hi there welcome back we are starting in class activity 7d and it's our last section of probability and we're going to be dealing with something called false negatives and false positives so let's get started um yeah oh good all right here we go so uh medical tests designed to detect uh disease are not always perfect sometimes people who get tested may be told that they have a disease when they do not actually have a disease so there's one bad thing that can happen so this right here if you're told that you have a disease and you don't that is what we call a false positive okay so that's one bad thing that can happen usually uh hopefully if it's a good test that is not what's going to happen so the other thing that can happen other times people who get tested may be told they're free of disease when in fact they have a disease so and that so you say oh you're absolutely fine so you're given a negative result but it's a false so this one is false negative you're given all clear when in fact you might have something wrong with you the goal is for medical tests to be correct as often as possible so that these types of errors don't happen but they do happen in your opinion so this is your opinion so there's no right or wrong answer but in your opinion what's worse telling someone they have a disease when in fact they do not or telling someone they do not have a disease when actually they do so just write down what your opinion is and pause it and come back to me well it really depends on the disease doesn't it i think one example i get strep throat a lot so if you get strep if you're given a false positive you're told you have strap so for false positive um you're told you have strep and you take antibiotics unnecessarily take anti biotics when not needed well that could be pretty bad um globally it's pretty bad to abuse um antibiotics that's gonna be a problem that we we're gonna deal with a lot and then you're in hopefully well we're dealing with it right now where you get strains of bacterial infections that are just very resistant but on an individual basis it's not the end of the world and strep is actually pretty it can be a very serious illness if it's not treated it can go down into your heart and so the false a false negative untreated untreated strep can lead to heart disease and even death and even eventually death so if you have strep you should really you should really go to the if you have a sore throat for more than a couple of days and especially if you run a fever you should go and get tested and i would rather be told that i have strep and take the antibiotics that i don't need than to be told i don't have strep when i do so that's one example another but there are lots of examples where it could go the other way where maybe the treatment is more dangerous than um than just writing out the illness so it really depends on the situation if you're told that you um well i can't think of it i mean i'm sure there are a million examples where you're told one thing and you go ahead and act and say oh i i have this illness and then maybe you deprive yourself of a lot of joy or you experience a lot of stress who knows so this is an opinion question so moving on in this section um we're going to really be taking some time to construct probability tables we're going to be looking at tables because they are so helpful in answering questions about probability especially conditional probability which can uh the notation can be intimidating and if you're not thinking in the most simple terms it can become overwhelming so i'm going to really rely on the tables for the most simplistic way of approaching conditional probabilities so we'll use those tables to calculate the conditional probabilities and uh probabilities can be constructed using a combination of other probabilities so i am not going to be relying heavily on the formulas i'm going to be relying heavily on the tables so let's go so here's an example so we're talking about covington so a false a false positive for covid um you're told you have coveted when in fact you don't have copic well you might miss a lot of work you um you might end up being isolated for a long period of time when it's not necessarily true that you were ill um so a lot of times people are asymptomatic but they're given these positive tests and they have to stay inside they miss a vacation um false negative you go out into the world and you spread that illness and it could actually be killing somebody else especially if you're college age traditional college age it might not be so bad for you but you inadvertently pass it on to grandma and grandpa could be bad news for them so here's some here's some test results um a testing facility reports that the following testing reliability results for covet 19. so 80 of the people tested in the facility do not have covet so it's 20 percent positivity rate um so 80 do not have covet i'm gonna use green for that because that's good news eighty percent do not have covid okay 2.25 of the people who do not have covet test positive so this is the what color did i use for that this is the false positive false positive okay i'm just making notes so 2.25 percent of people do not have covered but they test positive for it okay eighty percent of the people who have covid test positive for it okay eighty percent who have covet so i'm starting to get overwhelmed so i'm hoping oh yay look there's a table here um so eighty percent of