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5.3a - Conditional Probability | MrRzMath | YouTubeToText
YouTube Transcript: 5.3a - Conditional Probability
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This content introduces and explains the concept of conditional probability, demonstrating how to calculate it using both two-way tables and tree diagrams, and highlighting its application in real-world scenarios like drug testing.
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welcome to section 5.3 a where we're
going to tackle the concepts of
conditional probability now we did a
little bit of probability review at the
end of algebra 2 but this would not have
been one of the topics we talked about
there so this should be brand new for
you guys so let's go ahead and learn
what it means before we get to that so
we're going to jump in with an example
here and we're looking at the Arizona
Diamondbacks baseball team and then
apparently during the 2012 season there
was a kind of promotion going on by Taco
Bell where if they scored six or more
runs in the game then the Taco Bell
would give away free tacos the next day
so we want to take those two events
whether they win or not and whether
there are tacos or not and we want to
make a little two way table to represent
this problem so let's go ahead and
practice reviewing what we did the last
time this would be a good opportunity
for you guys as well to try this out now
that we've practiced two-way tables of
fair amounts so go ahead and fill it out
for yourself while I work it out over
here I'm gonna put wins going that way
taco is going this way but it really
doesn't matter which one goes each way
um it looks like there were a total of
81 games altogether they won 41 meaning
they lost 40 they gave away tacos in 30
so this guy over here is gonna be a 51
and then in 26 games they did both and
from there it's just easy math filling
out your little table so you guys should
hopefully be pretty comfortable with
that because it's an important thing to
be able to handle here but we have our
little two-way table representing what's
happening in the problem I didn't have
to define my variables because they did
it for me in the problem so now that we
have our two-way table in place we can
answer a whole bunch of questions using
our information so Part B asks for the
probability that the Diamondbacks win or
that there are three tacos and remember
the basic idea once you've taken the
time to put this all in a little table
right there you can just count up the
boxes that are good that you want and
you don't have to worry about using a
formula so you're just gonna add up
everything where they win or where
they're at
so in other words I'd want this box I
want this box because these are both
wins and I want this box over here if
you add all three of those boxes
together it looks like that's going to
be a 45 so it's gonna be 45 out of all
81 of the games you can make that a
decimal and do stuff there but that's
how you would set this up okay so or
probability we talked about it last time
if you make a table you don't have to
worry about overlap because everything
is nicely separated in our boxes now
these next two problems here are gonna
be the new stuff that we're getting into
so when you read through this
probability here in Part C it says find
the probability that the Diamondbacks
win given that there are free tacos this
word given is really important this word
given is a signal that we're doing
something called conditional probability
conditional probability we'll define it
on the next slide but it's basically
built under the assumption that one of
your events has already occurred so we
already know for a fact in this problem
there are gonna be free tacos if I was
gonna write this in symbols I'd be
looking for the probability of the
Diamondbacks win that was W the symbol
forgiven which again I'll give you on
the next page is like a big vertical bar
and free tacos is T so this is how I
would write Part C in words let's talk
about how you would actually find it we
already know for a fact there are free
tacos it's given to us in the problem we
don't have to worry about the games
without free tacos so what we would
basically do in this situation is all
these knots taco games over here on the
rights dead to us you can just cover up
that part of the table right there
normally on the board I like scratched
out and stuff but I have to use it again
for D so I'm not gonna do that yet what
I'll do instead is all circle the part
of the table that I actually care about
we already know there are tacos so that
games without tacos don't matter to us
anymore we're not looking at all 81
games anymore now we only have 30 games
that were too
from so there are 30 games where they
got tacos and they asked for the
probability that they win
well that's 26 out of the 30
so basically conditional probability in
a table is pretty straightforward
you just cover up the part of the table
you don't want and just do a regular
probability calculation from there and
it's not too bad to setup alright the
next one is another conditional
probability it says find the probability
