Transcript for:
Understanding Conditional Probability and Independence

hi so today we're going to start we're going to do in class activity 7c and um the way the information is presented it's kind of encouraging you to use formulas and you can do that and it'll work but i would rather you use concepts and i will always give you the information in a table or i'll ask you to create a table and so i'm going to approach it with how would i answer these questions if i have a table in front of me rather than some theoretical values floating in space so um so there we go and otherwise the formulas get a little sticky i'm not gonna lie all right so let's get started and i'd like you to go to the back of um of this worksheet so go to the very back because there really isn't enough room here i want you to have a little summary of like the big picture so the big big picture about probabilities i'm going to do purple for probability the probability of an event happening let's i'm going to make one up well in general the probability of an event happening a certain boy event happening equals number of those events over total number of all outcomes so um if you're looking at a two-way table um then this so when we were looking at the ducks it was gonna be you know being a winner was six stuck six ducks out of twenty this is going to be a subtotal or if we're doing standard probability subtotal over [Music] and this one right here is going to be the grand total and that is if we are dealing with basic probability so that's the most basic well today we're going to be dealing with um not basic probability so let's look at an example i live by examples example the probability the probability of being left-handed i always pick things that are meaningful to me so you should probably get things that are meaningful to you but this is my example is going to be equal to it's a fraction and if we're talking about a sample it's going to be a grand total of a sample if we're talking about um the whole population it could be all the people in the world total people in your study okay talk add up all those people and that's your denominator and your numerator is going to be all left-handed people in the study so probability is really simple total number of successes over total total if it's a basic probability but now we're going to be talking about a different situation and if you look so let's see so this is what we've done so far and this is a basic example so so the top right here would be the subtotal of lefties and the bottom is still going to be the grand total so i haven't done anything new this is just this just a summary that's an example right there but now we're changing things up just a little bit we're dealing with this thing called conditional probability and that's where you throw a condition on it so i'm not going to be saying what's the probability of being left-handed i'm going to be saying what's the probability of being left-handed given or that you already know that you have a female and so all of a sudden you're not looking at the grand total you're looking at a subtotal so your eyeballs go to that subtotal and that's your whole world okay so let's get so that's where we're headed um we're so we're doing this thing called conditional probabilities and we are also going to be introducing a couple definitions but before we do that let's do a simple simple example so go ahead and answer this um and unlike last times i think there are right answers here so go ahead and try okay answer the following questions if it is snowing what is the probability someone will ride a bike well i don't know the exact probability and i'm sure it would vary from i mean if it's snowing and you live in siberia probably snows almost every day and they probably know how to ride their bikes in the snow but for californians forget about it right i think that i'm going to imagine i'm in california and i'm just going to say p of bike riding and watch this there's a line here is going to be pretty low i'm guessing is low and this the way this translates is that that green little line right there that i drew this green line means given so i'm telling you um i'm telling you a condition so inside it's saying happy bike riding probability of bike given snow on the ground so it's a it's kind of a um it's a shorthand that green line right there is a shorthand and the green line that green line um right here just means given and so it's setting a condition okay and i'm just going to say so all of that the answer the actual answer that we want is oh it's probably pretty low we don't have a nice solid number on it okay so next simple question if you know someone holds one political belief how likely are they to hold a conflicting political belief so again i live for examples example um if you know that someone is an ardent roe v wade supporter if a person supports a woman's right to choose i think unlikely that they were happy with the last two supreme court nominees or not nominees but appointees to supreme court um appointments trump got to appoint three supreme court justices so i'm not going back to the third one the last two brett kavanaugh and um amy comet comey barrett i think is how you pronounce her name um they have a track record of not supporting um abortion and even though they said that they were going to uphold a precedent they didn't and so i'm just willing to bet that it's unlikely so it's very unlikely so if someone so i'm just gonna the answer over here is unlikely really unlikely that competing that things that are very republican will be supported by someone who's a democrat for example like gun control if i know that somebody is supporting gun control i'm going to guess they're probably republican it's not 100 okay so what we're thinking about the back of our minds is that probability changes if we have certain information and that information can be called conditions if the conditions change so um okay um you'll understand something called mutually exclusive events which we did talk about a little bit mutually exclusive events are events that cannot happen at the same time and that's an entirely different concept than independent events they're different but related but maybe not related in the same way two events are independent when one event has no impact on the probability of the other event occurring so i'm going to write out that that actually means that knowing a happened does not help you predict be happening we say a and b are independent of each other so that's the broad conceptual