just from your
gut on this like oh it doesn't seem like
one affects the other what you need to
do is show mathematically that the
events are independent so there is a
formula you will check to make sure the
events are independent or if they're not
what you will do is find the probability
of one of the events probability of a
and if the probability of a is equal to
the probability of a given B this would mean
mean
that a and B are independent so a and B
are independence if this is true
probability of a equals probability of a
given B you could also reverse it and do
it the other way probability of B equals
the probability of B given a basically
what this means I'll use the top one so
probability of it is the same thing as
the probability of a given B in other
words B happens well okay it's still the
same probability for a doesn't tell me
anything all right you have to be
careful about not mentioning cause and
effect when you do this because that's
not what's going on it's that one
variable gives you no information about
the other so if B happens that doesn't
tell us anything about the probability
of it and we're going to look at the
example from last time which will be a
really good one to kind of visualize
this which is that Diamondbacks example
where we had that free tacos and we had
the wins I'm gonna go ahead and put that
table back on here but you guys should
look back in your notes and flip to that
so you had that handy when we talked
about this or just recopy it when I
write it down okay so here's that table
that you guys had from your last slides
I believe it's the exact same one you
guys already have written down they
asked us if tacos and wins are
independent of one another and again you
have to be careful when you talk about
independence it doesn't mean one has to
affect the other we're not talking cause
and effect we're just talking if one
event gives us info about the other so
if we want to establish that these
events are independent if you start
saying well yeah the wins affect the
tacos or something is wrong no credit it
has to be mathematically and you do that
using what's written in red right here
so what I will do to verify these events
are independent or if they're not is I
will find the probability of one of them
doesn't matter which way you do it let's
say I go for the probability of tacos
probability of tacos is thirty out of
eighty one based on what I have written
down right here thirty taco games out of
eighty one then what I'm gonna do is I'm
gonna find the probability of the same
thing tacos given
the other events given that you win
so okay they want does that tell us
anything about the probability of tacos
and then if I know that they won I would
be looking at this guy right here
so my probability of tacos given win is
26 out of 41 you can look at those right
there and see that they're probably not
the same but I'm gonna go ahead and make
them decimals anyway to make it easier
to compare and we can make a point about
that so let's talk about this first guy
right here there's a 37% chance in a
random game that the Diamondbacks are
gonna win tacos or that you're gonna get
tacos because the Diamondbacks scored
six runs and there is a 63% chance 63.4%
chance that the Diamondbacks if they win
that you're gonna get tacos so in a
random game if you just walk in and said
hey are we getting tacos today that's
when you score six runs or more there's
a 37% chance that's gonna happen but if
you know they win the game you look only
at the games they win that chance of
getting tacos jumps to 63.4% in other
words if you know they won the game
there's a greater chance that they
scored six runs because those terrible
games where the scores euro runs in one
run they probably didn't win in the
first place so if you know then your
chance of getting tacos actually
increases does that mean the wind caused
you to get tacos
no it didn't you could win by scoring
two runs or three runs or four but
they're tied to each other and winning
knowing that piece made it more likely
that you would get tacos you could have
also opted to do these the other way you
could have done probability of win and
probability of win given taco it's the
same logic if you know they get tacos
there's at least six runs and if you get
six runs you're probably more likely to
win the game okay
now I need to actually answer the
question cuz I haven't done that yet
what I would have to do to get full
credit for this problem because I've
just shown some work
I haven't actually addressed what they
said I'm gonna say something to the
effects of
since the probability of tacos which is
37 percents
does not equal the probability of tacos
given wins which was 63% the events T
and W are nots independence in order for
them to be independent you have to get
the same thing that means they're not
independent then it's because of what we
talked about if you win you're more
likely to have scored six runs so two
events can be dependents without one
actually causing the other one so you do
not need causality to be able to talk
about independence here so this is a
good thinking page right here where you
probably should pause the video and
think about it for yourself a little bit
there's a difference in the two terms
that kids get confused a lot are the
