Hang tight while we fetch the video data and transcripts. This only takes a moment.
Connecting to YouTube player…
Fetching transcript data…
We’ll display the transcript, summary, and all view options as soon as everything loads.
Next steps
Loading transcript tools…
5.1a - Randomness and Probability | MrRzMath | YouTubeToText
YouTube Transcript: 5.1a - Randomness and Probability
Skip watching entire videos - get the full transcript, search for keywords, and copy with one click.
Share:
Video Transcript
Video Summary
Summary
Core Theme
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.
Mind Map
Click to expand
Click to explore the full interactive mind map • Zoom, pan, and navigate
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
impossible
mark can never happen probability of 1
Union means it will definitely happen
for example if I pick one of you guys
out of my class the probability that you
are a student at mrh is a hundred
percent it's assuming I'm doing this in
my classroom um etc you guys should know
very basic stuff about probability I
would think already this last problem
right here is giving us a probability it
says oh the probability of getting a sum
of 7 when rolling two dice is apparently
1/6 later on in this chapter we'll learn
how to find probabilities like this
ourself but for now it just gives us a
probability and we are being asked it
says over here to interpret this value
when you interpret a probability in AP
statistics it is important that you talk
about it being over many
or over the long run you need to make
sure you mention this concept of many
trials when you give an interpretation
of a probability kids forget it all the
time I'm gonna do it on your next quiz
and ask you to interpret a probability
if you don't mention that there were
many trials you will lose credit
audience so what I would say for this
we're getting a sum of seven when
rolling two dice over many sets of two
dice rolls approximately one-sixth or if
you wanted to do a decimal point one six
six ish oh the rules will have a sum of
seven if you do it many times about 1/6
of them will have that sum that we were
looking for in the problem so when you
interpret a probability always put it in
terms of like long run over many trials
one other comment on this it's a bad
mistake that kids sometimes make when
you look at probability if you do
probability that's kind of small which
happens especially second semester we'll
find some small probabilities let's say
your calculator goes like this that's
kind of hard to see two point three
eight one negative six or something
people will be like oh the probability
is two point three probability is always
between zero and one that doesn't make
any sense if you see something like this
you have to remember scientific notation
you would move your decimal six spots
over from there so you're gonna end up
having five zeros in front of that
decimal so be careful don't ever report
a probability that's over one if you do
that it's like automatically is wrong
you can't get any credit for it
so yeah the basics of probability here
not too bad we have a distinction which
I've talked to you guys about now there
were two a little bits between
theoretical and experimental probability
theoretical probability is what the
formula says it should be so what
determine the probabilities should be
cool um example of that the theoretical
probability of flipping a coin and
handing having it land on heads is 1/2
the probability of rolling and die one
through six and having land on one is
one out of six that's just using common
sense or math and you get more difficult
or complicated as well where you start
multiplying or adding probabilities but
if you did like math to get your answer
and you ran a formula of some sort
that's a theoretical probability it's
what it should be in theory but then we
have what's called experimental
probability which is probability
calculated or estimated estimated or
calculated by using a large number of
trials okay so if you're actually
collecting data yourself for something
that's not so easy to do um you would be
finding an experimental probability so
if I wanted to flip a coin and count the
number of heads out of a hundred flips I
get 53 my experimental probability of
getting heads on point could be 53 out
of 100 it does not have to match up with
the theoretical probability but if
you've lots and lots and lots of trials
it should level out at that value okay
the law of large numbers is basically
repeating what I just said right there
and it's talking about what we saw in
that little applets
which is that if many trials are conducted
the probability of an events will level
out at it's true probability and it's
true theoretical probability so
basically law of large numbers says if
you do lots of trials the probability
that you get from your data is going to
be like it'll level out and be the
probability that you were looking for
okay so those little spikes that we saw
at the beginning will kind of go away
and it'll just flat flatten into the
probability it's supposed to be so yeah
um two more things to kind of close out
this lesson here both relatively short
um I'm not gonna make you actually write
these down right here but I have read
about like a professor who actually does
this in his one of his college level
statistics classes their homework for
the night is to go home and flip 50
coins or just lie about it and write
down 50 results and then he'll kind of
walk around the room and decide oh yeah
you made it up or you actually did it
see things that people will do first of
all if you look at all 50 of these
trials right here people probably won't
have too many streaks of more than like
three or four in a row of the same thing
where like if you're flipping a coin
it's very possible you could get five or
six so the same thing eight or nine do
the same thing in a row people miss in
their head they misunderstand and they
feel like things need to level out or
balance out faster than they actually do
so most people when they do this right
here when you look at all 50 of these
there's probably about the same number
of heads and tails probably 25 to 25 or
27 to 23 pretty close to each other not
too many people would go like 35 to 15
or something like that but as we saw in
that thing with the babies in the
hospital if it took a couple hundred
trials before it really did kind of
level out where it was supposed to so
it's very well possible that you would
have a big streak of the same thing or
that you would have more of one than the
other when you do such a small number of
trials like 50 so that's the main thing
that that slide is getting your crops
in the last part of this lesson is just
two misconceptions that are kind of
opposites of each other just to watch
out for the hot hand is basically
assuming that you're basically on a hot
streak with probability so if I'm
flipping coins and I get like five that
are heads in a row somebody would look
at them like oh my gosh you're on a
streak you're just getting all these
heads the next point I'm gonna bet is
heads as well so they think that you're
hot so you're gonna keep doing the same
thing that you've been doing the law of
averages is like the opposite of that's
where if somebody looks at you and
you're flipping coins you flipped like
six heads in a row or something like
that oh it's been a lot 4-h those next
once we meet tail is because it's gonna
even out both of these are
misconceptions it takes lots of trials
for things to actually even know and you
can get things that look somewhat fluky
in the short term you can't make
predictions in the short term about the
very next one what you want to do if
you're going to be accurate is you're
gonna make predictions about the overall
um two quick things to mention right
here one in casinos um like there's like
the room that wheel that you can spin in
bed on like what it's gonna land on
stuff like that
they'll put like a little scoreboard on
the side where it'll show what the last
roll was that just happens and people
will look at that okay actually was read
so we must beginning black for this next
one or oh isn't even them there's we
must be getting an odd number that thing
that they have up there on the side has
nothing nothing whatsoever to do with
what's gonna happen next because one
trial has no impact on the next one but
people like to kind of read into it and
picture like it matters when it actually
doesn't it's also industries like um
insurance is a big one so I'm going to
use just auto insurance as an example
your rates for auto insurance as a
teenager are gonna be more expensive
than your parents are then like someone
who's like in the fifties or sixties
would have because statistically you're
more likely to be in an accident and
what companies like that will do is
they'll look at the long term of like
okay how many people if I insure a
million people how many of them are
probably going to have an accident over
the course of a year and they adjust
their rates to make sure that they're
always making money in situations like
that casinos do the same thing okay how
many people are gonna win if I have five
million people come in this mom
and then one year whatever died this
many people coming in and they set their
rates that they end up making money so
if you have the ability to have large
amounts of trials you can get a pretty
good sense of what's going to happen if
Click on any text or timestamp to jump to that moment in the video
Share:
Most transcripts ready in under 5 seconds
One-Click Copy125+ LanguagesSearch ContentJump to Timestamps
Paste YouTube URL
Enter any YouTube video link to get the full transcript
Transcript Extraction Form
Most transcripts ready in under 5 seconds
Get Our Chrome Extension
Get transcripts instantly without leaving YouTube. Install our Chrome extension for one-click access to any video's transcript directly on the watch page.