idea of independence okay and for mutually exclusive if a happens then b cannot happen another way no overlap no overlap or common shared outcomes so again i live by example so example i'm going to give i'm going to talk about independence first i'm going to test you on it a little bit if i tell you that i have a friend who um i've i have a friend who's african-american and uh what do you guess is how that person voted in the last election knowing that they're african-american does it help you guesstimate whether or not they voted republican or democrat uh it helps me a lot because i know that 89 of all black voters voted for hillary two elections ago and voted for biden in the most recent election so this is a perfect example of race at least for african-american race is not independent of voting status so that's kind of a counter example race um actually no i'm going to say being black or african-american and republican republican are not independent events tell me someone's a republican i'm going to guess they're not black tell me somebody is black i'm going to guess they're not republican so one piece of information helps me predict the other in both of the last two elections approximately 89 percent of black voters voted for the democrat um and it's very interesting that there's a lot of republicans now targeting states where there's more african-americans and those are the same states where there's a lot of voter changes in how votes are made so it's a little bit uh disingenuous there in that they do they really want everybody involved i don't think so so okay not all republicans just those politics certain politicians all right so um are black being black and being republican mutually exclusive think about it being black and republican are mutually exclusive is that true that is not true i'm going to put a big not in there are not mutually exclusive because i just said 89 support democrats well that means that 11 percent don't so there was an overlap 11 of people who identify as black also identify as republican or at least they voted republican that's hard to believe isn't it but it says overlap so you can't say that they're mutually exclusive so the two don't go totally hand in hand and the relationship between being mutually exclusive the concept of mutually exclusive is really simple no overlap the concept of independent is not as simple but you can get i don't in if i spelled that wrong indeed can you i missed a whole dependent okay there we go so i'm kind of throwing all this at you um but we're going to go through some nice examples and so those are the broad definitions of mutually exclusive and independent and now let's test them out okay so what we're going to be doing in this class is we're going to be calculating conditional probabilities and oops highlighter additional probabilities and we're going to be doing it all in the context of tables we're going to be um testing out if something is independent and i'm going to approach it uh conceptually but then i'm also going to give you a mathematical test for it um same thing we're going to be doing the same thing for mutually exclusive so it's a different slightly different color because slightly different concept um and then we're going to be calculating probabilities of independent events and probabilities of mutually exclusive and we're going to be then at the very end the punch line is going to be what is the relationship between mutually exclusive and independent so let's go all right the general social survey so a student did come up to me and say uh what's what are some really good resources if i'm doing a research paper what are some good resources for getting really good data well um the general social survey is an excellent resource it's operated out of nork which um work means something and it's really funny that i don't know what it means because my husband works for this institution and it's connected with the university of chicago so i don't remember it's it has an outdated acronym and i think i'm saying that wrong but it's outdated but what it is is it generates very um non-bi bipartisan no wait a minute non-biased uh data uh it's a great institution and the data is free and um it's uh it's great so and and so one of the things that does this is opinions i think the o stands for opinions but it does a lot of other things so it doesn't like to call itself that anymore all right so the gss asked a survey asked survey respondents whether united states um spending on alternative energy so i'm going to make that green so solar wind is it too little about right or too much so these are possible categories within that variable how do you feel about alternate energy do you think the government spends too little about the right amount or too much money on alternate non you know basically not oil then the next question that we ask people is uh what about child care um ask questions about child care and same question too little about right or too much so they're completely separate questions but do you think there's some relationship between the people who answer one way for one question versus another all right so here's the data it's a two-way table which i love so um the energy is the energy rows or columns develop alternate energy too little about right too much and these are the subtotals and how many people were asked so if you want to know if you want to ask about this you're going to be looking at these columns i'm sorry these rows rows rows so like movie theater rows so these are subtotals and this right here is the grand total so if i just flash to the subtotals i am completely ignoring the relate whether or not what their opinions are about child care now similarly so i'll go back to this i could ask the same group of people this question and we've got these answers right here how many people say too little for child care how many people say about right and how many people say about too much so these are different but possibly similar subtotals so these are subtotals and if you add them all up you get that same grand total so that's why it's called a two-way table because there's two different pieces depending on how you slice it up um so um so we love our two-way tables um and so if you remember the cell values give you intersection and if you unite all the row the row and a column value you get the union and interception is and and union is or so that was from last time use the table to estimate the following probabilities so the probability that a person thinks the united states spends