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5.2a - Probability Rules | MrRzMath | YouTubeToText
YouTube Transcript: 5.2a - Probability Rules
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Summary
Core Theme
This content introduces fundamental concepts and rules of probability, moving from conceptual understanding to practical calculations, essential for analyzing statistical situations. It defines key vocabulary like sample space, probability model, and events, and demonstrates their application through examples.
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so section 5.1 is probably not what you
were expecting when it comes to like
starting a unit on probability it wasn't
much in terms of calculations and like
traditional probability stuff it's more
just about conceptual designing a
simulation using long run to help you
out etc sardian section 5.2
and also in 5.3 we're getting to a lot
more of those traditional probability
calculations because it's really
important that you can handle the basics
when it comes to analyzing more
complicated statistics situations so
this is the beginning of that and we're
talking about some big probability rules
in some vocabulary as well first off
vocab terms that you probably heard from
me in algebra 2 the sample space is the
list of all possible outcomes outcomes
being something that can happen and the
variable we use for the sample space is
a capital S but we'll put a lot of times
like these little fancy I don't even
know what you would call that on the
edges of the s so you would say like the
sample space is and when you're making a
list you do squiggly brackets so if I
was going to roll a die the numbers 1
through 6 would be my only possibilities
if I was gonna flip a coin my sample
space would be heads and tails if I was
gonna pick somebody out of my class get
the list of your name this would be my
sample space the sample space is just
everything that can happen okay a
probability model is gonna be a very
useful thing to us here it contains two
things whenever you have a probability
model you're going to have a list of all
outcomes so basically you can have your
sample space everything that can happen
and then you're gonna have the
probability for each outcome the
probability for each thing that occurs
so if I was going to design a very basic
probability distribution um a lot of
times you'll see these as tables right
here I could have a Cohen example where
I have heads and I have tails
and then I have probabilities 1/2
probability as 1/2 so I'd have liked
probability and I'd have liked the thing
the events speaking of the last vocab
word on this page is called in events
and an event is the a subsets of the
sample space what do I mean by subset of
the sample space it's inside the sample
space but it's a smaller part of it so
it's like just a collection of outcomes
that are all in s so collection of
outcomes in s so on my ruling and I
example I could look at the events
getting a 4 or higher or I could look at
getting an odd number or I could look at
I don't know a prime number or something
like that if I was gonna pick somebody
in class at random I could look at the
probability you have class the event of
you wearing glasses are you being at
least 5 5 or something like that so an
event is just basically whatever you
want it to be you just define a
collection of things that you're
interested in finding the probability of
usually when we establish an events it's
because we want to know its probability
so there's a little bit of vocab for you
all and then what we're gonna do right
here is look at a basic example using
sample space to arrive at a probability
or probability distribution so we're
gonna picture flipping a fair coin three
times and we want to describe the
probability model which means all the
possibilities and the probability for
each so in a problem out like this
that's not that big in scale the best
thing to do is just write out all the
possibilities so I'm gonna picture if
I'm gonna flip three coins it could be
where they go heads heads heads just
like that and I'm just gonna write out
all the possibilities here trying to be
somewhat systematic with it I can go at
heads heads tails heads tails heads and
then I need tails heads heads so that's
all the ones with two and then I could have
have
adds eight of HTT I could have th T I
could have TT h TT t so this is a list
of all of the possibilities incidentally
thinking about like algebra 2 when we
look at the number of possibilities
total we had two choices for the first
one two for the second one and two for
the third one there should be eight
possible outcomes all together so I have
all the possibilities here this is what
can happen when you flip three points
they say to describe the probability
model for this chance process and use it
to find a probability of getting at
least one head in three flips so what we
care about in this problem is the coin
landing on heads we want to get at least
one of those so if I'm gonna make a
probability model those are generally
written as tables and we're gonna care
about two things when we make our table
we care about how many coins land on
heads and we care about the probability
of that happening so the least number of
coins landing on heads we can get a zero
or we could have one on heads or we
could have two or we could have three
and that's all the possibilities getting
zero coins landing on heads is only
possible one time one right there and
for probability you just take it out of
the total there are eight possibilities
that are all equally likely so the
probability of getting one
sorry no coins landing on heads is 1 out
of 8 next up I'll go for one coin
landing on heads that's going to be
these three options right here so this
is a three out of eight to coins landing
on heads is these ones right here that's
a three out of eights and then the last
one all three landing on heads is a one
out of eight so they asked us find the
probability of getting at least one head
in three flips at least one head could
be this or it could be this or it could
be this basically it's just not that one
and if I look at my
all those circled no not all the circle
guys all these circle guys have at least
one heads so the answer to the problem
we care about is seven out of the eight
possibilities here okay so if you make a