people who have covid test positive for it so twenty percent of people who have cova do not test positive all right so that's a different result altogether it's not a false negative it's something else we want to answer the following questions if someone tests positive for covid what's the probability they actually have it if someone tests negative what's the probability they truly do not have it so these are kind of like what what these are good things what here how often are the are the results absolutely spot on all right so um the first thing let's get comfy with tables so to begin begin by labeling the yellow parts and so what they're telling you here is um it's a really good idea to have all of your headers and um labeled so what are the possibilities i'm going to do a nice black pencil here so you can test positive what's the only other thing you can happen you can test negative so negative negative test if you go in those are your two possibilities and then the other two possibilities are you don't have covet yay and the other sad possibility is has coven you have it okay so those are the two realities and then what this row down here signifies is always the totals total and it's totals those are the subtotals and the grand total so the grand total is going to go here total always does okay so begin by labeling the highlighted rows so i did that um complete the table including rows and columns based on the previous information and we have a hypothetical 1000 people hypothetically so they all that's how many people wandered into the testing center so that is the grand total so i'm going to put that there and now i'm going to i it's kind of scary at first but i'm just going to have faith that my table actually i can answer all of the information with the given information up here so i'm and i'm going to go in order so 80 of the people tested in the facility do not have coven so i've got my totals over here i know there are a thousand people eighty percent so of the totals which are right here eighty percent don't have coven so that's going to be the do not have coven is going to be right here do not have covid so it's going to be 80 so i'm going to come over here and go 80 of how many of a thousand don't have coven so that translates directly to point eight of which is multiplication a thousand and i'm going to pop that number in right here so eighty percent of a thousand is eight hundred so yay one square down so if i know that one could i figure out this one if i know that 800 is here and a thousand is here well i know that this one has to be the grand total minus that subtotal so it's going to be 200. so this is nice it's just kind of writing itself here and i think that i have now exhausted this piece of information so i'm going to move on to the next one and figure out where that one goes so let's pick a different color 2.25 of people who do not have covid test positive so this to me sounds like there are two conditions do not have covid and test positive so that's going to be um the probability not do not have covid and test positive okay so we've got an and statement here and if you remember and is an intersection so it's going to be we're interested in the people who tests positive and do not have covid so we're really interested in that cell right here if that makes sense so we're interested in and we know that that probability is .0225 of the total [Music] of whoever the positive people are so i'm gonna just uh erase some of this so i know because i'm getting a little i'm gonna erase this so i just have it to look at oops well that's okay i'm interested in these are the people that i'm interested in oh shoot that was interesting back to me these are the people that i'm interested in they do not have covid and they tested positive and it's an and here and so i know that the result is 2.25 percent who do not have code with so two let me make this a little smaller 2.25 percent of people who do not have covet the 800 test positive so i'm going to put them right here so it's going to be 2.25 percent of the 800 is if i make that a decimal that's .0225 of the 800 do not end up testing positive run that through the calculator and i get 18. so 18 people are in that box all right so moving on to the next one 80 percent eighty percent of the people who have covid test positive okay so eighty percent who have covid test positive so i'm interested in the people who have covid so that's going to be this group right here and i know that 80 percent are going so 80 percent i'm just going to write this down as a eighty percent of how many people have coveted 800 test positive so here are the people right here who test positive and have covet and i know it's of the grand total it's going to be oh my bad hey this should not be an 800. this should be a 200. 