too much on child care okay is this a simple question from the last section yes we're not even we did you notice there is no mention of energy and so we don't care about the sub the green subtotals we're not interested in this at all the only thing we're interested in is child care so i want to use the notation probability of what too much on child care so all you need to do is oh well this is just a really simple simple simple question and down below is going to be all the people in the study and up above is going to be those who think it's too much child care so i'm only looking so i've got my subtotal my grand total is 200 and 219 that's the grand total and how many people fit this category well it's going to be these people so we don't care about how they feel about alternate energy so we skip all that and we just go right to the subtotal so it's going to be 279. no it's not it's gonna be i don't know what i'm saying two set one thousand two hundred and seventy nine so this is this um this is the subtotal oh too much oh i'm getting ahead of myself too much my bad so we're gonna fix that too much i'm getting too excited too much it's going to be 120. so 120. i was just making sure you guys are paying attention um subtotal who think too much okay so we've got 120 divided by 2000 so what you do is you get out your beautiful calculator and work it out and what i got as a decimal and please please please only answer in terms of decimals and not percentages because if you do percentages and then you round chances are you're rounding incorrectly because the decimal's shifted so i'm going to say the default is three places past the decimal so that's going to be one two three and that four bounces up to a five so it's zero point zero five five and now i'm gonna do or five point five percent so very few people in this study believe that we're spending too much on child care i'm going to box my answer and that was a review all right so now we're getting into something new the probability that a person thinks the united states spends about the right amount on both child care and developing alternate energy okay so i'm going to clean this off and what bounces out at me is we are looking at two characteristics and that they fit in this category right amount on child care and right amount on energy okay so let's see who are all the people who believe that we're spending the right amount on child care it's all of these people and that is this that is the subtotal of all the people who think it's just right um so that's that and also the right amount on developing alternate energy so i'm going to come over here here's alternate energy these are the people who are the right amount so all of these people so all together it's 756 people but i'm interested in who is in the green row and also the pink column so when we're doing and it's the cell value so this is the magical people who are both pink and green and this the way i've broken it down so let's do the proper notation p and i'm just going to use the colors i know i shouldn't but um right amount on alternate energy and energy and that's cell value which i hope you know by now right on childcare so they're both really happy and it's the basic definition of its total number over grand total so it's going to be 297 out of 2119 2000 119 yep and so um if i work so it's pretty exclusive and it's going to end up being when you round and i don't remember if it's right on or not but what i get once i'm done rounding is this so that's my decimal answer which i'm going to ask for probably in the but to help i'm like oh it's only 13.5 okay so not many people are completely satisfied with both um so moving on number three uh the pro and this is the probability that a person thinks the united states spends about the right amount on developing alternate energy and too much on child care so you go ahead and work that one out and see how you do okay so p and it bounces out at me that we're still talking about an and statement and we've got the right amount on alternate energy and what we have over here is too much on child care so two i'm gonna put a very different color too much on child care okay so i'm not gonna be looking at the pink column anymore but i am still looking at the green one right amount on energy so not interested in the subtotal because they have to fit that row and they have to fit this column so uh while they're there's a good chunk 120 who think we're spending too much on child care there's only 41 people who are in the green and the ground row column so that's going to end up being um 41 so it's kind of not surprising that if you're having children i remember when i started having kids um all of a sudden things that i never really worried about like the environment i mean not that i didn't worry about the environment but the reality that um in 70 years the ozone layer blah blah blah well i'm like well that's too that's sad but but then when i know it's going to be my kids and my grandkids are going to be seeing that it kind of brought it home for me so usually people who are having who want to see the environment and check they also want money right now for child care so i would not say those are independent events um so it ends up being 41 out of if you run it through your calculator and i did round i got 0.0186 um so i did actually go zero one eight six so you would on the calc um in canvas if i asked you to you would say zero one nine and that's less than two percent of the population holds those two characteristics it's very unusual all right um in the previous question asked about people in general so notice there's no i'm not i'm just saying what we've got all these people right here why were they responding in general that's what i'm asking and who has to of all these people who has both of these characteristics but now and i want to shrink this down so i can look at the table at the same time the previous question um asked about people in general so it asked about people in general all people so the denominator was the total it was the grand total now let's focus on people who think the united states spends too little on child care so we're going to now not look at the whole table we're going to look at people who believe we spend too little on child care so i'm going to come up here and clean up my clean up everything so you know these are the subtotals um we know these are the subtotals and we got a grand total okay so too little on child care if i only want to consider these people then believe it or not this is