there are free tacos given that the
Diamondbacks will win their game so if
it is given that they win we're looking
for probability of tacos given win now
we're looking at this part of the table
right here because we already know for a
fact that they want these games over
here where they didn't win those don't
matter to us anymore
those are stupid they don't care about
those we know they want and we're
looking for the probability there's
gonna be tacos well there are 41 games
that they won and there are 26 where
there were tacos so the key takeaway
with this kind of problem yeah is your
denominator is gonna change because you
already know one of the offense occurred
so let's formalize what we were just
talking about right there
and officially define a conditional
probability so conditional probability
is the probability or let's say given
we're going to be finding the
probability of another events so
basically when you knew conditional
probability one of the events already
happens and they'll tell you that in the
problem what's already locked in but
then we're finding a new probability
based on that previous example really
simple example that I don't even have to
show you guys a slide on I'm just gonna
talk to you through it right here let's
say I had a bag with
marbles I had five blue and I had five
red okay then I would say something like
given that my first marble was blue if I
don't put it back what's the probability
the second marble is also blue so if
it's given that the first marble is blue
that marble is already out of the bag so
when I say the answer to that there are
going to be four Blues left out of nine
marbles so in basic problems you can
just kind of logic your way into the
answer by thinking about it it can get
more complicated though which is why we
have a formula for conditional
probability this formula is on your
formula sheet here so it's not something
you have to memorize but with
probability formulas honestly I think
you should memorize them we use them
enough that you shouldn't have to be
backtracking every time we do a problem
like this to look up the formula but it
is there if you are iffy on it's during
the quiz or something like that um so
first of all the symbol for conditional
probability looks like this and you
would pronounce this the probability of
a given B the bar basically means given
and then you have your second event
right there
in other words B already happened in
this problem the way you calculate a
conditional probability through a
formula you're gonna find the
probability of both events occurring
those events is going to be a and B a
intersect B right here and you're gonna
divide it by the probability of your
second events so in my like Diamondbacks
example with tacos and stuff like that I
believe what I had going on um the
probability that they win the game was
like so let's say it was like tacos
given win win was like a 41 out of 81
tacos I think was 26 out of 81 if I'm
not mr. member in these if you divide
those you get 26 out of 41 the 80 ones
and canceling outs so it's just
probability of both over the probability
of the one that you care about okay
that's basically our set up there so
let's look at a fresh example Shannon
eats this news bar on our alarm clock on 60%
60%
school days if she doesn't hit this news
bar there's a 90% probability that she
makes the class however she does hit
this news bar there's a 70% probability
this 90 percents and this 70% are both
conditional probabilities they are
conditional because we already know what
happened with the snooze bar if she
doesn't hit it there's a 90% chance if
she does hit it there's a 70% chance so
that whether or not she hit it is
already kind of wrapped into the problem
when you have a situation like this
where there's a couple of events going
on so first thing that happens is
Shannon has to decide is she hitting
snooze or nods and then she either makes
it to school on time or she doesn't when
the last section we talked about those
nice to wait tables for probabilities
our second probability tool which I did
talk to you guys about a little bit in
algebra 2 is making something called a
tree diagram and tree diagrams organize
probability questions where you have
seek a sequence of events Colonna the
first thing that's gonna happen in the
context of this problem is whether she
doesn't hit snooze or she does so I'm
gonna have no snooze it doesn't matter
what order really I'm gonna have snooze
and it says she hits the snooze bar on
60% of school days so snooze is 60% of
the time that means that no snooze is
40% these should be complements of each
other if she doesn't hit snooze there's
a 90% probability that she's on time
there's a 90% probability she's on time
meaning that there's gonna be a 10%
chance that she's late if she does hit
snooze her chances of making it on time
gets smaller she's gonna have a 70%
probability of making it on time and
she's gonna have a 30% chance of being
late so when you make a tree diagram
like this this second wave of
probabilities right here our conditional
probabilities because they're based on
what happened before so they just asked
us to make a tree diagram for Part A and
