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