idea of mutually exclusive and
independence mutually exclusive comes up
in our or style probability independent
comes up more in our hand style
probability so when it says what is the
difference here maybe write that over
here mutually exclusive comes up in or
independence comes up in it but think
about how you can fill out these tables
I usually give groups a few minutes to
think about this one so I'm gonna go
ahead and talk about the answer pause
the video if you haven't tried to get if
two events a and B are mutually
exclusive it means they can't occur
together which means that a and B up
here is a zero and then in the rest of
the boxes are pretty easy you can just
use basic math to figure out what's
going on right there that's the only
possibility there that they cannot occur together
together
that's what mutually exclusive means so
mutually exclusive means they cannot
independent on the other hand this one
is usually a little tougher on kids to
figure out it goes back to that
definition of independence you can do it
in either direction but I'm going to do
probability of a first
probability of a given B and these mean
to be equal the probability of a I can
pick out of my picture right here it's
60 out of a hundred so that's fine I
need to compare that to the probability
of a given B so if B is given I'm out of
this box right here and it would go
probability of a and B which is this box
out of probability of B which is 30 so
essentially what I have here is like a
little cross-multiply problem or you can
just treat it like a decimal make this a
point six right here if you actually do
that that's going to be 18 for your
answer so this is gonna equal an 8 out
of 30 this box right here must be 18 y
again if you find probability of it
that's 60 out of a hundred any given B
is 18 out of 30 which is also a point
set so it equals the right thing and
then from there you can just fill out
the rest of the numbers using your basic
subtraction rules here and you get your
table so independent means that one
gives you info on the other we just
defined that on the last page so I'm not
gonna make you write it down again
yeah but make sure you can keep those
two terms straight so review here of all
this probability business that we've
been talking about in this chapter and
you have to look at whether or not in an
or problem they're mutually exclusive
and an N problem if they're independent
amounts if they are mutually exclusive
in an or problem all you do is you add
your two probabilities together done
they don't overlap Peters Adam and it's
over if they're not mutually exclusive
you still add them up but then
afterwards you have to subtract out the
overlap the stuff you double count which
is this right here
again if you do the true way table
strategy for problems you don't have to
worry about this step because you've
already broken it into mutually
exclusive boxes but that is your basic
setup for problems
and we have independence right here if
events are independent I flip a coin I
flip a coin
all you do define an ian's probability
is you multiply the probabilities
together one half times one half
whatever events you want to think about
if they're independent you just multiply
their probabilities together if they are
not independent or in other words if
they're dependents you're still gonna
multiply so you start out with the same
first probability but then what you do
is you adjust your second probability
like I did a marble example a video ago
if I wanted to get two blue marbles I
have five blue five red my first one is
gonna be a five out of ten but my second
one if I don't put it back is gonna be a
four out of nine so you have to adjust
that second fraction and the way that
you would do that is you would find the
probability for your second event given
your first events so basically you have
to account for whatever it is you
already took out I do wouldn't get too
caught up in symbols with these what I
would do is I would just think your way
through it and you should hopefully be
okay so let's look at a few other
smaller things right here this next
problem is actually a little bit
different than what we've been doing
here oh so we have a test during the
first trimester of pregnancy to decide
if there's some sort of abnormality
going on approximately five percent of
these tests will be false positive
so what we're doing here is we're
looking at 100 women and we're saying
out of those 100 women with no
pregnancies and what is the probability
there will be at least one false
positive there's a 5% chance you have a
false positive okay the first thing you
need to do when you're doing some sort
of probability question like this is
acknowledge that the events are
independent so it's reasonable to say
that each woman's tests first
is independence okay so each woman's
test is independent which means if we
need to we can just multiply their
probabilities together and we're cool we
do not have to adjust one does not have
any sort of bearing or tell us anything
about change the odds for another
probability all right so that's the
first thing the other thing about this
problem here is they're asking kind of a
tricky question what they're saying is
what is the probability out of a hundred
women at least one of them will be a