my whole world that yellow column is everybody i no longer it's it's like a mini study so i first want to focus on people who who who have this characteristic um what is the new denominator so i'm not looking so i'm going to i'll be kind of dramatic here i don't care about these people so this is all i care about so what was a subtotal becomes the new kind of mini grand total so what's the new denominator 1279. that's my whole world so here's my eyeball looking just at them um what is the probability that the person believes the united states spends too little on developing energy given that they think the united states spends too little on child care so so we're only looking at that row and now it's a new question um spend too little on energy so i'll take a slightly different bone okay so of that little world everything else is x'ed out too little is going to be this group right there so while 783 might be kind of small for the grand total of 2119 how does it compare to the new total so what i'm interested in so i'll do my fraction first and we want the probability so number of successes on top over total number success is what we're interested in so we're interested in um two uh that's a little hard to see 783 out of our subtotal and so i want to make sure so that's going to end up being if we do that we work that out that's 0.61 two two um in canvas i'll probably ask you go three places past um so that's what i'm gonna want you know data enter but in your brain over half well over half 61.2 percent of people who already believe we're spending too much on child care also think we're spending too little on alternate energy so that those two characteristics kind of track together really well so i want to make sure that your notation is right so it's the probability to little let me make this i'm going to run out of room here i'm going to write it a little bit sideways the probability of two when i'm interested in too little on child care given is that what you asked no i got it wrong um what is the probability that a person spends too little on developing energy given they spent too little on child care okay so the good news is i didn't screw that up i was screwing this up so it's actually this is a good lesson here i'm not going to backtrack the condition goes here initial condition and this green line means given and this is the final thing that you're interested in interest okay so um what is the probability that a person in the united states tends too little on alternate energy given that they are so the given is here given spend too little on child care so the two little on child care goes right in the back so two little on child care and sure enough there we go that's the condition so that you set your condition and and it's always going to be your denominator so subtotal of the condition and then what we said is uh spends too little on energy so order is really important and i'm not going to mess with you i'm always going to give the given second and it'll always go in the denominator so the given condition goes in the denominator oops okay so set your denominator first look for the given and then look for the cell value within that and you'll get it right every single time and i'm not going to mess with you so you focus on your row or your column and you go from it there okay so um same situation and problem number four i would like you to pause this and on the next midterm i'm going to have i'll probably have two problems associated with the table at least and one of the problems will be a basic probability and one of the problems will be a conditional probability where you're looking for a given and i'll use the word given so i want you to carefully read these two so if it is a general question you're going to be using the grand total as your denominator and if it's a given conditional question you're going to be using a subtotal as your denominator and then from there you'll know whether or not you should be using a subtotal or a cell value for your um for your numerator numerator being on top okay so pause this and answer these questions okay so welcome back um for problem number four let's see oh they both i'm looking at this and they both have givens in them so both of these are not i can tell right from the get-go that we're not going to be using the grand total because in both of these you're restricting your gaze to either a row or a column and then that's your whole question okay so the first one the probability that a random and selected person thinks that the u.s spends too much spends too much on child care given that they spend about the right amount on developing alternate energy so even though it comes second i'm going to focus on the condition first i'm going to focus on always focus this is the condition the thing that follows a given so um spend the right amount on alternate energy so i'm going here's alternate energy and right amount is about right here so um so i am only considering this row of information and i don't care about anything else and so within that row how many people so maybe i'll highlight it and i'll make it mustard color so here's my whole world these are i don't this is my whole study and so i'm going to get started because um i liked it so p of my given and the given is um right amount on [Music] energy so true or false can i treat that green line kind of as if it's a fraction i can and then what goes down here is what followed it so it's going to be this okay and so um what was i interested in i'm interested in um too much on child care so too much on child care is it too much on child care is this whole column but i'm restricting my gaze to just that must so it's just these 41 people fit both those so it's going to be 41. and um when i worked that and that's the hard work then you just go ahead figure out the decimal 0.0542 which is about equal to 0.05 or 5.4 that's the answer you're going to put in canvas so too much on child care and so that that shorthand notation with the slash line for given is pretty helpful because it actually reads beautifully this is uh this is going to be the top of your fraction and this is going to be the bottom and the bottom is always a subtotal and the top is always a cell value if you're dealing with given okay so if you've got that wrong pause and work out b again now let's do b together so for b i've got probability of something given something else and i'm going to focus on the given condition first they think the united states spends too much on child care so i'm going to come over