This content introduces the fundamental concepts of probability, emphasizing its crucial role in statistical analysis and the importance of using simulations and a large number of trials to accurately estimate probabilities.
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Chapter five marks the start of the
probability portion of AP statistics and
probability is really really important
we're gonna get second semester into all
the really big topics that talk about
how you would run analysis on data
things like confidence intervals and
being able to run a significance test
before we can get to those really
important concepts though we have to
have a good understanding of probability
probability kind of grounds us and lets
us figure out like how likely something
is um so we can decide if results we get
are statistically significant or not
chapter five is our first of two kind of
heavy probability chapters in this
chapter we're gonna focus on a lot of
stuff that you would have probably seen
before with me and algebra 2 so we're
getting the basics of probability rules
down in our units and to kick things off
here we're looking at a little bit of a
um just scenario here so you and three
other friends there are four of you all
together go and you're studying for a
statistics test so you guys are to go
study together
um and then while you're studying you
all have your own textbook with you but
nobody put their name in their book like
they were supposed to and the books all
get mixed up while you guys are studying
so everybody just kind of grabs one
takes a home with them cuz what
difference does it really make and at
the end of the year when you all go to
turn your textbooks back in it turns out
that nobody actually grabbed the correct
textbook so the problem we're analyzing
right here is what is the probability
that out of four of you none of you
would have taken your actual book back I
just randomly grabbing one so when you
enter a situation where you have a
probability and you're not really sure
how to calculate it
because we haven't done that much with
probability yet think back to the very
first day of school when we did that
simulation with the airline pilots um
that right there is an idea that's gonna
carry us through a lot of statistics
when you don't know how to find a
probability you do what we did on the
first day of school and you design a
simulation to try to mimic the problem
and you find the probability that way so
right now I haven't taught you any
probability tools yet that we can use
we're gonna turn through a simulation to
help us out next lesson we're going to
talk about designing the simulation
ourselves but for now we're just gonna
use a nice little
pewter applet here to illustrate a few
points so um in my problem we had a very
light-hearted Oh lost your textbook the
problem here is a little bit darker in
that there's a hospital and there was a
mix-up with babies and there are four
babies that are all trying to get back
to their actual house so if you look
right here I'm gonna run a trial there
goes the store to the hospital and here
come the babies and you can see like the
blue one went to the right house there's
a little son right there but then the
other three did not end up with it was
post two so there are rain clouds
instead again a little bit darker than
just missing a textbook but let's
pretend this is just like a textbook
right here we want to know what the
probability is that nobody matches there
are four kind of people in this problem
none of them are supposed to match
that's what we're trying to figure out
so one trial right there like we had a
match but that's not nearly enough data
to figure out what's going on but before
we get to the specifics of that I want
to show you guys what I'm tracking in my
graph down over here this picture on the
right this is gonna estimate for us what
the probability is of getting no matches
and we did one trial and we got a match
so right now after one trial it thinks
the probability of no matches it's
impossible to get no matches which
obviously isn't right but we don't have
enough data yet so I'll run another
trial and there it goes again there go
the babies their knots we're gonna have
a couple matches here we had one match
again so it still thinks that it's
impossible to have no matches and we
just keep going and repeating the
process here I'm gonna keep going until
I get no matches which I think just
happens yeah they all missed on that one
so that was the third trial and one out
of those three had no matches so now it
thinks the probability of no matches is
1/3 or 33% sweet a dot at 0 dot at 0
now we've got at 33 I don't do one or
two more just to give you guys an idea
here you can see what's going on
we had some matches there so now one out
of four trials has had no matches on and
that's a 25% I'm choosing to analyze the
probability of no matches I could have
easily done one or two or three it just
depends on what you are interested in
I'll do one more and then I will kind of
bump up the trials here um and again we
had matches so one out of five times if
we had no matches so right now after
five trials I would estimate the
probability of no matches to be about
20% but naturally we learned that having
just a little bit of data is not good
enough so what I did right there is I
quickly did ten more trials turning off
the animation and you can see how the
probability after that first match was
33% then it went down to like 20% now
I'm back up a little bit so I must have
happen again where there were no matches
etc I'll run another 10 trials and what
I want you guys to kind of observe right
now from this graph is the probabilities
kind of spiky it's like up down over it
can't really decide where it wants to go
just yet after this is like 30-something
trials it says it's around a 37% if I do
another 10 trials now I'm up to a 42%
etc if you do small amounts of trials it
takes a little while and the probability
we're trying to estimate fluctuates
quite a bit now I'm gonna start doing
these like a hundred at a time and again
still got some spikes as we're
collecting more data right here and it's
gonna keep on going we were at like 44
or 45 percent over there now we're down
to 40 percent and I'm gonna throw in a
thousand trials now and it's gonna keep
on going we were in the forties for a
while it looks like we're getting into
the 30s right here 38 and I'm up to like
3000 trials and what I want you guys to
look at at this graph at the beginning
you saw it was all over the place oh
it's 33% no it's 20% no its 45% and
there was a lot of that going on but as
we keep going with the number of trials
notice that the graph for this
probability right here is getting a lot
flatter and a lot more consistence you
can look at that number as it goes right
there it's not really changing a bunch
like it used to and what's gonna end up
happening if you do enough trials and
you repeat the process enough times you
end up leveling out at what the actual
probability is supposed to be okay so
judging by doing like seven or eight
trials right here I would say the
probability of no matches is around 37
or 38 percent just by looking at all
these trials you have to do enough
trials that it levels out if you do just
a couple there's a lot of fluctuation we
can't trust it just yet how many trials
is enough is a question that we're not
gonna answer right now but just do lots
and you can kind of see that it does
level out around 37.5% sir so okay so
that is the kind of big idea I want you
guys to have in your head as we're
talking about these right here so it
appears the probability is you know 37
or 38 percent in that range so let's
introduce some kind of basic vocab
that's you guys are probably pretty used
to already probability you guys all know
and how is probability measured most of
the time the vast majority of the time
in AP stats we want a decimal between 0
and 1 so usually if we have like a 38%
like in the last problem we're always
gonna want to turn that into a point 3/8
treat your probability as a decimal so
you can actually do math on it a
probability of 0 would mean something is impossible