that's good I got that taken care
on the next page they ask us for a few
calculations so the first one says on a
random day what is the probability that
she is late to class so if you look at
your tree there are two kind of paths
that Shannon can take that would result
in her being late it could be where she
does not hit snooze and she's late so it
could be this path right here that would
result in her being late to school or
she could hit snooze and she could be
late so it could be this path right here
when you go down a path in a tree diagram
diagram
you're gonna multiply the probabilities
to get or you add the probabilities
this is like an end situation where you
mean snooze and late or something like
that you multiply your probabilities so
for the 0.4 times 0.9 this top branch
for example would be a point 36 that's
really hard to see so we'd have like a
point 36 up here 0.4 times point 10 is
point O four point six point seven is
point 42 and then finally point 18 and
if you add up all the end probabilities
here to make sure you get it right they
should equal one okay so they asked in
Part B what is the probability that
she's late to class it's a good idea to
show the work I guess I have a tree
diagram so you probably wouldn't lose
credit if you see it end but it's just
smarter to show what you multiplied it
in this times this or it could be where
she hit snooze and she's late so you
multiply the individual guys you get
this you read this and then it's this or
this so you just add those two things
together if you do that you're gonna get
points 22 for your answer so good idea
to show your work right here there are
have been rubrics for AP problems where
you wouldn't use credit if you didn't
write that down so you got your little
answer with work shown all right so my
next problem here suppose the chin hit
this snooze bar this morning what is the probability
probability
she is late to school this part right
here suppose she hits the snooze button
this morning that is a dress the
conditional probability it didn't say
the word given but it does say hey she
hit snooze if something is locked in you
know it's a conditional probability so
if I was gonna write this in symbols I
would be looking for her using
probability probability she is late to
school given that we already know
she hit snooze so this is the
probability were being asked to
calculate so sometimes with conditional
probabilities you can use common sense
if you look at my tree right here it is
given that she snoozed
we all already know Shannon took this
path right here and she's already at
this point then we want to know the
probability she's late that's actually
really easy well if she already walked
to here and she's already made the
decision she's snoozing then she's going
to be 30% of the time ladies so we can
just use common sense and say oh the
answer is thirty seven thirty percent
their point thirty two that example okay
so if you're doing a conditional
probability where the first part of the
tree is given to you great all you're
gonna do is just write down the part
that they give you at the end right here
and you're good to go but let's talk
about the last one here and my picture
is getting all kinds of messed up right
now it's hard to see um so I'm gonna
erase what's in red right here at least
give myself some room to work but for
Part D it says that we are supposing
that Shannon is late to school and we
want to know the probability she hit
snooze this probability the probability
of snooze given lates is not as easy to
find us late is given but if you look at
the tree you can't just be like oh I'm
gonna walk to here in my tree doesn't
work like that late is on the end and
it's in a couple different spots so if
you have a situation like this where
it's the second part that's given you
have to use the formula
that I gave you on the last page and
what that formula says is that you take
when you do a conditional probability
the probability of both of those things
that's the probability of snooze and
lates over the probability of the one
that's given and the one that's given in
this problem is late so what is the
probability of snooze and lates snoozers
here late is here
neither is given just this and this you
do point six times 0.3 which is going to
be a points 18 that's the probability of
snooze and limits over the probability
of lates which happens two different
times but I already did that up here in
purple a lot of times probability
questions will make you use the answers
that you've already written down so
before you start making an answer from
scratch see if it's already something
you've answered the probability of late
we already found is a point 22 so let me
talk about what this is saying right
here Shannon's only late to school 22%
of the time so that means our
denominator are like terminal
possibilities here is no longer a
hundred percent we're only looking at
those 22% of days where she's actually
late and then out of those 22 percent of
days 18 percent of all days are where
she's actually hits limbs in his legs
and if you divide these outs point
eighteen divided by point 22 that's like