80 of all the of the 200 people who have coveted test positive so i'm going to go 80 percent 80 percent of the coveted people actually test positive so that's going to be 0.8 and of the people who have coveted it's 200. so when i run that through the calculator i will get 160. so it's just reading this and figuring out which box it's referring to and then it kind of the rest of it kind of works itself out so if do i have enough have i used i've pretty much used all the bits of information up here so um how do i figure out the other boxes remember it's the name of the game is fill in the damn boxes so if we're going to do this one i know that 160 plus whatever this is is going to be 200 so i'll go 200 minus 160 and that is going to give me 40. okay and then similarly this one down below right here if i know this is 18 and i have a grant i have a subtotal of 800 i'll know that 18 plus whatever this is is going to give me 800 so i'll go 800 minus 18 and i end up with 782 and so from there i think i can figure the rest out so this column right here i know that 160 tested positive for covet and uh i know that these are all and i've got 18 here from the doesn't have coven so everyone who tested positive is going to be the 160 plus 18. so it's going to be 178. similarly this if i now look at this column all the negatives i've got 140 here i'm sorry 40 here and 782 so the subtotal is going to be 40 plus 782 i'm going to end up with 8 22 right here and i have now filled in the damn boxes yay go ahead and get rid of that one and now so complete the table all right i did that um so for problem number three i'm gonna do problem number three and then i'm gonna want you to pause and i want you to try and do problem number four they're very similar so question two part a and b will help you so filling in the table is gonna help you answer the question um answer the following question the question is if someone tests positive for covid what is the probability they actually truly have it so this has to do with conditional probability which we did in the last section what you want to do is whatever comes first what is the condition what's the first thing that the person the researcher knows is it that they tested positive or that they have covet and the condition the thing that comes first is they are first they tested positive so that is the given so i'm doing the given in green so um tested positive that's what that was the first piece of information and so we're looking at all the people who test positive what is the probability they actually have it so after identifying that probability they truly have it actually have coveted so you want to kind of think backwards on these um and you always want to look at the condition first and my um favorite way to write this when it says find the probability this probability here i always go p um second information given first information so this is this is the condition so put in parenthesis here condition the thing that comes first look so tip look for [Music] the thing that happens first not that is necessarily mentioned first but happens first this is the condition and that's right here that we've got a group of people they tested positive so our condition is tested positive and over here we've got has covered so when i'm setting this up i always remember i know probability is always number of successes over total and the total is the condition right here i'm going to be looking at all the people who tested positive so i'm only looking at this column of data here is my eyeball looking right there so how many people tested positive 178. okay so my eyeballs are here this is all i'm looking at and nothing else matters so if we were meeting in person i'd cover up the rest of this i think what i'll do instead is i'll do a big bold black slash slash slash so i'm not looking at that part at all um so my eyeballs are only on the green column so if that's my whole world how many people within that subset subgroup have coven well how many people have covid 160. so it's going to be 160. and i've got my fraction go ahead pull out your calculator and you will get equals to point zero point eight nine eight eight seven six dot dot dot around to four places past one two three four so it's not going to stay an eight it's going to go up to a nine so that equals 0.8989 so if i ask you to calculate a probability in our class the default is to just give me the decimal if i ask you to round this to percent it would be 89.89 if i asked you to round it to a whole percent 89.89 so that would be closer to 90 if you round it to a whole percent but i didn't ask you to do that all right so that's practicing that one um and i'm just gonna now get rid of these big black lines because um we're going to ask a different question so i'd like you to pause and try to do the next one and remember the tip the tip is that you want to look for the condition and that's going to be your whole row or your whole column and it'll help you set up your denominator so go ahead and pause and look for that and to work it out okay welcome back all right let's see how we did um so the question is if someone tests negative for covin what's the probability that they truly have it do not have it do not don't skip that do not have it so the given [Music] is asking you what comes first that goes here and that's going to be and i think i'll pick a different color because let's see i'll do turquoise okay so if someone tests negative what's the probability that they truly do not have it so the first thing that happens is they test negative so we are interested in only the people that test negative so that means that we're ignoring everything over here and we're ignoring everything over here and we're only focusing on this column of data okay and so the given how many people so tested negative we're only looking at tested negative only looking at these people um and of those people do not have it do not have it okay so taking that apart what's the probability so i'm going to use my p of do not have covid given tested negative and this helps me because now i know