0:04 so we are in our final lesson of chapter
0:07 5 tackling the concept of Independence
0:09 in probability questions and
0:10 independence is a topic we've talked
0:13 about before um a little bit and I'm
0:14 gonna remind you guys of what that means
0:17 as we get things started here so first
0:19 of all before we answer this question
0:21 and I tell you how you tell if things
0:23 are independent I'll talk about how what
0:25 what even means for things to be
0:27 independent where would we have heard
0:30 this word before it was back in like
0:32 chapter 1 when we talked about the idea
0:34 of a conditional distribution
0:36 conditional distribution where we were
0:38 looking for if one variable had an
0:43 association with another so we said that
0:46 two variables have an association if one
0:48 variable knowing the value of one
0:50 variable gives us information about
0:52 another we like we're careful not to say
0:54 cause and effects well if two things are
0:57 independent it means that there is no
1:01 Association and if there's no
1:19 about the other in like a very simple
1:22 sense if I flip a coin and flip a second
1:24 coin knowing the outcome of my first
1:27 coin does nothing for me about my second
1:28 coin earlier in the chapter I talked
1:30 about that roulette wheel where they
1:32 show what the last spin was well the
1:34 spins are independent of one another so
1:36 knowing one doesn't help you at all with
1:40 the other now when we talk about
1:43 independence in AP stats you very very
1:45 rarely will be talking just from your
1:47 gut on this like oh it doesn't seem like
1:49 one affects the other what you need to
1:52 do is show mathematically that the
1:53 events are independent so there is a
1:56 formula you will check to make sure the
1:58 events are independent or if they're not
2:02 what you will do is find the probability
2:03 of one of the events probability of a
2:07 and if the probability of a is equal to
2:12 the probability of a given B this would mean
2:13 mean
2:17 that a and B are independent so a and B
2:21 are independence if this is true
2:23 probability of a equals probability of a
2:26 given B you could also reverse it and do
2:30 it the other way probability of B equals
2:35 the probability of B given a basically
2:38 what this means I'll use the top one so
2:40 probability of it is the same thing as
2:43 the probability of a given B in other
2:47 words B happens well okay it's still the
2:49 same probability for a doesn't tell me
2:51 anything all right you have to be
2:53 careful about not mentioning cause and
2:54 effect when you do this because that's
2:56 not what's going on it's that one
2:58 variable gives you no information about
3:00 the other so if B happens that doesn't
3:02 tell us anything about the probability
3:04 of it and we're going to look at the
3:05 example from last time which will be a
3:07 really good one to kind of visualize
3:09 this which is that Diamondbacks example
3:11 where we had that free tacos and we had
3:13 the wins I'm gonna go ahead and put that
3:15 table back on here but you guys should
3:16 look back in your notes and flip to that
3:18 so you had that handy when we talked
3:20 about this or just recopy it when I
3:23 write it down okay so here's that table
3:25 that you guys had from your last slides
3:26 I believe it's the exact same one you
3:28 guys already have written down they
3:31 asked us if tacos and wins are
3:34 independent of one another and again you
3:36 have to be careful when you talk about
3:38 independence it doesn't mean one has to
3:40 affect the other we're not talking cause
3:42 and effect we're just talking if one
3:44 event gives us info about the other so
3:46 if we want to establish that these
3:48 events are independent if you start
3:50 saying well yeah the wins affect the
3:52 tacos or something is wrong no credit it
3:54 has to be mathematically and you do that
3:56 using what's written in red right here
3:59 so what I will do to verify these events
4:01 are independent or if they're not is I
4:03 will find the probability of one of them
4:05 doesn't matter which way you do it let's
4:09 say I go for the probability of tacos
4:13 probability of tacos is thirty out of
4:15 eighty one based on what I have written