almost 82 percent of the time so if you
know she was late there's like an 82
percent chance that she hit snooze okay
so that's how you work your way through
a conditional probability so let's go
ahead and take a look at another problem
here actually we had a little ways to go here
here
um tree diagrams first in just a couple
rules and then we'll tack learning
problem when do you multiply two
probabilities together you multiply
probabilities together when it's and or
the on which not for but and slash the
problems which means and so we talked
about how our problems or is the Union
you're supposed to add probabilities
when it's and your intersect that's when
you multiply your probabilities together
is the multiplication formula on the
formula sheets no it's not so you do
have to remember that though I gotta
multiply them together now really
quickly before I keep going with this
I see common errors with and problems
ones kids remember oh I got a multiply
if you've got two different events going
on and you have to multiply them that's
great that's what you're supposed to do
with the probabilities but if you're in
two way table lands like earlier we were
doing a problem where like it was T and
W like this if they say what's the
probability of T and W and it's in table
format like this you can just find the
one that applies to look at them so you
would say it's 26 out of 81 so if you're
looking at a table use common sense and
pick the one that applies to both if you
have different probabilities going on
like what's the probability I roll a six
on a die and then a four it'd be 1 over
6 times 1 over 6 like that but if you
have a little table you oftentimes can
just look at it and get the answer that
way but in general you are supposed to
multiply your probabilities so are two
big tools that we use for complicated
probability questions are the two way
table and the tree diagram if you see a
probability question where the answer is
not just like there there's a very good
chance you're gonna do one of those two
things either tree diagram or to a table
when do you use each one gonna give you
a few options here so it says when
should you use a tree diagram instead of
a two-way table two-way tables are for
our problems or problems with overlap
when should you use a tree diagram when
you have conditional probabilities that
can be a giveaway so like Shannon her
snooze like face like that
affected how late to school she was off
and she was like swore um or more
frequently you're probably easier to
remember when you have sequential events
what do I mean by that one thing has to
happen before the other so Shannon had
to decide if she was gonna snooze or not
before she was late to school or not so
if there's a clear order to how your
events happen a tree diagram is one
you're wondering a lot of it is so those
are gonna be our two big strategies here
all right so we have a big problem to
kind of finish off our slides here will
gets practice what we've been talking
about so a lot of employers require that
employees take drug tests before they
start working um and there's such a
thing as a false positive and false
negative so if a test comes back
positive if you take a drug test and it
comes back positive that means the test
says there are drugs in this person
system positive means they found drugs
in your system negative means that they
did not find drugs in your system it's
the same thing with like diagnosis of
stuff like that you test positive for an
illness you have the illness but then
there's a false positive and a false
negative so if you think just logically
about what that means a false positive
means the test said positive but it was
wrong so the test said hey you did drugs
when you actually did this and a false
negative would mean hey the test comes
back says yeah this person is good
they're the drugs but they actually did
so you have to kind of think in your
head through the words right there when
you talk about like a false positive
there's a hidden conditional probability
in that statement if you say false
positive what's assumed or what's given
in the problem
well the test said positive but the
person was actually negative so it's
given that the person was negative
what's the probability the test says
positive so anyway these are hidden
conditional problems good news right
here so what we're gonna have to do with
this problem
it doesn't say I'll make a tree diagram
but it would be logical to do so it says
4 percent of employees use drugs and
then you're gonna take a test which has
false positive and false negative rates
the first thing that's gonna happen if
you think about this
when something's right here the employee
is either going to do drugs or they're
not that's going to happen before the
test comes into play so the first thing
we would want to branch off for is
whether or not the employee does drugs
and it says that that happens in four
percent of the time on our problems
today with our tree diagrams have only
had two possibilities here it's very
possible you could have a tree where
there are three or four options the key
is that all the branches along the same
like little end right here have to add
