hi welcome to the second section of chapter 7 7d in class 7d today we're going to talk about political affiliation and vaccination status so is there a relationship between those two things or not that's not what we're exploring today we're just going to be looking at those two variables to understand probability rules so you're going to get one pretty famous rule today on probability but i would really strongly say think about the conceptual big picture of what that rule is and if not you might make a mistake when applying it and two the best way to deal with probability is to keep it simple what's the simple idea about probability number of successes over total number bam no formula just the definition so let's get started first questions a little bit delicate um i'm asking about flu vaccine but you know what i'm gonna i'm just gonna cross that out because i i want to get to the interesting stuff coronavirus peron or let's say coven covet and it is it's probably going to end up being annual booster i don't know but did you get your cove a covid vaccine do you think vaccines in general are definitely safe so what's a good color for definitely safe are they definitely safe are they probably safe little gray there are they probably unsafe it's a that's a good unsafe color i guess yeah that was a good probably unsafe so warning warning warning or do you just know everything in the world and you think they're definitely unsafe what do you think about coronavirus so virus vaccine um state your opinion and explain why you feel that way okay so what do i think i am a trained statistician my husband is a scientist with medical research as his emphasis and we both think that shock shock you can't actually say anything is definitely safe but i'm gonna say which one am i going to say i'm going for this one but it's probably absolutely almost definitely safe for almost all people um and what i'm doing is i'm weighing the benefits of the vaccine and the risks to the benefits and the risks of not getting the vaccine so if you get the vaccine it could make you feel really crappy for a number of days you could even get significantly ill but for almost all people it diminishes the chances that you're going to have to go to the hospital and i've read literature that says well we haven't had enough time to actually see how it is on people well that we've now replication number of trials leads to more stable results almost everybody on the planet who except really poor people in poor countries have gotten the vaccine we've got data on that now and also so many people have gotten a virus and the people before we had the vaccine people were dying dying dying dying and they're still dying so for me i'm not going to say definitely safe because you may qualify for a medical exemption for a valid reason but i'm saying probably safe and i've explained my my reasoning but probability we to really understand this stuff we have to look at the probability of results so let's get to it and i respect if you want to tell me a different opinion but this one really is i really like different opinions but this one please get vaccinated for not it's you know individual good it's we're talking about the greater good here for society for people who can't themselves get the vaccines those of us who can get vaccinated to protect those that can't you really i think it's the right thing to do okay um probably okay uh after this section we're gonna look at probabilities not just straightforward probabilities so what does probability mean what does probability mean the probability of getting an a is going to be number of a's over total number total number of grades but actually we use p of a to mean any event so the probability of any event happening is the number of events number of outcomes like that over the grand total so that's probability but today we're going to look at it for more complicated events it's not just one straight event it might be a couple of events all mixed together and that that probability can be found thinking about the individual events and combining them in some way and that's going to be the focus for today um okay and what we're going to do is specifically we're going to focus on and statements and we're going to focus on or statements i'll do one for o and then the not statements are not that hard um what color do i want for the not statements let's do green okay so generally speaking so i love to do it by simple examples let's think about here's the whole world there's the whole world all the people in the world and what we have those are all the women in the world okay so what are the who's on the outside here out here so i'll just put a little somewhere for women um um this represents women so that's a symbol for women who besides women what other kind of women are other kind of people are in the world who's on the outside here outside here notice i'm not putting glue outside here these people who live outside the pink circle they're not women it's not that they're men because we now are pretty so gen i'm doing gender i suppose not not sex assigned at birth but the people in the green area are men or people who are gender fluid um and there's i guess there's a spectrum so that's everybody else is on the outside but what i'm now gonna draw what's near and dear to my heart um is and it's not i'm gonna make it actually a little smaller guess who the the turquoise circle is these are left-handed people well it's not drawing interesting like i guess it's so light that it's not showing up so i'll pick a different color okay these are left-handed people okay those are left-handed people so if you look at the way i've drawn that circle most of the women are not left-handed there's only a tiny group so guess who just what this letter stands for that's me i am a woman and i am i'm i'm