4:17 down right here thirty taco games out of
4:21 eighty one then what I'm gonna do is I'm
4:23 gonna find the probability of the same
4:26 thing tacos given
4:29 the other events given that you win
4:31 so okay they want does that tell us
4:33 anything about the probability of tacos
4:37 and then if I know that they won I would
4:39 be looking at this guy right here
4:42 so my probability of tacos given win is
4:47 26 out of 41 you can look at those right
4:48 there and see that they're probably not
4:50 the same but I'm gonna go ahead and make
4:51 them decimals anyway to make it easier
4:53 to compare and we can make a point about
4:56 that so let's talk about this first guy
5:00 right here there's a 37% chance in a
5:02 random game that the Diamondbacks are
5:04 gonna win tacos or that you're gonna get
5:06 tacos because the Diamondbacks scored
5:13 six runs and there is a 63% chance 63.4%
5:16 chance that the Diamondbacks if they win
5:18 that you're gonna get tacos so in a
5:20 random game if you just walk in and said
5:21 hey are we getting tacos today that's
5:24 when you score six runs or more there's
5:27 a 37% chance that's gonna happen but if
5:30 you know they win the game you look only
5:33 at the games they win that chance of
5:37 getting tacos jumps to 63.4% in other
5:39 words if you know they won the game
5:42 there's a greater chance that they
5:43 scored six runs because those terrible
5:45 games where the scores euro runs in one
5:47 run they probably didn't win in the
5:49 first place so if you know then your
5:52 chance of getting tacos actually
5:55 increases does that mean the wind caused
5:56 you to get tacos
5:58 no it didn't you could win by scoring
6:00 two runs or three runs or four but
6:03 they're tied to each other and winning
6:05 knowing that piece made it more likely
6:09 that you would get tacos you could have
6:12 also opted to do these the other way you
6:13 could have done probability of win and
6:17 probability of win given taco it's the
6:19 same logic if you know they get tacos
6:21 there's at least six runs and if you get
6:23 six runs you're probably more likely to
6:24 win the game okay
6:26 now I need to actually answer the
6:27 question cuz I haven't done that yet
6:30 what I would have to do to get full
6:31 credit for this problem because I've
6:32 just shown some work
6:34 I haven't actually addressed what they
6:37 said I'm gonna say something to the
6:39 effects of
6:45 since the probability of tacos which is
6:47 37 percents
6:51 does not equal the probability of tacos
7:00 given wins which was 63% the events T
7:08 and W are nots independence in order for
7:09 them to be independent you have to get
7:10 the same thing that means they're not
7:12 independent then it's because of what we
7:15 talked about if you win you're more
7:17 likely to have scored six runs so two
7:20 events can be dependents without one
7:23 actually causing the other one so you do
7:25 not need causality to be able to talk
7:29 about independence here so this is a
7:31 good thinking page right here where you
7:32 probably should pause the video and
7:34 think about it for yourself a little bit
7:36 there's a difference in the two terms
7:38 that kids get confused a lot are the
7:40 idea of mutually exclusive and
7:44 independence mutually exclusive comes up
7:48 in our or style probability independent
7:50 comes up more in our hand style
7:53 probability so when it says what is the
7:54 difference here maybe write that over
7:56 here mutually exclusive comes up in or
7:59 independence comes up in it but think
8:01 about how you can fill out these tables
8:02 I usually give groups a few minutes to
8:05 think about this one so I'm gonna go
8:07 ahead and talk about the answer pause
8:09 the video if you haven't tried to get if
8:10 two events a and B are mutually
8:13 exclusive it means they can't occur
8:16 together which means that a and B up
8:18 here is a zero and then in the rest of
8:19 the boxes are pretty easy you can just
8:22 use basic math to figure out what's
8:23 going on right there that's the only
8:26 possibility there that they cannot occur together
8:26 together
8:29 that's what mutually exclusive means so
8:32 mutually exclusive means they cannot
8:42 independent on the other hand this one
