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5.2b - Two Way Tables & "OR" Probability | MrRzMath | YouTubeToText
YouTube Transcript: 5.2b - Two Way Tables & "OR" Probability
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Summary
Core Theme
This lesson introduces and reinforces the concept of "or" probability, emphasizing the use of two-way tables as a highly effective strategy for organizing data and simplifying calculations, especially when dealing with overlapping events.
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so in this lesson we are going to tackle
the concept of or probability you should
remember or probability from algebra 2
we did talk about it kind of briefly
towards the end of the class but we'll
review it here and we're gonna actually
use a new strategy that I didn't teach
you at the time which is using something
called a two-way table which we have
talked about in this class but using
tables actually makes or probabilities
very very straightforward so you'll see
how that works is we jump in now when we
do probability questions on we talk
about or and and in algebra 2 and I
didn't use these symbols with you at the
time but maybe you've seen them before
looking at geometry class or something
like that we have what's called the
intersection and the Union intersection
is this little symbol right here looks
like an upside-down U or an N and then
Union is this symbol right here which is
like a u U for union should be pretty
easy to remember intersection as like an
N in its and I think I make a note about
that at the bottom for you guys to but
you can one way you can look at these is
through a Venn diagram the intersection
of a and B is what they both have in
common the intersection of a road is
where the two roads meets so the
intersection is what they share is part
right here in green and the union when
you unite two things you put them
together and it's all of the stuff that
you see in either one of them or in both
of them so intersection works very
naturally with the concept of and in
probability you need a and B and it's
this part in the middle Union goes right
along with the concept of or probability
this or this both so Venn diagrams can
be a helpful way of visualizing this
stuff um and we'll talk about them a
little bit I personally don't care for
Venn diagrams a whole lot I'd rather use
a table but I'm gonna show you how they
works you're at least familiar with them
and there's the little hints you free
Union and for intersection so let's look
at a data set here which is looking at a
two-way table of
versus whether or not your ears are
pierced and there's a bunch going on in
this slide right here
first of all the two strategies you can
use for problems like this you've got
multiple events usually we're gonna
focus on two events at a time here we
have gender and then ear status we want
to either analyze them in a two-way
table which we've done before back in
Chapter one or in a Venn diagram
personally I like to wait tables a lot
better I find that when I make Venn
diagrams that's like think really hard
about where everything goes and I spend
too much energy just making the thing
and then even when I make it I get
confused so I like tables better but
then diagrams are perfectly good if you
like them too so what you would do the
two events that we care about in this
problem are gonna be we've our two
variables of gender and pierced ears but
our events that we're establishing our
male and pierced ears a is male B is
pierced ears so this yellow circle right
here is all the males you can see in
this problem that there are 90 of them
and they're broken into the ones who
don't have pierced ears and the ones who do
do
because B is pierced ears and these
nineteen people have both things so
they're in the middle right here then
with pierced ears if you look at these
there's a hundred three of them all together
together
there's the 19 that are male and then
there's the 18 four that are female
right here and you'll also in a Venn
diagram problem have the number where
neither of these things applies so those
are going to be females without pierced
ears evidently and that would go on the
outside of the circles and you can look
at each of these events right here the
little intersection symbol basically
means ends so this is a and B male and
pierced ears a and not B etc etc etc so
you can look at those and analyze that
then diagrams are great if you like them
but I don't care for them so I will
usually opt to make a table instead of a
Venn diagram when I analyze these sorts
of probabilities so if two events are
mutually exclusive this is our addition
rule for our probability in algebra two
I would have told you guys that or means
you add up your probabilities if they're
mutually exclusive it means that they don't
don't
overlap there's no overlap possible and
if I want to know the probability of A
or B or AP stats we'll use the Union
symbol for it see if they're kind of
interchangeable and you got to be able
to use both all you do is you add up
your probabilities and you're done like
we did in the last problem with the AP
scores three or four yeah add them
together and you're done
but a lot of times in real life events
are not mutually exclusive in other
words they overlap each other so if they
overlap what you're gonna do you still
need to add your two probabilities
together but after you do that you have
to subtract out the stuff you double
counted the stuff you double counted
would be a and the the intersection so
what this means right here is that you
have to subtract the overlap this
formula is on your formula sheets
written just like this on the first page
so you'll have that there but hopefully
this is something you can just remember
and not have to look up because we use
it quite a lot
let me quickly show you in our last
example what that we um back on my
little table right here my little bun
diagram whatever if I wanted to find the
probability of male or pierced ears I
would add up male I would add up pierced
ears but then I would be double counting
the males with pierced ears so what I
would need to do to offset that is I
would subtract out these 19 people that
I double counted because if I did all
the male's that's 9ye all the pierced
ears that's 103 that's gonna be too much
I double counted this 19 or just add
these three separate numbers which are
mutually exclusive and get your answer
generally I don't even mess around any P
stats or I would advise you not to mess
around with the formula because you can
use a table that makes the problems so
much easier so we're gonna go ahead and
I'm gonna show you guys how to set that
up so in this problem right here fresh
example we're looking at home ownership
versus whether or not you are high
school graduates so we have two events
that we care about here and they don't
define them for us I'm
go ahead put letters just so I can not
have to write out the words every single
time so G is gonna be that you're a high
school grad and then o is gonna be that
you're an owner like you own a home
so with the two-way table we're gonna
have two events and it doesn't really