i'm identify as female and i'm left-handed and i'm rare most left-handed people in the it are not women and most women are not left-handed um it's i think it's one in 10 people are left-handed but one in a hundred or something ridiculous like that are when our of left of women are left-handed so this intersection right here i'm going to shade it in this intersection i don't want you while the color okay this intersection right here is an overlap that right there is all the people who are female and left-handed so if we do an and statement it's very exclusive you have to live in both circles so and means intersection visit visually means intersection so if i say all the women in the world it's about 51 if i say all the left-handed people it's about ten percent but if i say all the people in the world who are both women and left-handed it's more like one in a hundred it's a tiny little sliver of people um we're rare so if we were in class together it would be so fun oops that didn't work it would be so fun to see who my partners are um screwing this up my buddies are who are both women and left-handed very unusual but also even left-handed is unusual so um so i want you to imagine the two circles and i want you to imagine that it's the overlap so we've got two circles that and is really exclusive okay so now the or is going to be if i have almost the same picture all the people in the world and we're interested in the people who identify as female which is about 51 i probably should make that circle bigger and it's also the lefties so i'm sure you know where i'm going with this or is going to be all the people who you can find in either circle so it's going to be all of these people i'll shade that in plus all of these people so or means union so you unite both groups so you get a very big group in this case so i'm just going to go ahead and it's going to be all these people okay but not everybody in the world because you're going to leave out the gender fluid people who are not left-handed if they're left-handed they no oh sorry female or left-handed so um so anyway it's it's going to be union so you want to consider both circles and what i'm drawing here are called venn diagrams and um it's just a very helpful way for most people to understand this okay find the probability of one event and not another okay so visually what that looks like same picture and i'll and what i'm going to i'll just focus on the lefties so if 10 percent of the world is left-handed how much of the world is not left-handed so not it means the compliment so probability probability of not turquoise not left-handed okay so what it's going to look like visually is it's going to be literally everything but so it's gonna look like you cut out your you cut out that circle and you throw it away so that's a dotted line and then it's going to be everybody else so we'll throw that 10 away and who do we have left do you need to gather the data on who the right-handed people are and who the ambidextrous people are or can you just say it's 90 usually what people do is they say oh we'll just take everyone everyone in the world minus the lefties and that doesn't mean righties it's everyone but so it's going to be in this case it would be about 90 and you don't need the information on how many people are lucky enough to be ambidextrous so it'll be 100 of everybody you start with the whole world and then you throw away the lefties i think it's about 10 um so and uh when we're doing percents i really would like you to do the proportion rather than the percent so that would be 1 minus 0.1 which would be 0.9 just as an example um so those are the formulas so this the formula um we'll get to the formulas so right now that was just concepts so i really would like you to hold on to the concepts so and is intersection or is union and not means everything but so you just throw out the what you the one thing that you you know about and you'll have the not okay so here we go um in may of 2021 we surveyed we surveyed 750 california adults and asked a number of vaccinated related questions the survey respondents were considered representative of all california adults so we're think well that's a signal saying this is a good sample so what we have what we know about the sample can probably be extended to everybody in california one of the questions was based on what you know are you currently um are the currently available coronavirus vaccines definitely safe so this was in 2021 so it's quite a while ago so definitely safe probably safe uh probably unsafe or definitely unsafe so i think i want to use the same colors that i had up here so green is like a go so we've got definitely safe probably safe so definitely safe probably safe probably not safe or probably unsafe so that's getting him out of the gray area and into the red area and danger danger danger oh that's disappointing the red seems not very i'll fix that we're definitely unsafe so these people are really sure of themselves and um here are the results so this is real data so what you see here you've seen this before this is a this is called a two-way table or a contingency table and what uh here's the question we're just asking based on what you know and the information is over here and these are the subtotals um oh that's weird let's go back oh i guess i lost my blue so here that's the question and here are the subtotals so uh definitely safe probably oh i skipped probably safe oh i guess that's gray all right probably safe probably unsafe and [Music] definitely unsafe okay so there they are oh and then we've got the people who say they're just they don't know i should i like that that's a good that's a good i can't forget those people i don't know we'll wait and see that's probably i i might so my camps are the the dark purple or the dark gray though i'm running