8:44 is usually a little tougher on kids to
8:46 figure out it goes back to that
8:49 definition of independence you can do it
8:50 in either direction but I'm going to do
8:52 probability of a first
8:55 probability of a given B and these mean
8:59 to be equal the probability of a I can
9:01 pick out of my picture right here it's
9:07 60 out of a hundred so that's fine I
9:10 need to compare that to the probability
9:17 of a given B so if B is given I'm out of
9:19 this box right here and it would go
9:21 probability of a and B which is this box
9:26 out of probability of B which is 30 so
9:27 essentially what I have here is like a
9:29 little cross-multiply problem or you can
9:31 just treat it like a decimal make this a
9:33 point six right here if you actually do
9:35 that that's going to be 18 for your
9:38 answer so this is gonna equal an 8 out
9:43 of 30 this box right here must be 18 y
9:45 again if you find probability of it
9:48 that's 60 out of a hundred any given B
9:51 is 18 out of 30 which is also a point
9:53 set so it equals the right thing and
9:54 then from there you can just fill out
9:57 the rest of the numbers using your basic
10:00 subtraction rules here and you get your
10:03 table so independent means that one
10:04 gives you info on the other we just
10:06 defined that on the last page so I'm not
10:07 gonna make you write it down again
10:09 yeah but make sure you can keep those
10:15 two terms straight so review here of all
10:17 this probability business that we've
10:18 been talking about in this chapter and
10:21 you have to look at whether or not in an
10:23 or problem they're mutually exclusive
10:25 and an N problem if they're independent
10:29 amounts if they are mutually exclusive
10:32 in an or problem all you do is you add
10:34 your two probabilities together done
10:36 they don't overlap Peters Adam and it's
10:39 over if they're not mutually exclusive
10:43 you still add them up but then
10:45 afterwards you have to subtract out the
10:47 overlap the stuff you double count which
10:48 is this right here
10:50 again if you do the true way table
10:52 strategy for problems you don't have to
10:53 worry about this step because you've
10:55 already broken it into mutually
10:58 exclusive boxes but that is your basic
11:01 setup for problems
11:03 and we have independence right here if
11:05 events are independent I flip a coin I
11:06 flip a coin
11:09 all you do define an ian's probability
11:11 is you multiply the probabilities
11:14 together one half times one half
11:16 whatever events you want to think about
11:17 if they're independent you just multiply
11:20 their probabilities together if they are
11:22 not independent or in other words if
11:25 they're dependents you're still gonna
11:27 multiply so you start out with the same
11:30 first probability but then what you do
11:33 is you adjust your second probability
11:35 like I did a marble example a video ago
11:38 if I wanted to get two blue marbles I
11:40 have five blue five red my first one is
11:42 gonna be a five out of ten but my second
11:43 one if I don't put it back is gonna be a
11:46 four out of nine so you have to adjust
11:49 that second fraction and the way that
11:51 you would do that is you would find the
11:53 probability for your second event given
11:57 your first events so basically you have
11:58 to account for whatever it is you
12:00 already took out I do wouldn't get too
12:02 caught up in symbols with these what I
12:04 would do is I would just think your way
12:06 through it and you should hopefully be
12:10 okay so let's look at a few other
12:12 smaller things right here this next
12:14 problem is actually a little bit
12:15 different than what we've been doing
12:19 here oh so we have a test during the
12:21 first trimester of pregnancy to decide
12:23 if there's some sort of abnormality
12:27 going on approximately five percent of
12:31 these tests will be false positive
12:33 so what we're doing here is we're
12:36 looking at 100 women and we're saying
12:38 out of those 100 women with no
12:39 pregnancies and what is the probability
12:42 there will be at least one false
12:45 positive there's a 5% chance you have a
12:48 false positive okay the first thing you
12:49 need to do when you're doing some sort
12:51 of probability question like this is