matter which one goes each way I'm gonna
make grab go this way so I have high
school grad and I would have not a high
school grab a little compliment symbol
and then I would have home owner and I
would have not a home owner and then
what I'm gonna do is break the problem
up the numbers in here and put them in
the different boxes it says there's a
random sample of 500 people
the grand total usually goes bottom
right outside the box like this kind of
like a marginal distribution where you
put it on the outside and then what I'm
gonna do is start filling out the
problems with the numbers they give me
three hundred forty people were
homeowners owners this and this together
our homeowners there's three hundred
forty of them uh there are three ten
high school graduates and it says to 21
we're both homeowners and high school
graduates so the way these problems
usually work is they'll give you the
outside numbers they have to give you at
least one on the inside for the problem
to be possible so there's two hundred
twenty one that are both right here and
once you know that you can you start
subtracting things like these up to five
hundred this would be a one ninety so
you get five hundred I can do three ten
minus two twenty one and that'll get any
of that box etc so basically you just do
math no no and you can fill out the
whole inside of this picture
yeah 71 that's one nineteen yes so you
get your whole inside filled out takes a
tiny bit of effort to set up a little
two-way table but once you have it you
can answer it all sorts of probability
questions super quickly so what I'm
gonna do is leave my little table right
there and I'm a tab over to the next
page so what's the probability you're a
high school graduates
graduate was gee so you're a high school
graduate if you're here or here that's
actually just the three times and they
actually straight up told us that in the
problem so that's not very exciting
that's gonna be 310 out of 500 and you
could make that into a decimal
I think that's 62% that's not that bad
but you don't have to you can also just
leave it as a fraction
honestly AP test usually doesn't even
care if you reduce your not so don't
stress yourself too much over that
unreduced traction that's what you want
it is cool alright next problem says
that they are not a high school
graduates so not a graduate is gonna be
these guys right here and they own a
home owns a home is here so not graduate
and a home is gonna need these people in
my table so I have a hundred nineteen
people who qualify out of the five
hundred total and then you can make that
into decimal I'm not even gonna bother
with this one so we just have my answer
right there okay this next one is where
the whole table thing is actually gonna
help us because so far it's been kinda
just like looking at the picture and
it's not that difficult to do um but
with this last one here they're finally
asking us about or probability and when
they bring in or probability this is
where the two-way table kind of shines
for us so there are two events that we
care about is is I school graduate or
they own a home now you could do the
high school grads which was I think
three ten plus the homeowners which is
340 if you do that that's already over
500 and you got to subtract out the ones
you double count subtract out to 221 and
you would get your answer but if you
have a table the tables are already
separated so each box on the inside is
mutually exclusive if you're a grad and
a homeowner you're not a grad and a not
homeowner you can only fit into one of
these boxes right here so if you take
the time to make the two-way table all
you need to do is just circle up all the
stuff you care about high school grad
okay that's you that's you or OH
well I already circled that one so I'm
not gonna do that again but this is also
a homeowner right here and if you add
these guys straight up you will get your
answer for the problem always good to
show work of some fashion if you add
those up
I think that's 429 out of your 500 so if
you make a table you don't have to worry
about subtracting out the double
counting because it's separated so there
is no double counting or you can use the
formula if you prefer either way it's
kind of up to you what you do you can
even use a Venn diagram if you want to
do that so we have one more example
right here with a bunch of parts to it
fresh problem we have 60 percent of
households in the US with a landline
phone but that's gone down since 2012
and 89 percent with cell phones 51
percent have boats so our two events in
this problem that we care about Mia
L is that they own a landline and then
we all do C for cell phone doesn't
really matter what letters you choose
that's just as long as you can finance
and in my little two-way table right
here I'm gonna have L I'm gonna have not
L I'm gonna have C and I'm gonna have
not see this time they didn't give me
wrong numbers like the last one they
gave me percentages so if they give me
percentages then obviously the total is
gonna be out of a hundred percents or
one so it says sixty percent had
landlines so 60 percent landlines that
means this is already gonna be a forty
percent right here eighty-nine percent
had cell phones so 11 percent goes here
and it says the number on the inside is
that fifty one percent have both once
you know a number on the inside you just
start subtracting and doing easy math
and you just have your problem basically
and it's pretty straightforward getting
this all filled out so I've got all my
inside numbers and that took a little
bit of work in the first place to get it
all ready but once it's there it's
really easy to answer a bunch of
probabilities pretty quickly so it says
for Part B find the probability that the
household has it
one of the two kinds of phones so again
all you need to do if you took the time
to make a table is circle the stuff you
care about that could be this one or
this one those are both cell phones or
it could be this one over here so you
just add up to these three boxes or you
could also do the complement and
subtract out the one thing you don't
want however you set that up you're
gonna end up getting the answer of 98%
for this problem so it's pretty quick to
get that answer let's go on to the next
page and answer a few more quick
probabilities here I'm gonna keep my
little picture next up find the
probability that the household has
neither type of phone well actually I
just identified that in the last one
that's gonna be this box right here you
don't have a cell phone and you also
don't have a landline that's just the
simple two percents really no work to
show there to get that answer and then
finally find the probability this house
is a cell phone only when it says cell
phone only there's like that implication
there that they have a cell phone and
they also don't have a landline so that
is actually just gonna be this box cell
phone and no landline which is just 38%
so when you're dealing with more
complicated probability where it's not
just a very simple quick answer a
two-way table is a great technique for
organizing probability and making it
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