to get my vaccine and if you add so these right here these are subtotals subtotals and if you add them all up there you have your grand total grand total okay and that's what the researcher really cared about but they asked another question and that question was what's your political affiliation and pro so this is a whole other question and we've got the republicans we've got the democrats we've got the independence what's a good color for independent oh i'll make them green and then we have them now we're not really involved in any politics so we'll make them i don't know what's a big color i i like to sing orange for other things okay they're just kind of the three spirits okay so again these are subtotals and if we add these up we're gonna get that 750 so it's called a two-way table because you slice it two ways and if you look at the rows you are looking at vaccine status responses and if you look at the columns you're seeing political affiliation and so columns tell you politics and rows tell you how people feel about vaccines and you know that there's almost certainly going to be some relationship between the two but that's not what we're exploring today we're just playing around with probability so um so i'm going to answer these questions and then i'm going to set you loose on another problem where you're going to answer it and see how you do so i'm going to use my keep it simple which is the probability of anything is all those things that you're interested in over the total grand total so there goes uh what is the probability that a randomly selected a california adult believes that the coronavirus is definitely safe so i want probability of definitely safe and it's not whether it's safe it's whether people believe that and so what you want to do is you want to find the definitely safe row or column and in this case it's going to be it's all these people right here and but you don't care about political affiliation you care about all of them you don't ask them their politics so so you're in this case it's just broken down by i know the grand total is 750. oh sorry i know the grand total is 750. how many people of all the people that were asked um said it was definitely safe and it's going to be 278 and then you get out your beloved calculator and you divide top by bottom and you're going to get 0.37066 going on forever round to the nearest thousands on this homework if you don't read the directions it's gonna be so frustrating because you will know what to do but you'll be getting it wrong so the thousands is three places past so it's going to be one around this one so does it stay zero or does it go up to a one it goes up to a one so zero point three seven one so i want you to do probability as a decimal unless i say otherwise but you could write it as a percent 37.1 percent but i'm if i'm asking it around i'm asking you to round it as a decimal not as a percent so that's a little confusing and that's why make all of your answers decimals okay moving on to the next one what is the probability that the randomly selected adult's political affiliation is republican so again you want to ask yourself oh republican so i'm going to i want the probability of what republican so we're a blue state there are not that many republicans in california um and i don't think a republican i think even ronald reagan didn't win i'm not sure to be honest but so what you want to do is you want to you you go oh i love this two-way table but we're not interested we're not interested i haven't asked how you feel about vaccines so i'm not going to be interested in any of the inner i'm going to be interested in a subtotal over the grand total so i know the grand total is 750. so i got that right for sure so i'm interested in republicans and these are all the republicans so you yes you you do have these are all republicans but that's not what we're interested in we're not interested in breaking it down we're just interested in republicans as a whole so it's 108 i'm sorry 180. and when you work that out you get exactly 0.24 and they still say to the nearest thousandth so technically the right answer is this but if if on the homework your decimal is smaller than what terminates before it's okay just to put that in there and so the answer then would be or 24 but i didn't ask you for percent okay um what is the probability the randomly selected california adult does not believe the coronavirus is safe so this is um basically p not safe not and is it just straight up safe what is the probability the randomness does not believe the quran virus is definitely safe not definitely safe so you could answer this one of two ways um you could just i'm so we've got here the green is the definitely safe so the not safe is gonna be everybody else so it's gonna be well the um so we're dealing with this question again right here we're not asking about political affiliation we're just asking about this so i'm going to look at the subtotals and decide well these people said probably safe that works this works this works this works and that's the grand total but i could just say number of events over the grand total and i could just say well it's everybody but so it's the 293 plus the 83 plus the 45 plus the 51. it's all those people so it's the people who think it's probably safe the people who think it's unsafe maybe the people who are sure it's unsafe and it's the people who have almost no opinion so you could do it that way and totally get it right um it does require a little bit more math to do it that way but that's okay it's you're being very conceptual you're keeping it simple so that's gonna end up being um 278 i believe so if we add up all of these together it's 278 over the grand total of 750 and then you end up getting um 62.629 or 62. uh it was three 0.629333 but i've rounded it so you