12:53 acknowledge that the events are
13:00 independent so it's reasonable to say
13:09 that each woman's tests first
13:14 is independence okay so each woman's
13:16 test is independent which means if we
13:17 need to we can just multiply their
13:19 probabilities together and we're cool we
13:21 do not have to adjust one does not have
13:23 any sort of bearing or tell us anything
13:25 about change the odds for another
13:27 probability all right so that's the
13:30 first thing the other thing about this
13:31 problem here is they're asking kind of a
13:33 tricky question what they're saying is
13:35 what is the probability out of a hundred
13:38 women at least one of them will be a
13:41 false poundin these words like at least
13:44 or at Lowe's are tricky words that you
13:46 should like signal to you that you're
13:47 gonna use a little trick I'm about to
13:50 teach you at least one false positive is
13:53 crazy because what that would mean is I
13:55 would need to find the probability that
13:58 exactly one woman out of a hundred has a
14:00 false parlament then I would need to
14:02 find the probability that exactly two
14:04 women out of a hundred have a false
14:06 positive and that I defined the
14:08 probability for three and four four and
14:10 four five and four seventy eight and
14:11 four seventy nine and four eighty eight
14:14 489 490 one four ninety five and four
14:17 ninety nine and for one hundred I would
14:19 have to calculate literally a hundred
14:22 different probabilities to answer that
14:24 red question at least one false positive
14:28 could be all of those options now that
14:30 is awful and it is not something you
14:32 will ever be asked to do because what
14:33 we're gonna do to answer this question
14:36 is use our good friend the complements
14:38 that we learned about earlier in this
14:42 chapter instead of finding all of this
14:44 awful at stuff right here hundred
14:46 probabilities think about what you
14:50 haven't found in the rep what we haven't
14:54 found is the probability of no false
14:57 positives at all so what you can do with
15:00 a problem like this is find this guy way
15:03 easier than all that red stuff and then
15:05 afterwards just do a compliment to get
15:08 what you actually care about so that's
15:10 the basic set up right there when you
15:13 see words at least or at most you're
15:15 probably being set up to do a compliment
15:18 type of problem so I need to find the
15:21 probability that in a hundred women none
15:22 of them
15:24 none of them have a false-positive
15:27 that's what I'm being asked to do well
15:29 what's the probability for an individual
15:31 for one out of those 100 women that they
15:33 don't have a false positive
15:37 if 5% of pregnancies have a false
15:39 positive that means the probability they
15:42 exult have a false positive is going to
15:46 be 95 percent that is woman number one
15:51 okay woman number two is also gonna need
15:55 to not have a false positive so that's a
15:57 95 percent curve woman number three is
15:59 gonna have a 95 percent chance no false
16:03 positive etc etc etc for each of those
16:05 different women so an easier way of
16:09 writing this one out is to say 0.95 to
16:12 the 100th power I mean the first woman
16:14 and the second and the third in the
16:16 fourth in the fifth and the hundredth to
16:19 not have a false positive so if I
16:22 multiply that out point nine five to the
16:24 100th power it's kind of small
16:33 this ends up being 0.005 9 so I've been
16:35 purposely trying to color code this
16:36 right here because this can be a little
16:39 tricky too retro head around the green
16:41 guy the probability of everybody being
16:43 cool and not getting a false positive is
16:45 less than one per sentence but I didn't
16:48 want the green guy I wanted the red part
16:50 or pink part right here which is at
16:52 least one of them being a false positive
16:54 so what I'm gonna do to finish this one
16:56 off is do the complement I'm gonna take
17:00 1 minus the probability of nobody having
17:04 a false positive and if you do that it's
17:10 gonna be 0.99 4 so that's the
17:11 probability that we're actually asking
17:13 us for the problem which is all these
17:17 other hundred options so we talked about