here and too much on child care this is child care so too much on child care is this whole row so that's my whole column that's my whole world um and i'll just too much on child care so i don't care about the other people i only care about these people so it's almost as if i've blacked everybody else out that's my whole world um and so before i get too much in the thick of it i know that 120 is all the people that i care about so probability total number of at an event over the total total now we're looking at actually a subtotal and now i'm going to backtrack the probability that a randomly selected person thinks the united states spends the right amount right amount on developing alternate energy so within that pink column right amount is going to just be okay it's the same 41 but it's a different denominator so right amount on energy and so it's going to be 41 people but that 41 now is out of 120 so um you get a whole different fraction and i got 0.3417 which rounds to 0.342 and that's 31.2 percent that just helps me with my brain um and so that's how you do conditional probability and i'm not going to try to trick you i'm going to give you the condition and then that's going to be your row or column focus on that and then just apply what you know and that's the whole enchilada but for those of you who want to go on um i'm going to be introducing some formulas but you do not need to do the formulas if you're comfortable with what i've just gone over now um i may and i definitely will on the next on the next homeworks i'm going to ask you to put to actually read a situation and fill in the table so actually i mean in the real world you'd make your own table so um i'd like you to pause this and try and carefully read this and try to decide what goes where and tip i would look for the grand total first that usually helps me so tip for these kinds of problems uh look read the whole thing through but then look for grand total then subtotals and cell values and remember cell value is going to be and statements okay and that so it's helpful to know that when you're looking so go ahead all right let me do let me just read it through and see how i do and so oftentimes the first time i read it through i'm like whoop and then i find one little piece of information i know and then i go back and backtrack so every day jade recorded whether she drank bubble tea cafe or both okay or neither okay on five percent of the days and i'm not going to put the percents in i'm just gonna keep them as they are um on five percent of the day she drank bubble tea i'm going to make the tea green bubble tea and cafe so i'll make that brown anything to help me not screw up okay and sure enough if i look here they've already started me with that one that value um is something she's an intersection of here's all of the bubble tea these are all the results of the bubble tea with 20 people 20 times she did bubble tea and these are all the results of cafe which we don't know much about yet um on 35 of the days she drank cafe and did not drink bubble tea so that's an and statement again so uh did not drink bubble tea and drank cafe oops i'm going to do the color right and i know it's an and statement so right here that actually makes a lot of sense to me that this right here is all the time she did both the cafe and not bubble tea and they're saying it's 35 so i'm going to put a 35 there yay um and these are time consuming i would really slow down on those if i were you um okay so the next one on twenty percent of the days jade drank bubble tea okay so they're not really um giving me they're not mentioning anything else on twenty percent of the days she during two well oh it's right here did you know that okay so not too conditioned so that's a subtotal um on 80 of the days jade did not drink bubble tea did we really if we didn't really need to they didn't have to tell us that one because if she drank bubble tea twenty percent of the days then she did not drink bubble tea 80 of the days so this one was they didn't have to tell us that because we know we're talking about percentages so what's the grand total going to be it's going to be 100 but you could you can always add this a cell plus a cell within a row or column equals the total and in this case it's a grand total so i call this game fill in the damn boxes so if you have two cells within a row or column you can figure out the other one so i know for example um the subtotal here is 20 and this little piece is 5 so therefore this has to be 15 because the 5 plus the 5 plus the 20 15 has to be the 20. that's the that's how the subtotals work so similarly i can figure this one out right here um and i will get rid of the color um i can just go oh well i know my total is 80 minus 35 is going to give me what's inside there because this is the subtotal and this is a piece of the subtotal so that has to be so 50 45 and do use your calculator so it just kind of writes itself you just take along and it's like a little puzzle so now can i figure out this subtotal given that i know what she does when she drinks tea and when she doesn't drink tea so all together um this one has to be 40. and 40 is all the time she drinks cafe um here she drinks the tea with the coffee here she drinks she doesn't drink the tea with the coffee and similarly this one down here is going to be 45 plus 15 so a piece plus piece equals subtotal within that row so that's 45 55 60. and just a quick check does it add up yes that's how you know you did it right so um so it's just just it kind of unravels on itself and the game's called fill in the damn boxes so you have to have faith and i'm going to give you enough information to fill in the boxes okay so phew now that we've done that this problem is going to be no new information you just need to remember that if i'm asking you a straight up probability you're going to use the grand total as the denominator and if i'm asking you oh sorry it's my mother decline right now so that i get screened back so i have to be a good daughter okay sorry about that i can't not answer my mother's phone call so um so read through these and hunt your you know what's coming some of the problems will be straight up no givens and you just answer them the way you did in the previous 7b and 7a are you on 7c yes and then some of them are going to have the word given and that's when we're looking at the subtotals and you're ignoring the whole table and you set up with a new denominator so go ahead and pause this and do these and see how you do okay um b uh so a