could do that now if you're sharp and slick you can do a shortcut which is well i already found out that um 37.1 of people think it's definitely safe so the not definitely safe the shortcut would be p of not safe not definitely safe is equal to one minus p of safe of definitely safe and i already figured out that 37.1 percent of people say it's definitely safe so i'll just do 1 minus 0.371 and when you do that you get the exact same thing 0.629 so if you know the formula you can do a shortcut and then there's a little less math but so you decide what camp you're in keep it simple just add up everybody who fits the description and divide by the grand total or use this handy-dandy formula okay just erased handy dandy formula which is this one right well i didn't write it down but it's this concept right here okay um what is the probability that a randomly selected california adult believes that the coronavirus is sick is definitely safe and their political affiliation is republican so if we know all the definitely safe people are 37.1 percent i'm suspecting that of those who are also republican it's probably a much smaller number so um so what i want to look for and remember what they said about and and his interception so who are the republicans these are all the republicans and there's the grand total of republicans so it's actually a subtotal every red number is a republican and definitely safe is every green number so the question is who is both red and green and the answer to the question is going to be right here these are the people that fall into both the red category and the green category and sure enough it's the intersection so and means intersection and there's nothing that you really need to do but let's make sure we get the notation right it's the probability of being republican and safe uh definitely safe so if i say and and you're dealing with a table i'm looking for a cell value and it's the cell so if this is an excel spreadsheet and means cell value so i'm just going to come up here and make a note of that note for tables it's the cell value divided by grand total okay and you can take that to the bank so now um oh i didn't answer the question i got all excited and i didn't answer the question so the cell value the thing that is both green and red is so it's kind of 3d there um but don't forget that you're dividing by the grand total is going to end up being and i don't know at this point i'm not paying attention to this so it's going to be 0.081 so while 37.1 percent of people think it's definitely safe for republicans those that are also republican so they both say it's safe and they identify as republican it's only eight percent of the population in california but that might be because there aren't a lot of republicans so um okay what is the probability that the randomly selected california adult believes the the coronavirus is definitely a vaccine my bad is definitely safe or probably safe so remember or or means union so you are going to add up all of the values in both rows not or both row and a column both areas you're going to identify who we're talking about and then you just take everybody and or statement i am a woman or an elephant is a true statement because i fit into at least one of those categories so um i'm gonna wipe this out so that i have a clean slate to work from i do have the subtotals i'll just let's make it all clean so we get the idea but i always keep in mind the grand total that that helps me there's the grand total okay so we're interested in definitely safe so we're interested in all these people okay um or probably safe so so we're just interested in all the people in two rows and we have the two subtotals so we should just go ahead and and rather than adding up everything we've got the totals over here the subtotal so these are all the definitely safe voters they say it's definitely safe so we'll just we're going to say oh it's 207 plus and this is where it gets a little confusing because you've been taught that and means addition but in probability or means addition add everything up um the and means you've got to fit both those categories so we're going to go ahead and add those together but we're going to remember what probability means it's the probability of definitely safe i'm just going to abbreviate it or probably save so it's going to be those two rows oh but probability is that over the grand total so i can't forget that so over the grand total is 750 and yeah probability better be a decimal or i really screwed something up so it's going to end up being um if i add those two up i get 571 which was a lot of people so that's 76 percent one three three three so round to three pages pass i didn't say two but i'd like you to so 0.761 and that's where i fit i'm in the it's probably safe and i'm going to go get it because it's not gonna kill me and the coronavirus could actually kill me so um i'm gonna go get it okay um so the next question are those two categories is the green category and the great category are they mutually exclusive so we got to remember what mutually exclusive means mutually exclusive means that you can't be in both categories at the same time so a really good example of mutually exclusive is pregnant and not pregnant can you be pregnant and not pregnant at the same time you can't you're one or the other you can't be both can you be um can you have curly hair and wavy hair yes you can i can have my eye hair challenged and some days half my head is wavy and half my head is curly so you can have both those characteristics but you can't be kind of pregnant so mutually exclusive means it what i just said but i remember no overlap so they can't have anything in common and what i see is that people here fit into either this row or this row so they are mutually exclusive yes people clearly fit into