up to one alright so after you do drugs
or you don't do drugs you're gonna take
a test and the test is either gonna come
back positive or the test is gonna come
back negative I'm gonna work the false
positive and false negative in two
things when I actually write down where
the numbers go but we're gonna do drugs
or not your test is gonna come back
positive or it's gonna come back
negative if you do drugs and the test
comes back positive that's not a false
anything that's the tests making the
right call but if you do drugs and the
test comes back negative
think about what that would be is that
false positive is that a false negative
it is a false negative for this purpose
so there's a 10% chance that if you do
drugs the test isn't gonna catch you
that means there's a 90% chance it will
correctly catch you and then a false
positive that means no drugs but it says
you did drugs
that's 0.05 meaning this guy down here
negative like it supposed to is 0.95 and
what I'm gonna do is I'm gonna multiply
along my paths right here and get all my
little probabilities for the end it's
0.4 times 0.9 is a point 36 sometimes
it's helpful if you don't want to show
that work later you can just show it
like right here so I can do point 4
times point 10 is 0.04 as long as the
work is somewhere on your problem you're
in pretty good shape
0.04 eighths and then point nine six
times point nine five is point nine one two
two
so now that I have my little tree work
though I can actually answer the
question they asked me which is what is
the probability a person will test
positive so that could be this path
right here and it could also be oh shoot
my probability was off on this one but I
caught it early in that late a point
four point oh four times points nine is
0.036 I was off by a decimal spot there
omma that's also a point oh four so I
screwed those up as I was going through
here this is maybe point zero zero four
why had I caught it sooner rather than
later sorry about that so we've got our
new adjusted probabilities here anyway
we were finding probability of positive
which is gonna be that one I have
circled and also this one over here
alright so when you look at these two
probabilities what's actually kind of
crazy right here we're about to find the
answer for this one there's a point O
three six chance three point six percent
chance that a person did drugs and test positive
positive
there is a point O four eight chance
that a person didn't do drugs and came
back positive so if you think about what
that's saying if you get a positive test
it's actually crazily enough more likely
that you didn't do drugs than that you
did in this problem okay why does that
make any sense at all that's so crazy
like why would it be more likely you
test if you test positive that you
actually didn't do drugs
it has to do with the weights of your
different groups right here most people
fell into this no drug category on the
way that the problem was structured
right here so five percent of a big
number is going to be more than ninety
percent of a small number so if your
groups are fairly different in size you
can actually have stuff like that happen
so if you get a positive test like in a
situation like this and you didn't do
drugs ask for a second test like how do
you fix it you take a second test and
that one should clear you okay
so that's actually kind of crazy that if
you think about it that it's more likely
your positive test would mean you didn't
do drugs so what percent of prospective
employees who test positive actually do
drugs I assume that's what that says
here it's covered under that but it's
kind of what we were talking about right
here this is a hidden conditional
probability it's the probability who
test positive actually do drugs so you
want to find out the probability that
you do drugs given that you test
positive so we know already that they
test positive what I would do for this
is I would take the probability of both
drugs and positive which is 0.036 this
times this is 0.036
and I would take that out of 0.08 for so
if I actually divide these right here
I'm basically just figuring out what
percent of the positive tests are people
who did drugs and the answer to that is
0.42 nine if you actually round your
answer right there so as I said if you
get a positive test it's actually more
likely and still that you didn't your
drugs just because of those false
positives and false negatives when
companies are figuring out like how
sensitive their test is going to be they
have to take that into account false
positives and false negatives because if
they move it too far one way where it's
too sensitive you're gonna have more of
one type of area where if they make it
less sensitive you have more theater
this plays in the stuff we'll talk about
more second semester but that idea of a
false positive or false negative
basically if you're ever in a situation
where you're in testing for an illness
or tested for drugs or whatever it is
and the test comes back wrong or like
you're not thinking the result is
correct that's why they wouldn't have
you do a second test to make sure that's
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