that's my denominator it's kind of like a fraction there so i'm going to put in the 8 22 and then do not have covid do not have coven is going to be so this whole column do not have covet is going to be this one i'm worried about obliterating that the information so i'll put it in do not have covet is 8 7 82 7 8 2. so within that group do not have covet and then if i pull out my calculator the decimal you're going to get is 0.951 338 dot dot we go to four places past one two three four that three stays a three so zero point nine five one three so there's my answer if i asked you what it would be as a whole percent well that's 95.13 which is about 95 so about 95 of the people who um tested negative actually don't have covet um so five percent are going to be wandering around that have coveted and i think actually this these results were taken a long time ago before so many people had access to the rapid test and i think the rapid tests have just a terrible uh false uh negative result my daughter just had coveted and she took four tests before one came back um and it really had to wait for her to have symptoms okay so that's problem number four we're gonna switch gears now talk about something a little bit more pleasant and that is pregnancy i don't know maybe i actually i can't i hope maybe it's a horrible thing to get pregnant it depends on who you are um so in a certain pregnancy test a certain pregnancy test detects 90 of pregnancies so 90 of the time that you're pregnant the test will say you're pregnant okay so i'm going to highlight that uh pregnancy let's do that in pink so 90 of pregnancies so it's right ninety percent of the time if you're pregnant um it gives a negative result saying you're not pregnant 99 of the time it gives a negative test result let me make sure oh no 99 of the time if you are not pregnant it tells you you're not pregnant that is a different piece of information negative test result 99 of the time for people who are not pregnant okay two different pieces of information i'm going to write this up i'm going to keep track of this so 90 percent 90 percent positive for pregnant people okay and then it gives a 99 negative for not pregnant people we're going to assume women i don't know what it does for men okay that's the second piece of information and then the third piece of information is a scientist decides to study pregnancy probabilities of this test in her study 30 of the participants are actually pregnant okay 30 percent 30 percent of people of total are actually pregnant okay i think i'm ready to go okay construct a table out of and we have a grand total of a thousand people okay so what i'd like you to do is i would like you to pause and try this out on your own but before you do that let's set the table up so we're looking because it's arbitrary how the table can be set up and i'd like us all to have the same setup and i think what would be easiest on everybody's eyes is if we had a similar setup to what you see right here so we have positive test negative test total here has covered i'm going to say is pregnant here is not half of it i'm going to say not pregnant so let's set up our table so it's very very similar so it's positive first right yeah positive first so that we have so okay so this is pregnant not pregnant and moodle okay and then this way we have positive negative and total right great okay so um what's the one thing they told us so far is that there are a thousand the one thing we know for sure is there's a thousand people so let's start with that and i think i'm gonna um i wish i made my lines a little shorter so so pause it and well let's do this i'll just make these a little shorter so there we have a little room to work with here okay pause it and try and work it out on your own okay here it goes um welcome back so i know i'm going to start with actually the orange piece of information first this one right here i know that 30 of the total are actually pregnant so 30 are actually pregnant so here's pregnant of all so some people are going to be positive they're going to be pregnant and they're going to be told they're pregnant some people are going to be told they're pregnant but they're going to be told they're not pregnant and there is but we know it all in all thirty percent of total are are actually pregnant and so that's going to be thirty percent of the total and we know that the total is a thousand are pregnant so that's going to end up being um 300 people so that's for sure and so if we know how many people are pregnant can we figure out how many people are not pregnant well i'm gonna just uh go ahead and say here that for this one right here i know that 300 plus whatever this is is going to be the grand total so to figure this one out all i need to do is do a thousand minus 300. so 700 people are not pregnant and this in this particular example um so this is this filling in the damn box this is going pretty well so i have just i believe i have exhausted this 30 so moving on and it doesn't matter which one i want to start with i think i'll just start with this one 90 positive for people who are pregnant so how many people are pregnant 300. so if i know that 300 people are pregnant and i know in general of all the people who are pregnant 90 will be getting the correct result then i can come over here this is 90 percent of pregnant of all pregnant people are going to get that positive result so it's going to end up being 90 percent