one of these two categories with no overlap people are in one of these two categories with no overlap okay so um what are some other mutually exclusive things you can't be i mean can you be a good person and do bad things yeah you can i'm a good person but if if push came to shove and i had to um you know do something despicable to save my children i would i would totally do that people who say oh i would never steal under any circumstances probably haven't been hungry or they probably haven't imagined their own children being hungry so those are not mutually exclusive but um you know right now can you be a republican and a democrat no right now our country is really polarized um and we we fit into really hot excuse me so but this is an example so row anything that's a row all rows they're mutually exclusive all columns are mutually exclusive but rows and columns are not usually exclusive there's usually some overlap all right moving on what is the probability that a randomly selected california californian adult believes the coronavirus vaccine is definitely safe or their political affiliation is republican so i'm going to shrink this a bit you have here to look at because i want us to see it at the same time so um let's see if i can get it a little bigger there we go okay a randomly selected definitely safe so i'm using green for definitely safe or is a republican okay and i remember or means union so i'm interested in all the people who identify as definitely safe so i'm going to do a clean slate here okay so definitely safe is all of these people and there's the grand total sorry the subtotal of all those people they said they rushed to go get the vaccine i did too and then the republicans are all right here and there are 180 republicans out of 750 we're not a republican state this um this uh sample reflects that so um how can you so i know i can i'll do a shortcut the probability of being both of those things one or the other not both p of republican or definitely safe is everyone who falls into both the red column and the green row so i can see i'm going to be clever and i'm going to do a shortcut that's 278 people plus 180 people is that right and then it's divided by the grand total of 750. is that right did i do that right no some of you are probably ripping your hair out and you're going wait a minute when i did the two the 278 people i did all these people there they are 278 bam but then i did the republicans and i'm like well wait a minute there's a group it's a pretty unique group that is being counted twice so i should do this one this one this one this one but it's not 180. it's not because this group right here you don't want to count this group twice these people already got counted counted in green rho so what you have to do is you have to subtract the intersection so minus the overlap or intersection so that the the group right there that is both red and green so i'll make them they were first counted as green but then they were counted as red that 3d group we we can't count twice so we'll subtract them so they were already included and so that's an interesting formula and i'm going to go back up to the top and write that formula down to calculate this is the intersection to get oh this is the worst statement my dad to get this you're doing the probability of the small circle plus the probability of the big circle the other circle minus the probability of the intersection so when you're asked for and this is the formula now having said that i usually don't rely on the formula i just do it i just look at the cell values and for and um sorry this is an and this is ore see i i kid you not that is um or the or this is p of a or b okay so if you want the union you get the one circle the other circle but then you take away the overlap and i wonder if i can do that over here so um we'll get to that in a minute so so you could just say um i'm gonna just say i'm interested in all these people bam bam bam bam and all these people down bam bam so you could just add all those up that's one nice way of doing it um and what you would do is you're saying oh i'm just gonna um or means union so you would just write them all out but i just did a shortcut okay so what does that end up being it ends up being um 397 over 750 and that ends up being 52.5 repeating all within threes repeating so round it up that's what it would be so 52 52.9 almost 53 of the population are either republicans or they think that the vaccine is safe so that's an important formula but what i remember is or is union so i just add up all the cell values that fit that description i would just literally add up all the brain cells and all the red cells that i didn't already count you'll get it right every time okay are the two events mutually exclusive and the answer there is it's mutually exclusive means is there an overlap is there an overlap between that row and that column overlapped or are there people in both groups are there people in both groups um well if we're asking is there let me fix that because that's are there no people no people in both groups so in this case we're going to go and say not mutually exclusive because there was an overlap and i didn't ask you to explain but 61 people in the sample were republican and believed that the vaccine was completely safe okay so not enough room there all right so we're breaking it down and what they're saying is you can do the um you can do the um hear the republicans that's the probability of being republican and this is the probability of definitely safe um and when you add it up it's 61 not 52 percent where's the flawed reasoning you can't just add up the two probabilities to get it because there is this overlap the flawed reasoning is that there's these people right here these magical people that are both republicans and they believe the