0:03 so section 5.1 is probably not what you
0:05 were expecting when it comes to like
0:07 starting a unit on probability it wasn't
0:09 much in terms of calculations and like
0:11 traditional probability stuff it's more
0:13 just about conceptual designing a
0:15 simulation using long run to help you
0:18 out etc sardian section 5.2
0:21 and also in 5.3 we're getting to a lot
0:23 more of those traditional probability
0:25 calculations because it's really
0:27 important that you can handle the basics
0:29 when it comes to analyzing more
0:32 complicated statistics situations so
0:34 this is the beginning of that and we're
0:35 talking about some big probability rules
0:39 in some vocabulary as well first off
0:42 vocab terms that you probably heard from
0:44 me in algebra 2 the sample space is the
0:51 list of all possible outcomes outcomes
0:55 being something that can happen and the
0:57 variable we use for the sample space is
1:00 a capital S but we'll put a lot of times
1:02 like these little fancy I don't even
1:03 know what you would call that on the
1:06 edges of the s so you would say like the
1:09 sample space is and when you're making a
1:11 list you do squiggly brackets so if I
1:13 was going to roll a die the numbers 1
1:17 through 6 would be my only possibilities
1:20 if I was gonna flip a coin my sample
1:23 space would be heads and tails if I was
1:25 gonna pick somebody out of my class get
1:26 the list of your name this would be my
1:28 sample space the sample space is just
1:31 everything that can happen okay a
1:33 probability model is gonna be a very
1:37 useful thing to us here it contains two
1:42 things whenever you have a probability
1:45 model you're going to have a list of all
1:51 outcomes so basically you can have your
1:53 sample space everything that can happen
1:54 and then you're gonna have the
1:57 probability for each outcome the
2:00 probability for each thing that occurs
2:02 so if I was going to design a very basic
2:05 probability distribution um a lot of
2:06 times you'll see these as tables right
2:10 here I could have a Cohen example where
2:13 I have heads and I have tails
2:15 and then I have probabilities 1/2
2:17 probability as 1/2 so I'd have liked
2:21 probability and I'd have liked the thing
2:24 the events speaking of the last vocab
2:26 word on this page is called in events
2:31 and an event is the a subsets of the
2:34 sample space what do I mean by subset of
2:36 the sample space it's inside the sample
2:38 space but it's a smaller part of it so
2:41 it's like just a collection of outcomes
2:48 that are all in s so collection of
2:58 outcomes in s so on my ruling and I
2:59 example I could look at the events
3:02 getting a 4 or higher or I could look at
3:04 getting an odd number or I could look at
3:07 I don't know a prime number or something
3:09 like that if I was gonna pick somebody
3:10 in class at random I could look at the
3:12 probability you have class the event of
3:14 you wearing glasses are you being at
3:17 least 5 5 or something like that so an
3:19 event is just basically whatever you
3:21 want it to be you just define a
3:22 collection of things that you're
3:24 interested in finding the probability of
3:26 usually when we establish an events it's
3:28 because we want to know its probability
3:31 so there's a little bit of vocab for you
3:33 all and then what we're gonna do right
3:35 here is look at a basic example using
3:39 sample space to arrive at a probability
3:42 or probability distribution so we're
3:44 gonna picture flipping a fair coin three
3:46 times and we want to describe the
3:48 probability model which means all the
3:51 possibilities and the probability for
3:54 each so in a problem out like this
3:56 that's not that big in scale the best
3:58 thing to do is just write out all the
4:00 possibilities so I'm gonna picture if
4:03 I'm gonna flip three coins it could be
4:07 where they go heads heads heads just
4:09 like that and I'm just gonna write out
4:11 all the possibilities here trying to be
4:13 somewhat systematic with it I can go at
4:20 heads heads tails heads tails heads and
4:23 then I need tails heads heads so that's
4:26 all the ones with two and then I could have
4:26 have
4:34 adds eight of HTT I could have th T I
4:41 could have TT h TT t so this is a list
4:45 of all of the possibilities incidentally
4:47 thinking about like algebra 2 when we
4:49 look at the number of possibilities
4:51 total we had two choices for the first
4:54 one two for the second one and two for
4:56 the third one there should be eight