list of everything that can happen
it makes probability questions not that
bad in wise so let's go ahead and make a
little list right here of just basic
probability rules some of which you guys
have already heard our first rule is
something we've already talked about in
the last set of slides for any event a
usually with events and again an event
is just whatever you want it to be we
use a capital letter to represent it's
the probability of an event is always
between 0 and 1 and we'll use this
notation P of n you write down your
event right here so the probability of a
is between 0 & 1 in other words you
can't have a probability of 2 or of
negative 7 or something like that
probability of everything in your sample
space is the probability of s occurring
is 1 so basically if you're gonna have a
sample space it has to be the
probability of 1 it's everything that is
possible in the problem so third rule if
all outcomes are equally likely then
what you do is what we did on the last
example to find the probability of an
events you take the number of outcomes
in your events over the number of
outcomes in your sample space I don't
know maybe that looks complicated it's
really simple though all it's saying is
you take what you're looking for out of
the total to find a probability the last
example was like hey out of the 8 times
how many had at least one
on heads will it would be seven out of
eight that's all that means
two more you have what's called the
compliments of an events and the
complement of a is a and you put like a
little it looks like an exponent of C um
the complement means everything knots in
the original and maybe that's worth
writing down if you didn't already know
that's so compliments when they talk
about that is everything knots in a so
let's say my event a is ringing I want
to find the probability it's gonna rain
tomorrow the compliment would be
everything knots in it so everything
that's not raining snowing foggy sunny
tornado whatever else as long as it's
not raining that would be the complement
so instead of listing out all that stuff
it's easier just to say not a sometimes
now let's say tomorrow there's a 20%
chance it's gonna rain well the
probability it won't rain so the
probability of the complement if there's
a 20% chance it does rain there's an 80%
chance it doesn't you just do 1 minus
0.2 so you take 1 minus your original
probability and that gets you the
probability of the complement if there's
a 30% chance something happens there's a
70% chance it doesn't that's all that's
really saying and then finally my last
what is that mutually exclusive it means
so let's say I was gonna roll a die
rolling a to a rolling a 3 those are
mutually exclusive like I can't say 2 &
3 at the same time if it's something
that cannot occur together it's called
mutually exclusive and if you want to
find the probability of A or B so the
first thing or the second one this
should look a little familiar because we
did talk about this in algebra 2 all
you're gonna do when they're mutually
exclusive is you're just gonna add the
two separate probabilities together kind
of did that with you on the last example
where I added up the 3 over 8 the 3 over
8 the 1 over its if there's overlap you
have to deal with that but that'll be
the next video so those are our rules
for probability and let's just look at
an example to close this video outs so
at the time I'm recording this video
this is my full AP data sets of scores
that kids get on the AP test and the
probability associated with those
outcomes out of all the kids who have
taken the class and you can see the
breakdown here by score which
incidentally should give you guys like
make you feel good or motivated that hey
if I put in the work and I study I will
do well on this test and I will get
credit for the class because we'll cover
everything you so stick with it even if
it feels like shaky at times you'll do
well if you keep putting in the work it
says to show why this is a legitimate
probability model I didn't explicitly
say this earlier but you need to do two
things when you show it's a legitimate
probability model you need to show that
all probabilities are valid by valid I
mean they're between 0 & 1 so if you
look at these yes yes yes yes yes
they're all between 0 & 1
good you also need to show that all the
probabilities add up to 1 so these don't
add up to 1 that means that something is
missing and it's not a full probability
model because it doesn't have everything
in the sample space so they all
probabilities and to one check so what I
would have to do to verify that is I
would just add these up
and I would make sure that they actually
do equal one to make sure I didn't leave
anything out and they do so that is a
good call okay so if I ask you to verify
that something's a probability model
make sure there's no like weird negative
probabilities or probabilities of like
seven and then also make sure the
problem is at what they're supposed to
then once you've done that they just ask
you for a few basic things here what's
the probability that if you pick a kid
at random who's taking my EP class
they've scored a three or better
well three or better these are mutually
exclusive you can't get a three and also
get a 4
your score can't be both at the same
time so all I need to do is add up all
the ones I care about add up the threes
out of this add up this and that's going
to get you your total probability so if
I do those and when you write these down
guys show your work always show your
work no matter how basic it is it's
so it looks like the probability that if
you're you pick a random kid that they
pass the AP test it's going to be about
eighty-five percent eighty five point
five percents okay so that is just what
you do you add those up and then for
this next one here it says the
probability if you pick somebody they
did not get a four so it could be like
anything else but not a four there are
two ways of doing this one is just add
up this this this this and that will be
totally cool that's everything that's
not a four you can also use that
complement rule which sometimes is
easier and just do one - the one that
you do care about it so that you don't
want or whatever if you do that you're
gonna get a probability of 0.8 1/8 so
that is how you calculate basic
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