false poundin these words like at least
or at Lowe's are tricky words that you
should like signal to you that you're
gonna use a little trick I'm about to
teach you at least one false positive is
crazy because what that would mean is I
would need to find the probability that
exactly one woman out of a hundred has a
false parlament then I would need to
find the probability that exactly two
women out of a hundred have a false
positive and that I defined the
probability for three and four four and
four five and four seventy eight and
four seventy nine and four eighty eight
489 490 one four ninety five and four
ninety nine and for one hundred I would
have to calculate literally a hundred
different probabilities to answer that
red question at least one false positive
could be all of those options now that
is awful and it is not something you
will ever be asked to do because what
we're gonna do to answer this question
is use our good friend the complements
that we learned about earlier in this
chapter instead of finding all of this
awful at stuff right here hundred
probabilities think about what you
haven't found in the rep what we haven't
found is the probability of no false
positives at all so what you can do with
a problem like this is find this guy way
easier than all that red stuff and then
afterwards just do a compliment to get
what you actually care about so that's
the basic set up right there when you
see words at least or at most you're
probably being set up to do a compliment
type of problem so I need to find the
probability that in a hundred women none
of them
none of them have a false-positive
that's what I'm being asked to do well
what's the probability for an individual
for one out of those 100 women that they
don't have a false positive
if 5% of pregnancies have a false
positive that means the probability they
exult have a false positive is going to
be 95 percent that is woman number one
okay woman number two is also gonna need
to not have a false positive so that's a
95 percent curve woman number three is
gonna have a 95 percent chance no false
positive etc etc etc for each of those
different women so an easier way of
writing this one out is to say 0.95 to
the 100th power I mean the first woman
and the second and the third in the
fourth in the fifth and the hundredth to
not have a false positive so if I
multiply that out point nine five to the
100th power it's kind of small
this ends up being 0.005 9 so I've been
purposely trying to color code this
right here because this can be a little
tricky too retro head around the green
guy the probability of everybody being
cool and not getting a false positive is
less than one per sentence but I didn't
want the green guy I wanted the red part
or pink part right here which is at
least one of them being a false positive
so what I'm gonna do to finish this one
off is do the complement I'm gonna take
1 minus the probability of nobody having
a false positive and if you do that it's
gonna be 0.99 4 so that's the
probability that we're actually asking
us for the problem which is all these
other hundred options so we talked about
false positives and false negatives in
the last example but think again what
this is saying even if for one person
there's only a 5% chance of a false
positive when you test a lot of people
sooner or later like it's almost certain
we look a 99.4% chance somebody will
have a false
so most of the time when you go through
some sort of testing for whatever
they're gonna test you twice just to
make sure that you're not one of those
like false positive or false negative
situations okay
so problems like this where you have to
do at least or at most and do a
compliment are common especially on
multiple choice questions so that's
something you should expect to see me on
your chapter tests are it's our next
little problem right here is more of a
trick question kind of spoiling it for
you in advance we're gonna say that for
today there is a 70% chance of rain in
Maplewood and then there's also a 70%
chance of rain in Richmond Heights
what's the proof for tomorrow I guess
what's the probability it will rain in
both towns tomorrow so the word Bo's
kind of applies and so if you see and
you would be like oh you multiply the
probability is 0.7 times 0.7 but if you
multiply them straight up that is
assuming that one event is independent
of the other that's not true if it's
raining in Maplewood it is almost
certainly raining in Richmond Heights as
well so basically look if I was gonna do
this if the probability from Maplewood
is 0.7 of rain the probability of
Richmond Heights given that it's raining
in Maplewood is probably something like
0.99 like it's actually really really
high because um if it's raining in
Maplewood the town next door is probably
gonna be running as well there's always
that situation where it's not the case
but it's pretty unlikely so you cannot
just multiply these together straight up
because they're not independent
because the events aren't independent so
be careful before you start multiplying
stuff that it actually is independence
so otherwise it's like a trick question
this next slide I'm not gonna go over
with you guys here but it's kind of an
interesting article right here mine
Mason's covering it a little bit but