4:57 possible outcomes all together so I have
4:59 all the possibilities here this is what
5:01 can happen when you flip three points
5:04 they say to describe the probability
5:06 model for this chance process and use it
5:07 to find a probability of getting at
5:11 least one head in three flips so what we
5:13 care about in this problem is the coin
5:15 landing on heads we want to get at least
5:17 one of those so if I'm gonna make a
5:19 probability model those are generally
5:23 written as tables and we're gonna care
5:25 about two things when we make our table
5:28 we care about how many coins land on
5:32 heads and we care about the probability
5:34 of that happening so the least number of
5:36 coins landing on heads we can get a zero
5:41 or we could have one on heads or we
5:43 could have two or we could have three
5:45 and that's all the possibilities getting
5:49 zero coins landing on heads is only
5:52 possible one time one right there and
5:54 for probability you just take it out of
5:55 the total there are eight possibilities
5:58 that are all equally likely so the
6:00 probability of getting one
6:02 sorry no coins landing on heads is 1 out
6:05 of 8 next up I'll go for one coin
6:06 landing on heads that's going to be
6:10 these three options right here so this
6:13 is a three out of eight to coins landing
6:16 on heads is these ones right here that's
6:19 a three out of eights and then the last
6:21 one all three landing on heads is a one
6:25 out of eight so they asked us find the
6:27 probability of getting at least one head
6:31 in three flips at least one head could
6:35 be this or it could be this or it could
6:38 be this basically it's just not that one
6:40 and if I look at my
6:42 all those circled no not all the circle
6:46 guys all these circle guys have at least
6:48 one heads so the answer to the problem
6:50 we care about is seven out of the eight
6:53 possibilities here okay so if you make a
6:55 list of everything that can happen
6:57 it makes probability questions not that
7:03 bad in wise so let's go ahead and make a
7:04 little list right here of just basic
7:07 probability rules some of which you guys
7:10 have already heard our first rule is
7:12 something we've already talked about in
7:17 the last set of slides for any event a
7:20 usually with events and again an event
7:21 is just whatever you want it to be we
7:23 use a capital letter to represent it's
7:26 the probability of an event is always
7:31 between 0 and 1 and we'll use this
7:33 notation P of n you write down your
7:35 event right here so the probability of a
7:37 is between 0 & 1 in other words you
7:39 can't have a probability of 2 or of
7:42 negative 7 or something like that
7:54 probability of everything in your sample
7:57 space is the probability of s occurring
8:00 is 1 so basically if you're gonna have a
8:01 sample space it has to be the
8:03 probability of 1 it's everything that is
8:08 possible in the problem so third rule if
8:20 all outcomes are equally likely then
8:21 what you do is what we did on the last
8:23 example to find the probability of an
8:28 events you take the number of outcomes
8:34 in your events over the number of
8:40 outcomes in your sample space I don't
8:41 know maybe that looks complicated it's
8:43 really simple though all it's saying is
8:45 you take what you're looking for out of
8:48 the total to find a probability the last
8:51 example was like hey out of the 8 times
8:52 how many had at least one
8:55 on heads will it would be seven out of
8:57 eight that's all that means
9:00 two more you have what's called the
9:05 compliments of an events and the
9:09 complement of a is a and you put like a
9:11 little it looks like an exponent of C um
9:15 the complement means everything knots in
9:17 the original and maybe that's worth
9:18 writing down if you didn't already know
9:21 that's so compliments when they talk
9:31 about that is everything knots in a so
9:34 let's say my event a is ringing I want
9:36 to find the probability it's gonna rain
9:38 tomorrow the compliment would be
9:41 everything knots in it so everything
9:44 that's not raining snowing foggy sunny
9:46 tornado whatever else as long as it's
9:48 not raining that would be the complement
9:50 so instead of listing out all that stuff
9:53 it's easier just to say not a sometimes
9:56 now let's say tomorrow there's a 20%
9:58 chance it's gonna rain well the
10:02 probability it won't rain so the
10:05 probability of the complement if there's
10:08 a 20% chance it does rain there's an 80%
10:10 chance it doesn't you just do 1 minus