vaccine is safe that actually that number isn't going up it's going down with our country getting more polarized so the flaw in the reasoning is we are double counting the people in both groups so the formula is if you want to do union and i'm not a big fan of the formula i just add up all the people who fit both categories keep it simple or means everybody who fit in either one of the categories but what you can do is you can say okay it's the probability of being in the a group plus the probability of being in the b group minus the probability of being in the a and b group and visually if you want to figure out or well you know or is union so it's this circle plus this circle right that it's union so if you've got a red circle if you've got a red circle that goes like that it's the probability of that circle plus and maybe you've got a blue circle which is this one the probability of the blue circle but there's that sliver in the middle what is blue and red bank it makes purple and you counted it twice so you've got to minus the sliver that you already counted so it's minus the intersection minus the overlap so if you want to just rely on the formula you can but i highly recommend you just add up the values in the cell and union means every cell that that fits one or two of those groups explain why this property holds for mutually exclusive well if this mutually exclusive means no overlap so if you want the probability it's union right so it's the red circle plus the blue circle red circle plus the blue circle well it dot it equals i really hate that one people say do that to me but it's the probability of the red circle plus the probability of the blue circle minus nothing there's no if there's no overlap mine is zero i'll write that there's no overlap well it doesn't change so this rule right here if you are very math inclined it works no matter what and the beauty is you don't have to think about it at all but if you're very math inclined you probably already just know to add everything up and that's what that's what or means is just add everything up so all right survey asks so this is number three and i've now exposed you to all the new material but i am going to come up here and um what we've been talking about is this formula and it applies to this but like i said keep it simple that's the general definition add all the number of successes up by the total number and for or you add everything up and for and you add up just the intersection i get through it without doing any formulas all right um here is number three and what i want you to do survey says asked have you been vaccinated for the coronavirus now this was in may 21. so probably most college students were not vaccinated in may of 2021. the data was that i think 30 of young people had been vaccinated by that time so just we got now skip forward it's a little different but when this worksheet was made a lot of people weren't yet vaccinated so and did it run by party line now this is before employers were able to force people to get vaccinated um you know it's your personal right but an employer can say get vaccinated or quit i think i think we did that with police for example all right so um we have here's the data um and it's the same group of people you'll see we've got our 180 republicans we have our 338 democrats so it's blue state we're a blue state do i have a color for independence what color oh i said green for independence and then we have the people who have no opinion and then p for population so this is everybody that's the grand total so we can break it down that way new information is um have you been vaccinated yes no not sure i'm glad that only three people are not sure because they really should know so we've got these three people are unusual people how they would not know i don't know so we've got the people who said yes i'm absolutely vaccinated so what's a good color for vaccinated let's do brown beautiful brown these people are vaccinated and um i already use green uh i love the color turquoise i'm gonna uh we'll do this pink over here okay so 269 had not yet been vaccinated so probably a lot of college students in there um okay which is more likely the probability that a randomly selected um oh so what i'd like to do is i would like you to do a b and c on your own see how you do and then join me back here and we'll go over the answers but probability keep it simple probability is probability of a equals number of a and a doesn't mean grade over total number total of everything so that's all you need to remember and and since we're dealing with the table equals cell value and or means so it's and is intersection and it's the one cell the the value in the one cell and the or means um everything every value in both row and columns so or you unite everything and is very particular okay so go ahead and and i'm not doing any formulas keep that in mind when i explain this i'm not going to do formulas okay so turn the posit and come back okay which is more likely the probability that a randomly selected california adult is vaccinated republican or vaccinated democrat so if you're a vaccinated republican you're that's an and statement so the first one um the first one is what's the probability that you are vaccinated and republican so p of vaccinated and republican that means you live in both the red you live in this column because that's all the republicans and you live in this row so it's overlap so that's going to equal so you're both brown and red that's this guy right here it's and is cell value so it's going to be 104 but don't remember forget out of the grand total so um what does that end up being that's only 13.87 and i don't think i gave any i didn't give any rounds instructions so it's almost 14 of people and now we're going to do the probability of vaccinated and democrat