how many people are pregnant 300 so if you pull out your calculator on that 90 of 300 is 270. so 270 people are pregnant and told they're pregnant because there is a 90 correct rating on that so can we now figure out how many people who actually were pregnant got the result that they were not pregnant well we know that this plus this is going to be this so to figure this one out we'll just take the total pregnant which is 300 and subtract those that were told that they were pregnant and we get 30 people are given a false negative could that be bad it could be because you could if you don't think you're pregnant you could continue smoking you could continue drinking you could continue doing a lot of things a lot of pregnant women in the first they don't know they're pregnant in the first few weeks even months and they feel terrible about it trust me it turns out okay just do the best you can there's a lot of guilt associated with being pregnant and you just want to do the best you can so once you find out then don't smoke and drink but before that give yourself a break all right so i think i have exhausted this one oops i think i use the 90 so moving on to the 99 99 negative for the not pregnant people so that's a good result you are actually not pregnant and you are told that you are not pregnant so this is we know there's a 99 correct not pregnant when you're not pregnant uh negative when you're not pregnant so this is 99 of not pregnant are told they're not pregnant and that's the correct result and it goes right here so it's going to be 0.99 of all those how many people are not pregnant 700. so if we do 99 0.99 of 700 we will get 6 93 okay so that's that one so if we know that one can we figure out this one well we know of all the people who are not pregnant some are told that they're not pregnant some are told that they're not pregnant but some are actually falsely told that they are pregnant so what are the consequences of that well if you really want to be pregnant and you're told that you're pregnant it could be just devastatingly heartbreaking if it could be if you don't want to be pregnant and you're told that you're not pregnant and you are pregnant you could um you could be falsely relieved or you could have a rude awakening later on but anyway we know that that this number right here is going to be the difference between the total and those who were told that they were not pregnant and so when i work that out it's going to be seven people seven people were told that they were not pregnant oh we're told that they were pregnant when they were not pregnant okay so i think that i can now go ahead and figure out everything else okay so this the subtotal should be pretty straightforward now if um if we know that 270 people tested positive pregnant seven tested positive not pregnant this is going to be 200 whoops this will be 277 and with that same logic to get this one we're just going to add 30 plus 6 93 and we'll get 723 so that's the hard part is making the table so on an exam if i asked you to make a table and then i asked you to do probabilities you get you would get graded to make the table but then based on the table your probabilities uh you get full credit for whatever you put into the table so hang in there do your best the best you can okay so for number six if a randomly selected study participant receives a positive test what is the probability that she's truly pregnant so the first thing that happens is she receives a positive test so this is the given this is the condition or condition um so i'm interested in the probability that from those who test positive what is the probability that she was actually pregnant okay so we're only interested in looking at the people who tested positive so that's going to be all of these people and from that group how many actually are pregnant it's gonna be i better pick a different color a lighter color actually pregnant is going to be these people right here so my denominator is the 277 and my numerator is the 270. and when i work that out the decimal i got was [Music] 0.974729 dot dot dot so if we want to round to four places past one two three four the seven stays a seven so 0.9747 or um if you wanted to round to a whole percent that's 97.47 percent which would be around 97 closer to 97 than 98 but this is what they asked for so the next one for number seven if a randomly selected participant receives a negative test what is the probability they are truly not pregnant so the condition that we're interested in the thing that comes first is that they received a negative test so that is their condition given or condition so i am only interested in the people who are who received a negative test so i'm only looking at that column of data so p it's going to be actually received a negative test tested negative so from that group of people right here what is the probability that they're not pregnant so the not pregnant is going to be these people right here so from that orange column of data we're interested in the not pregnant tested not pregnant so it's going to be actually whoops oh my god back actually not pregnant so i'm going to start that goes there and that goes there so we've got 723 on the bottom and on the top we have 693 so 693 were not pregnant who tested negative and so when i worked this out in the um decimal i get that's about right and if i want to go to four places past it's going to stay it's not going to round up this