10:14 0.2 so you take 1 minus your original
10:17 probability and that gets you the
10:18 probability of the complement if there's
10:21 a 30% chance something happens there's a
10:23 70% chance it doesn't that's all that's
10:26 really saying and then finally my last
10:48 what is that mutually exclusive it means
11:02 so let's say I was gonna roll a die
11:05 rolling a to a rolling a 3 those are
11:08 mutually exclusive like I can't say 2 &
11:10 3 at the same time if it's something
11:11 that cannot occur together it's called
11:14 mutually exclusive and if you want to
11:18 find the probability of A or B so the
11:21 first thing or the second one this
11:22 should look a little familiar because we
11:24 did talk about this in algebra 2 all
11:26 you're gonna do when they're mutually
11:27 exclusive is you're just gonna add the
11:30 two separate probabilities together kind
11:32 of did that with you on the last example
11:34 where I added up the 3 over 8 the 3 over
11:37 8 the 1 over its if there's overlap you
11:38 have to deal with that but that'll be
11:42 the next video so those are our rules
11:43 for probability and let's just look at
11:46 an example to close this video outs so
11:47 at the time I'm recording this video
11:51 this is my full AP data sets of scores
11:53 that kids get on the AP test and the
11:55 probability associated with those
11:57 outcomes out of all the kids who have
11:59 taken the class and you can see the
12:00 breakdown here by score which
12:02 incidentally should give you guys like
12:05 make you feel good or motivated that hey
12:07 if I put in the work and I study I will
12:09 do well on this test and I will get
12:12 credit for the class because we'll cover
12:14 everything you so stick with it even if
12:16 it feels like shaky at times you'll do
12:19 well if you keep putting in the work it
12:21 says to show why this is a legitimate
12:24 probability model I didn't explicitly
12:27 say this earlier but you need to do two
12:29 things when you show it's a legitimate
12:31 probability model you need to show that
12:37 all probabilities are valid by valid I
12:40 mean they're between 0 & 1 so if you
12:43 look at these yes yes yes yes yes
12:45 they're all between 0 & 1
12:49 good you also need to show that all the
12:51 probabilities add up to 1 so these don't
12:54 add up to 1 that means that something is
12:55 missing and it's not a full probability
12:57 model because it doesn't have everything
13:00 in the sample space so they all
13:06 probabilities and to one check so what I
13:08 would have to do to verify that is I
13:10 would just add these up
13:12 and I would make sure that they actually
13:14 do equal one to make sure I didn't leave
13:16 anything out and they do so that is a
13:19 good call okay so if I ask you to verify
13:20 that something's a probability model
13:22 make sure there's no like weird negative
13:23 probabilities or probabilities of like
13:25 seven and then also make sure the
13:28 problem is at what they're supposed to
13:30 then once you've done that they just ask
13:32 you for a few basic things here what's
13:34 the probability that if you pick a kid
13:35 at random who's taking my EP class
13:37 they've scored a three or better
13:40 well three or better these are mutually
13:43 exclusive you can't get a three and also
13:43 get a 4
13:45 your score can't be both at the same
13:48 time so all I need to do is add up all
13:50 the ones I care about add up the threes
13:54 out of this add up this and that's going
13:56 to get you your total probability so if
13:58 I do those and when you write these down
14:00 guys show your work always show your
14:03 work no matter how basic it is it's
14:13 so it looks like the probability that if
14:15 you're you pick a random kid that they
14:16 pass the AP test it's going to be about
14:18 eighty-five percent eighty five point
14:21 five percents okay so that is just what
14:23 you do you add those up and then for
14:25 this next one here it says the
14:26 probability if you pick somebody they
14:29 did not get a four so it could be like
14:32 anything else but not a four there are
14:34 two ways of doing this one is just add
14:37 up this this this this and that will be
14:39 totally cool that's everything that's
14:42 not a four you can also use that
14:43 complement rule which sometimes is
14:47 easier and just do one - the one that
14:49 you do care about it so that you don't
14:51 want or whatever if you do that you're
14:55 gonna get a probability of 0.8 1/8 so
14:57 that is how you calculate basic