and democrat is the blue party so just the same it's gonna be everyone so you're gonna i would really so it's everybody in the brown so it's all those people but wait a minute oh no you also have to be in the blue row so it's the overlap and beans overlap and literally the two colors are overlapped so it's going to be [Music] 247 people out of um out of 700 out of all the people that were surveyed and that's going to end up being worked it out one divided by the other i got .329.3293 so there's my information and so from that i better answer the question which is more likely to be 13.87 percent of the population or be 0.3 3293 oh it's this this is bigger so i'm gonna go it's more likely is more likely to be [Music] a democrat and vaccinated why because the probability is a bigger number okay so that's pretty cool good um b what is the probability that a selected california adult is vaccinated does that seem like a much simpler question for did i ask anything about political affiliation for this one no i didn't so actually we don't care about any but we don't care about political affiliation we're not interested in this breakdown all we care about that's a little confusing the way i did that oh let me back up i just want to clean this up okay so here's everybody there's everybody what's the probability of being vaccinated vaccinated have you been vaccinating yes all you care about is vaccinated versus not vaccinated versus not sure so the probability of being vaccinated is 4 78 number of successes over total keep it simple keep it simple and that's point six three seven i think it's even i don't know but that's what if it's not that's i rounded these three places past so probability being vaccinated even in 2021 was almost 64 now other states it was much lower and if you remember by the time the summer came around there was huge surges in some areas and california was doing pretty well all right um last one i think i'm getting excited yes um so it's this one was the simplest question of all so far now what i see i read something i go is it simple is it an and or is it an or this one was an and so i look for the intersection this is simple so it's just straight definition and this one is or so i know i'm going to be dealing with union so what is the probability that a randomly selected adult is vaccinated or is democratic okay so basically we have to get all the democrat we have to get all the vaccinated people and then all the college students were left over because they probably most democrats that were older were already vaccinated but the young people are the ones that are left out so what's the probability so it's an or statement vaccinated adult is vaccinated so i'm gonna go well vaccinated um were you vaccinated yes so it's gonna be this person this group this group this group and this group and they all add up to this but i know it's definitely all those people or republican democrat my bad it's going to be well it's this group but i already have them included so i won't include them twice but i'll also do those so it's basically i'm going to add up all these numbers and i'm going to add up these numbers and i'm not going to worry about the formula i just know union means i mean or means everybody who fits that description so yes it's 478 plus 91 plus zero but um i'm just gonna be really i don't like to use formulas that i don't fully understand so p of v or d is i'm just going to add up all those numbers and i'm going to because by the end of the semester i'm going to be completely overwhelmed so it's 104 plus 247 plus 91 plus 36 and i know that's 478 but i'm going to keep going plus i don't want to count 247 again because it's already been counted so i'll do 91 plus 0. what's the zero not sure okay so um and so all of that divided by the grand total of 750. so when you add that together it's 556 all together that is 556 divided by 750 and that ends up being as a decimal 0.759 i'm rounding to three places past so um we're out to the nearest thousand so i did okay what is the probability i answered it or 75.9 percent but i think for canvas i'm going to want you to do decimals all the way so um if for those of you who want to do the formula um we know it's the probability of being vaccinated plus the probability of being democrat minus the overlap so it'll be um vaccinated is 478 over 750 plus democrat is 3 38 and the overlap is brown and green it's the 4 247 over 750 and when you work that out run that through the calculator it's still going to be 75.9 percent but i would prefer the decimal because otherwise around it gets a little confusing okay so we are done with that um let's see how we did and if we covered everything i was supposed to so this is a little messy um the complicated events are and and or and when you see and i want you to think about the intersection and there's no real formula for it you just have to look if it's a table you're going to be just looking at a few you're going to be looking at one cell value that is both in a row and a column it's literally the overlap um for the or i want you to think oh it's going to be everyone in the small circle and everyone in the big circle so it's going to be all the row cell values and all the column cell values but we don't want to count the overlap twice so um the formula is here and it's circle plus circle minus the overlap so that you don't count the area of the circle twice okay and find the probability of one event and not another that's the shortcut if you want to find all the area out here and you know the area inside it's just going to be a hundred percent minus the area on the inside so it's a little shortcut you can go your whole life without doing that without that shortcut okay um so we're done with 7b um take a little break and um take a little break and then go ahead and do the practice