time oh it didn't last time either 0.9585 and if i wanted to know the whole percent it's move the two decimal places over it's closer to 96 um okay so that was number seven so um again this is your last practice so turn the um set your table up the exact same way so it's going to be positive negative pregnant not pregnant um and then go ahead and try and make the table yourself so pause the camera and i'm gonna do the same okay i think we're recording again let's see [Music] yep we're recording so um i started my table and um the um everything is very deja vu a certain pregnancy test detects 90 of the pregnancies exactly like before and just like before 99 of the people who are not pregnant receive a negative test result so that's good the thing that's different is that now instead of 30 of the people actually being pregnant 80 of the people are actually pregnant so i'll make a little note of all of this up above here um 90 of pregnant people receive positive and that's a good thing positive test and what that means that tells me right here that 90 of the grand total are going to receive that positive test and then the second one [Music] tells me that 99 percent of non-pregnant people receive right before you hope that's right a negative test so that tells me that they receive a negative test if they are not pregnant so it's going to be right here that's where that cell value is going to be and last but not least eighty percent of people are actually pregnant and that is going to go right here because they're actually pregnant and so it's 80 percent of total are actually pregnant and so that's a little that's the only thing that is different um in the previous problem the only information that was different is that 30 of the total is actually pregnant so let's go ahead and fill in these boxes and we just like before we have a grand total of a thousand women so i'll pop that right here so there's a thousand so from that i can figure out that i know my total is a thousand and eighty percent so it's going to be point eight of and so that's going to be 800 people now are pregnant in this group so way more pregnant ladies than this one and so from that i can figure out that this right here is the grand total take away the pregnant women are going to leave me the non-pregnant women which is 200 people so they kind of swap places here a little bit here it was 300 pregnant and 700 not pregnant okay and moving on um did i you did i i fully okay i'm going to now look at this information here 90 of the pregnant people receive a positive test so i know that that's this result right here so it's ninety percent ninety percent of and it's now eight hundred so ninety percent of the pregnant women so that'll be 0.9 times 800 and so it'll be 720 and if that's 720 and this is 800 i know that this will be 800 minus 720 which will be 80. so that means that 80 pregnant women are told they're not pregnant so maybe they'll be smoking and drinking and doing other things or being really sad because they're not pregnant and that's going to be a negative consequence so moving on to the next group um if i look here 99 of non-pregnant people receive a negative so that's good because it's accurate so that's going to be 99 of the non-pregnant which is 200 0.99 times 200 um is going to be 198. [Music] 198 of the people who are not pregnant will be told they're not pregnant which means that two women are going to be told that they're pregnant when they're not pregnant and i got that by subtracting 198 from 200 and now i can go ahead and figure out the totals so here this is going to be i'll add them together 720 plus 2 is 722 and similarly here 80 plus 198 is going to give me 278. so that was the hard part so for number nine um if you randomly select a study participant and she's positive what is the probability she's actually pregnant so what's the probability she's pregnant when she tests but it's given that she's tested positive so the condition that we're interested in is she tested positive so we are only interested in this column so our eyeballs are just looking at that column right there nothing else exists but the orange column so if those 722 people who tested positive how many are actually pregnant is going to be 720 and that is 0.99722 dot which rounds to 0.9972 which is almost if we round it to two places past whole percentage it would be a hundred percent which never happens okay and the probability for this one if a randomly selected participant receives a negative test what is the probability that they are not pregnant so the condition here is that's what happened first is they received a negative test so we're only interested in the blue column and so [Music] tested negative so we're only looking at that pool of people so what is the probability that given that they tested negative what's the probability that they're actually not pregnant so not pregnant so my eyeballs are on this column and are not pregnant is going to be 198. so setting it up we're only looking at these people 278 people tested negative and of those people who are actually not pregnant is going to be 198 and when we work out that decimal 198 it turns out to be 0.712230 which rounds to 7.7122 so 71 if you test negative then the probability that you're actually not pregnant is only 71 which is really quite low and um so that's it um we're gonna stop there and uh go ahead and do your practice and um and i'll see you next time okay bye bye you