was set up the table b the probability so i'm going to try to get rid of these colors because i really like to mark them up the whole i hope i'm not erasing i'm gonna make my eraser really tiny so i can just erase what i want marking it up is really helpful for me you know a lot of people i went to stanford i got a full ride to stanford and it's not because i'm smart it's actually because i know my limitations and i can break things down and make them really simple for me so that i can handle them okay the probability of jade not drinking bubble tea on a randomly selected day is 80 so we knew that we figured it out what is the probability of jade um drinking bubble tea on a given day so drinking bubble tea well so drinking bubble tea we don't care about whether she's coffee drinking so it's going to be that 20 so actually that's a little disappointing i thought it was gonna be a little harder than that so 20 there's nothing and the percent is built into the table so you don't have to since this is a hundred it's all percent it's out of a hundred um on a randomly selected day what is the probability that jade drinks cafe whatever so i do not answer these questions until i have filled out the whole table like don't even look at these questions you know you need to have the whole table filled out so now um what's the probability on a random day that that just seems like a simple straightforward so here is uh drinks caffeine we're not interested in bubble tea so we just go straight to here so it's going to be 40 so but don't answer these questions till you fill in the damn boxes i don't get in trouble with the team for saying damage still so good okay what is the probability that jay of jade drinking bubble tea and i love that and cafe blah blah i don't know how to speak french on a randomly selected day so she's drinking bubble tea she's drinking bubble tea and i said bubble tea was green so she's drinking the tea and she is drinking coffee so i know we want both those characteristics so she's drinking coffee and she's drinking bubble tea so i know from experience oops that we want both those characteristics so it's an and so it's going to be intersection so there are five times that she fits both of those so it's going to be 5 okay that's super easy so far here's the new one what is the probability that she's drinking bubble tea give them that she already drank coffee we already know that she drank coffee okay so in that case probability of something given that she drank coffee at already or will we already know she already told us give already no coffee's a yes cafe she's fancy is yes so that means that this is our whole world so i know that the given that line this is going to be our denominator our whole world is the days that she drank coffee we that's our mini study and now i want to know did she what what are the chances that she also drank tea so it's going to be this is our whole world this is the time oh yeah this is the time that she so we're interested in that coffee row and so it's going to be 5 out of 40. for part e not 5 but 5 out of 40. and so that ends up being 0.125 [Music] or 12.5 percent that's what you're going to put in i just do the percent because it it's meaningful to me so um okay so that's the old stuff i mean now that's the new then this was the old and this is the new all right shrink this so now we're talking about conditional probability and i haven't done a single thing with math formulas i am using the old idea that um the probability of something happening is just simply the total number of of possibilities and then the events that fit that certain characteristic and um it can be us so uh the example i had was the probability of being left-handed is all the people who are left-handed over all all people but i'm now gonna i'm gonna use an example um what is the probability that you're left-handed given that you're female or born female at birth or a female so that was your assigned sex or i guess you don't have to say that earth so we know that the probability of being left-handed is about 13 so i you can google it 13 of all people are left-handed but we also know that and i wish i'd googled it ahead of time i'm just gonna say you know what i really want to know so i am going to i'm going gonna look it up because it's about me so i find it okay so there's lots of different answers out there so i'm gonna stick with 13 for men and the most common thing i saw was about eight percent for women um i mean 13 for all 8 for women um so i'm asking when this study is saying jet first look at all the women consider only women so you're looking at about 51 of the entire population and then from there then count up then within that group women count up lefties and it's less it's so it's eight percent so so um that's the conditional probability but it's a different example i have no new math formulas i just train myself to look at a subtotal rather than the grand total that's what i do because i'm not a formula driven person but if you want to have a formula here it is um so have at it it'll work and it kind of um it totally backs up that this is a cell value cell value isn't that and this is a subtotal i'm not giving up the table i like the tables use the formula um above to check part e so we did party without the formula so i gotta look at the table um so what they're saying is p this is not my favorite way to do this p of t given let's give him coffee what the formula says it's going to be p of t and coffee over p of coffee okay so the probability of tea and coffee is going to be i'll just go ahead and clean this up just a little bit [Music] the probability of tea and coffee is 5 out of 100 okay 5 out of 100 divided by that's the given i probably should have used green for that the probability of coffee and the probability of coffee is 40 out of 100 and i know that you guys probably i certainly am not you're not a big fan of fractions within fractions but if you want to be formula driven go for it so that's going to be .05 divided by and that's going to be 0.4 and then the whole thing is again gonna be point one two five so the formula works so if you wanna go on to higher level probability and you don't want to look at your tables and you just want a number crunch then go ahead and use that formula but i would not i i don't think i think fractions on top of fractions usually leads to mistakes so i would really do cell value divided by subtotal and i would i would refer to the table if i were you rather than the formula um but that formula if you