17:19 false positives and false negatives in
17:20 the last example but think again what
17:23 this is saying even if for one person
17:25 there's only a 5% chance of a false
17:27 positive when you test a lot of people
17:29 sooner or later like it's almost certain
17:32 we look a 99.4% chance somebody will
17:33 have a false
17:36 so most of the time when you go through
17:37 some sort of testing for whatever
17:39 they're gonna test you twice just to
17:41 make sure that you're not one of those
17:42 like false positive or false negative
17:44 situations okay
17:46 so problems like this where you have to
17:47 do at least or at most and do a
17:49 compliment are common especially on
17:51 multiple choice questions so that's
17:53 something you should expect to see me on
17:58 your chapter tests are it's our next
18:00 little problem right here is more of a
18:02 trick question kind of spoiling it for
18:04 you in advance we're gonna say that for
18:08 today there is a 70% chance of rain in
18:10 Maplewood and then there's also a 70%
18:12 chance of rain in Richmond Heights
18:14 what's the proof for tomorrow I guess
18:16 what's the probability it will rain in
18:19 both towns tomorrow so the word Bo's
18:22 kind of applies and so if you see and
18:24 you would be like oh you multiply the
18:28 probability is 0.7 times 0.7 but if you
18:30 multiply them straight up that is
18:33 assuming that one event is independent
18:36 of the other that's not true if it's
18:38 raining in Maplewood it is almost
18:41 certainly raining in Richmond Heights as
18:44 well so basically look if I was gonna do
18:46 this if the probability from Maplewood
18:51 is 0.7 of rain the probability of
18:55 Richmond Heights given that it's raining
18:57 in Maplewood is probably something like
18:59 0.99 like it's actually really really
19:02 high because um if it's raining in
19:04 Maplewood the town next door is probably
19:06 gonna be running as well there's always
19:08 that situation where it's not the case
19:10 but it's pretty unlikely so you cannot
19:13 just multiply these together straight up
19:15 because they're not independent
19:30 because the events aren't independent so
19:32 be careful before you start multiplying
19:34 stuff that it actually is independence
19:35 so otherwise it's like a trick question
19:38 this next slide I'm not gonna go over
19:39 with you guys here but it's kind of an
19:41 interesting article right here mine
19:43 Mason's covering it a little bit but
19:45 basically what it talks about is the
19:46 idea that for
19:49 name and last name are not independent
19:51 of one another so your first name and
19:53 your last name actually have connections
19:55 here you see it a lot in names of like
19:57 certain ethnic origins where like
19:59 different first names and last names go
20:00 together more often
20:02 even though they may not be the most
20:05 common name in America separately when
20:06 you put them together they are so
20:07 basically it's an article about how your
20:09 first name and your last name aren't
20:12 necessarily independence and then the
20:14 final thing I'm going to do right here
20:16 this is unrelated as well to our main
20:19 lesson right here but simulation style
20:20 problems are one of the toughest parts
20:23 of AP stats for kids and I wanted to do
20:24 an example of this here because there
20:26 will be something like this on your next
20:29 chapter tests so we have a girl who was
20:32 a sixty percent free-throw shooter
20:34 shooter last season she wants to get
20:36 better and she trains and practices and
20:38 all that and then in the first three
20:41 games of the season she had 12 of her 16
20:47 shots which is a 75% rate so we want to
20:49 see based on that evidence whether
20:52 that's convincing enough that oh yeah
20:54 she actually has gotten better or cuz
20:57 she still be a 60% shooter who's just
20:59 got lucky in her first couple of games
21:01 and had a sample where she did better
21:05 than usual okay that's the basic premise
21:07 of this problem is the evidence from her
21:09 first three games enough that we can be
21:12 confident she's no longer a 60% shooter
21:15 so what we would do in a situation like
21:17 this is design a simulation and we would