basically what it talks about is the
idea that for
name and last name are not independent
of one another so your first name and
your last name actually have connections
here you see it a lot in names of like
certain ethnic origins where like
different first names and last names go
together more often
even though they may not be the most
common name in America separately when
you put them together they are so
basically it's an article about how your
first name and your last name aren't
necessarily independence and then the
final thing I'm going to do right here
this is unrelated as well to our main
lesson right here but simulation style
problems are one of the toughest parts
of AP stats for kids and I wanted to do
an example of this here because there
will be something like this on your next
chapter tests so we have a girl who was
a sixty percent free-throw shooter
shooter last season she wants to get
better and she trains and practices and
all that and then in the first three
games of the season she had 12 of her 16
shots which is a 75% rate so we want to
see based on that evidence whether
that's convincing enough that oh yeah
she actually has gotten better or cuz
she still be a 60% shooter who's just
got lucky in her first couple of games
and had a sample where she did better
than usual okay that's the basic premise
of this problem is the evidence from her
first three games enough that we can be
confident she's no longer a 60% shooter
so what we would do in a situation like
this is design a simulation and we would
try to figure out basically we can do
something with a calculator like we did
earlier in this chapter where we would
make it so she's still a 60% shooter so
maybe the numbers 1 through 16 would be
she makes it the numbers 61 200 or 99
whatever it would be nine hundred would
be that she doesn't make it and we would
try to figure out if we make 16 shots
happen we would look to see in what
percent of the time she got 75% more
could 75% happen in 16 shots given that
she's still a 60% shooter so they went
ahead and gave us a dot plot right here
and you can expect to see a dot plot
like this somewhere on your chapter 5
test what you do with a picture like
this this picture assumes she's still a 60
60
and shooter and we have to figure out
how rare what actually happened in real
life is so assuming that she is still a
60% shooter
how rare is it to get 12 or more of her
shots correct or to make 12 or more of
her shots so when you do a problem like
this you find what you got in the
problem but you always count more
extreme than that as well because if
you're gonna be impressed by a 12
naturally you'd be even more impressed
by a 13 or a 14 of your shots and then
what we're gonna do is count up those
dots I count 14 right here I might be
off by like 1 but I'm pretty sure that's
14 months so in a hundred simulations 14
of them had a 12 or better given that
she's still a 60% shooter computer
assumes she's still 60% and they looked
at what happens
so the probability so assuming Ashley is
a 60% shooter there is a 14 out of a
hundred which is like 14 percent
obviously chance my face here so there
is a 14 percent chance that she could
make that's a really bad could could
make 12 or more out 16 shots so you have
to think about what that means if
there's a 14 percent chance that she's
still a 60% shooter that's basically
what we found right there is that high
enough or low enough that we should be
like okay confident she's actually
better now
14 percent is usually considered to be
too high for us to be certain that we
were wrong in our initial assumption so
there's a 14 percent chain
she's still a 60% sugar basically is how
you can think about that right there and
14% is probably too high for us to be
like certain so what should we do
well we should collect the little more
Deena if she kept not shooting that we
had bigger samples well that tells us
okay yeah she's probably not a 60
percent shooter anymore
the most common cutoff for problems like
this is five percents but you can use
10% or 1% and that can vary a little bit
from problem to problem but basically
what you do is you figure out what
happened in real life plus anything more
extremely mats and you figure out if
that probability is low and if that
probability like let's say same exact
situation but instead of hitting 12 out
of 14 shots sorry
12 out of 16 shots she actually hit 14
out of 16 shots well 14 out of 16 shots
that's 13 right here she never got 14
out of 16 so the probability of that
happening would be about zero out of a
hundred and if I saw that I would be
like whoa she's gotten better she's
definitely better there's no way that's
gonna happen or it's very unlikely to
happen if she's still a 60% shooter but
in our pink problem right here getting
12 out of 16 was like kinda rare kind of
surprising only a 14% chance of that
happening but that probability is too
high to be certain so that's how you use
one of these simulations to analyze a
probability and it's something we'll
practice in class as well because it is
something that kids do struggle with but
hopefully that makes at least a little
bit of sense and hopefully you
understand this concept as well as that
rest of that stuff we've talked about in
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