manipulate that formula um you get this thing called the test of independence and i think that's why statisticians like it it's a quick test and you rearrange the formula where you uh but i i have a better test of independence and that is just use the definition so um i want you to have this for your notes down here so um this was the new piece of information um no not that well there's no new information there but we're gonna now have tested independence test well let's first say what's what's independence again it's been a while independence two events are independent a and b are independent events if knowing a doesn't help you you predict so you can think of independent as being unrelated in a way um they're just unrelated events so a perfect example is flipping a fair coin flipping a fair the probability of tails equals 0.5 or 50 you can flip that if it's a fair coin you can flip it 100 times doesn't matter what happened those hundred times the next flip is a clean slate and you can't knowing what happened before it can help you predict what's going to happen next so but it's really hard to come up with things that are independent events when it comes to people usually all kinds of things are related so a counter example counter example knowing a person's race is not independent of political party for most races some more than others so the example i i told you was being black and republican are not independent okay if you're black you're almost certainly not republican and i can use the one piece of information to predict the other if you're republican it's likely that you're not black but having said that hispanic is interesting because i notice on the media they say oh the hispanic voters biden didn't have control of the hispanic voters blah blah blah hispanic is such a broad umbrella of people hispanic could be mexican-american and that you can do a profile on mexican americans mexican americans are socially very liberal as a group so this is trends it's not individuals but then they're also catholics so you can't you can't totally put them in a box but they're more likely to vote democrat way more likely to vote democrat but cuban americans are way more likely to vote republican so it's too broad an umbrella and i'd say it's more independent but you can still do a little bit of predicting um and and white if you're white in the right if you're a white male it's it's more likely that you voted republican so it's hard to come up with true examples of independent independent means unrelated unrelated in a way you're in the dark i tell you that somebody is female can you guess that person's colorize no you can't well i'm going to put that one on there so another example um sex at birth and eye color i'm happy with that one there are not more brown eyed women than men so that's a good one um curly hair and gender i don't think so well gender you know nowadays it's kind of fashionable to have curly hair and so it does take a lot of work as you can see i didn't work on it today um so maybe you're more likely to be female i don't know because it's more of a hairstyle but length of hair and gender not independent if i nowadays if i know someone has long hair they almost certainly are female identified because the fashion right now if this was the 70s it'd be different but right now i can use one to predict the other so it is not independent that's the that's the concept and i want you to have the concept but the test for independence um test if i ask you to prove it for independence there's two so you can have the probability of a given b is still equal to the probability of a whether i know b happened or not the probabilities are the same are the seven and that comes from just the straight up definition of what independence is there is another test um i'll put it in a lighter color or is that too late no probability of a and b cell value equals the probability of a times the probability of b so that is cell value subtotal um it's actually it will it's probability so um i won't write that down because that's not exactly right so if i know the probability of having long hair and being female if it's equal to the probability of having long hair times the probability of being female if it passes that test then you know they're independent it's not going to pass that test though because most things having to do with human beings are related um the probability of sex that you're assigned at birth and whether you're left-handed or right-handed not independent you can use one to predict the other the probability that you are obese if you are obese it is more likely that you are female so if you are up tall it is more likely that you are male if you are white okay i think color and gender probably are independent um so that is the test of independence but the one i would recommend if you are asked to prove would be this one because we just spent this is a sim this one right here is a simple probability straightforward probability from um section 7a and 7b and this is the new material new from 7c so that's what i would i would do so that's independence so let's go up let's go back okay so i i did highlight this but really the one that i would use would be this one because that's a straight up definition the probabilities don't change okay um can we say that drinking bubble tea and coffee are independent events explain well i'm going to use i'm going to go with the orange um intuitively do you think if you know that someone's already had coffee you can predict if they're going to have tea i think so i think having coffee probably decreases the chances that they're going to have teeth so just from a conceptual point of view i think they're not independent i think there is a relationship between them so the opposite of independent is they're related so um but the way i can test it the probability of drinking um oh i'm down here p i'm asking you to kind of prove it so the probability of t given coffee we could work it out but i think we already did probability of t given coffee oh it's right here is equal to we already worked it out so you don't have to do that again 12.5 percent okay so that's that's that one what's the probability of just t the probability of just t straight up simple one and did we work that one out i'm gonna see could you did we well the pro this is since it's out of a hundred um it's just gonna be the answer so the probability of drinking tea is here's t [Music] 20 20 times i don't know why i did that 20 times out of 100 that's 20 so it's 20 out of 100 equals 0.2 