21:19 try to figure out basically we can do
21:20 something with a calculator like we did
21:22 earlier in this chapter where we would
21:25 make it so she's still a 60% shooter so
21:26 maybe the numbers 1 through 16 would be
21:29 she makes it the numbers 61 200 or 99
21:32 whatever it would be nine hundred would
21:34 be that she doesn't make it and we would
21:36 try to figure out if we make 16 shots
21:38 happen we would look to see in what
21:41 percent of the time she got 75% more
21:45 could 75% happen in 16 shots given that
21:48 she's still a 60% shooter so they went
21:50 ahead and gave us a dot plot right here
21:51 and you can expect to see a dot plot
21:53 like this somewhere on your chapter 5
21:56 test what you do with a picture like
21:59 this this picture assumes she's still a 60
21:59 60
22:01 and shooter and we have to figure out
22:05 how rare what actually happened in real
22:09 life is so assuming that she is still a
22:10 60% shooter
22:14 how rare is it to get 12 or more of her
22:16 shots correct or to make 12 or more of
22:19 her shots so when you do a problem like
22:21 this you find what you got in the
22:24 problem but you always count more
22:26 extreme than that as well because if
22:28 you're gonna be impressed by a 12
22:30 naturally you'd be even more impressed
22:33 by a 13 or a 14 of your shots and then
22:35 what we're gonna do is count up those
22:42 dots I count 14 right here I might be
22:43 off by like 1 but I'm pretty sure that's
22:50 14 months so in a hundred simulations 14
22:54 of them had a 12 or better given that
22:56 she's still a 60% shooter computer
22:58 assumes she's still 60% and they looked
22:59 at what happens
23:08 so the probability so assuming Ashley is
23:17 a 60% shooter there is a 14 out of a
23:19 hundred which is like 14 percent
23:27 obviously chance my face here so there
23:35 is a 14 percent chance that she could
23:39 make that's a really bad could could
23:49 make 12 or more out 16 shots so you have
23:51 to think about what that means if
23:54 there's a 14 percent chance that she's
23:56 still a 60% shooter that's basically
23:58 what we found right there is that high
24:01 enough or low enough that we should be
24:03 like okay confident she's actually
24:03 better now
24:06 14 percent is usually considered to be
24:09 too high for us to be certain that we
24:12 were wrong in our initial assumption so
24:13 there's a 14 percent chain
24:17 she's still a 60% sugar basically is how
24:18 you can think about that right there and
24:20 14% is probably too high for us to be
24:22 like certain so what should we do
24:24 well we should collect the little more
24:26 Deena if she kept not shooting that we
24:28 had bigger samples well that tells us
24:30 okay yeah she's probably not a 60
24:31 percent shooter anymore
24:33 the most common cutoff for problems like
24:35 this is five percents but you can use
24:38 10% or 1% and that can vary a little bit
24:40 from problem to problem but basically
24:42 what you do is you figure out what
24:45 happened in real life plus anything more
24:47 extremely mats and you figure out if
24:49 that probability is low and if that
24:52 probability like let's say same exact
24:55 situation but instead of hitting 12 out
24:57 of 14 shots sorry
25:00 12 out of 16 shots she actually hit 14
25:05 out of 16 shots well 14 out of 16 shots
25:08 that's 13 right here she never got 14
25:12 out of 16 so the probability of that
25:14 happening would be about zero out of a
25:17 hundred and if I saw that I would be
25:19 like whoa she's gotten better she's
25:21 definitely better there's no way that's
25:22 gonna happen or it's very unlikely to
25:25 happen if she's still a 60% shooter but
25:27 in our pink problem right here getting
25:31 12 out of 16 was like kinda rare kind of
25:33 surprising only a 14% chance of that
25:36 happening but that probability is too
25:42 high to be certain so that's how you use
25:44 one of these simulations to analyze a
25:46 probability and it's something we'll
25:48 practice in class as well because it is
25:50 something that kids do struggle with but
25:51 hopefully that makes at least a little
25:52 bit of sense and hopefully you
25:55 understand this concept as well as that
25:56 rest of that stuff we've talked about in