and i can say these two values these two [Music] values are not equal so these events are not independent instead there's a relationship and i can exploit that relationship to make a prediction and it seems to me just like i suspected if she already drank coffee the chances of her going on and drinking tea are smaller then yeah i don't know so the chances of her drinking tea on any given day is 20 but if she's already had coffee that percentage plummets by almost half so they are not independent um so and feel sorry for me i've got so much grading to do you want to really um you want to explain yourself but you kind of want to box the answer too so it pops out at me for me can we say the two events on here drinking bubble tea and drinking cafe are mutually exclusive so we're now relating the two ideas so let's go and make notes about mutually exclusive so this was all about independence right here um and now mutually exclusive is a simpler idea mutually exclusive means the two events can't happen at the same time another in other words no overlap if you're dealing with rows and columns uh you got to have and the test so we have we have here the test of independence that i recommend the test for mutually exclusive is even easier it's simply the probability of a and b it can't happen at the same time equals zero okay so that's the test so you just look in your table and see if when you do row and column if they intersect and there's a number in there they're not mutually exclusive if it's row and row it's they're going to be mutually exclusive or if it's column and column they'll be mutually exclusive but so now that we've reviewed what mutually exclusive is let's go up here and answer the question can we say the two events drinking tea and drinking coffee are mutually exclusive so here's my table um drinking coffee these are all the coffee drinkers bam there's a 40 of 40 events all together and uh this is all the tea drinking so on f there's a five percent chance that she's gonna do both so there's days when they both happen so since p of t and a and coffee equals 0.5 not equal to zero not mutually exclusive so you've answered the question you've shown me that you know what you're talking about i'm sorry this is so messy can't leave enough room there we go um uh so true or false if something's mutually exclusive it's automatically independent if i know that someone um what's something that is mutually exclusive [Music] if i'm so being pregnant and being assigned male at birth are mutually exclusive if i know that somebody is male can i predict whether they're pregnant you sure can if so this is completely counter intuitive and it's gonna be on the next test um being mutually exclusive blank means also independent and sometimes b ohms c never and i'm just going to use an example pregnant if you're pregnant and you're male that doesn't happen p of p and m equals zero so we've got mutually exclusive that one's for sure so can't if you know somebody's pregnant can you predict whether they're male i'm not saying whether they're male i'm saying can you predict it you absolutely can you can predict that they are not male you know for a hun you know for sure what the answer is so this is totally independent so sorry totally not independent which is confusing so being mutually exclusive never means never if they are mutually exclusive they are absolutely related which is the opposite of independent and that's a bit of a mind twist i think it's a double negative it's a little tricky all right so going back up here last question consider two events a and b and so both events have a z they happen that the probability of a is not zero the probability of b is not zero it's going to happen um with both of them can they be mutually exclusive and independent and the answer here is no if they are mutually exclusive and knowing about one helps predict i'm going to do something here we'll get a little bit of space helps you predict the other you're predicting that the other is not going to happen but you can make that prediction with a lot of confidence so um so the events are related and related is the opposite of independent and not in the steps okay so we're done um that was a lot i'm not gonna lie but let's review what i really care about um so you learned that mutually exclusive the test for mutually exclusive is p let's make this a big bold one p of a and b equals zero that's mutually exclusive no overlap that you they have those two events have nothing in common and you learn independent which is knowing that one happens doesn't help you predict that the other happens so if something is independent the test for that is going to be p of a given b equals p of a so the probability of being female is equal to the probability of being female given you have green eyes there's no green eyes doesn't help you okay so those are two different concepts the relationship between the two is that if they are mutually exclusive they are related and therefore they are not independent which is kind of counterintuitive so you just got to keep thinking about that we've talked about calculating conditional probabilities you set your denominator to be the subtotal and you set your numerator to be the cell value um i don't recommend the formulas i recommend you just look at the table and you pick you keep with the old definition of probability but you can do what you like um we have a test for independence and i've written it right above there and we know what mutually exclusive means no overlap and we've spent some time calculating these probabilities and these are two separate concepts that are literally related in that mutually exclusive means not independent it means related okay so um have fun with this um it's just a little bit more complicated than the last two sections but with some practice i think you'll be fine and i'm not going to throw you any curveballs i'll be really clear i'll use given as the oh here's my condition um so that will be your subtotal okay good luck with that and i will so take a little break give yourself a real chance to forget a little bit so that forgetting a little bit and then 20 minutes later trying to revisit it is gonna make it go make the information go more easily into long-term memory in a real way if you don't take a break you might just be relying on your short-term memory and if you take a five-hour break then you have to kind of lean out the wheel so that 20 minutes to an hour is a sweet spot so do the practice in 20 minutes to an hour from now okay bye you guys