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# Addition rule for probability

AP.STATS:
VAR‑4 (EU)
,
VAR‑4.C (LO)
,
VAR‑4.C.2 (EK)
,
VAR‑4.E (LO)
,
VAR‑4.E.4 (EK)

## Video transcript

let's say have a bag and in that bag I am going to put some green cubes in that bag and in particular I am going to put I am going to put eight green cubes I'm also going to put some spheres spheres in that bag let's say I'm going to put nine spheres and these are the green spheres I'm also going to put some yellow cubes in that bag so I'm yellow cubes I'm going to put five of those and I'm also going to put some yellow spheres in this bag yellow spheres and let's say I put seven of those I'm gonna stick them all in this bag and then I'm going to shake that bag and I'm going to pour it out and I'm going to look at the first object that falls out of that bag and what I want to think about in this video is what are the probabilities of getting different types of objects so for example what is the probability what is the probability of getting a cube a cube of any of any color what is the probability of getting a cube well to think about that we should think about what are or this is one way to think about it what are all of the equally likely possibilities that might pop out of the bag well we have 8 plus 9 is 17 17 plus 5 is 22 22 plus 7 is 29 so we have 29 objects there are 29 objects in the bag that I do that right this is 14 yep 29 objects so let's draw all of the possible objects and I'll do this as I'll represent it as this big area I'll represent it as this big area right over here so these are all the possible objects there are 29 possible objects so there's 29 equal possibilities for the outcome of my experiment of seeing what pops out of that bag assuming that it's equally likely for cube or sphere to pop out first and how many of them meet our constraint of being a cube well I have eight green cubes and I have five yellow cubes so there are a total of 13 cubes and so let me draw that set of cubes so there's 13 cubes I could draw it like this there are 13 13 cubes this right here is the set of cubes this area and I'm not drawing it exact I'm approximating represents the set of all the cubes so the probability of getting a cube the number of events that meet our criteria so there's 13 possible cubes that have an equal likely chance of popping out over all of the possible equally likely events which are 29 that includes the cubes and the spheres now let's ask a different question what is the probability of getting a yellow a yellow object either a cube or a sphere so once again how many things meet our conditions here well we have five plus seven there's 12 there's 12 yellow objects in the bag so we have 29 equally likely possibilities I'll do that same color we have 29 equally likely possibilities and of those 12 meet our criteria so let's or by so let's say let me draw 12 right over here and do my best attempt so let's say it looks something like so the set of yellow objects they're 12 there are 12 objects that are yellow so the 12 that meet our conditions are 12 over all the possibilities 29 so they're probably going to cube 13 29 probability of getting a yellow 12 29 now let's ask something a little bit more interesting what is the probability what is the probability of getting a yellow cube so I'll put it in yellow so we care about the color now so this thing is yellow what is the probability of get whereas my son would say hello what is the probability of getting a yellow cube well there's 29 equally likely possibilities there's 29 equally likely possibilities and of those 29 equally likely possibilities five of those are yellow cubes or lellow cubes five of them so the probability is 529s and where would we see that on this Venn diagram that I have drawn in this Venn diagram this is just a way to visualize the different probabilities and they become interesting when you start thinking about where sets overlap or even where they don't overlap so here we're thinking about things that are members of the set yellow so they're in this set and they are cubes so this area right over here that's that's the overlap of these two sets so this area right over here here this represents things that are both yellow and cubes because they are inside both circles so this right over here let me write it let me write it over here so there's five objects that are both yellow yellow and yellow and cubes now let's ask and this is probably the most interesting thing to ask what is the probability what is the probability of getting something that is yellow or that is yellow or a cube a cube of any color a cube of any color probability of getting something that is yellow or a cube of any color well we still know that the denominator here is going to be 29 these are all of the equally likely possibilities that might jump out of the bag but what are the possibilities that meet our conditions well one way to think about it one way to think about it is well probably there's 12 things that would meet the yellow condition so that would be this entire circle right over here 12 things that meet the yellow condition so this right over here is 12 this is the number of yellow that is 12 and then 2 that we can't just add the number of cubes because if we add the number of cubes we've already counted these 5 with these 5 are counted as part of this 12 one way to think about it is there are 7 yellow objects that are not cubes those are the spheres there are five yellow objects that are cubes and then there are eight cubes that are not yellow that's one way to think about it so when we counted this 12 the number of yellow we counted all of this so we can't just add the number of cubes to it because then we would count this the middle part again so then we have to essentially count cubes the number of cubes which is 13 13 so the number of cubes number of cubes number of cubes and we'll have to subtract out this middle section right over here let me do this so subtract out the middle section right over here so minus 5 so this is the number the number of yellow cubes it feels weird to write the word yellow Green the number of yellow cubes or another way to think about it and we could just you could just do this math right here 12 plus 13 minus 5 is what it's it's 20 right it is did I do that right 12 - yep it's 20 so that's one way you just get this is equal to 20 over 29 but the more interesting thing then even did the answer of problem of the probability of getting that is expressing this in terms of the other probabilities that we figured out earlier in the video so let's think about this little a little bit we can rewrite this fraction right over here we can rewrite this as 12 over 29 plus 13 plus 13 over 29 minus 5 over 29 minus 5 over minus 5 over 29 and this was the number of yellow over the total possibilities so this right over here was the probability of getting a yellow this right over here was the number of cubes over the total possibilities so this is plus the probability of getting of getting a cube the probability of getting a cube and this right over here is the number of yellow cubes over the total possibilities so this right over here was minus the probability of yellow and and a cube and I'm going to write it that way minus the probability of yellow I'll write yellow and yellow yellow and yellow and getting a cube yellow and getting a cube and so what we've just done here and you could play with the numbers the numbers I just used as an example right here to make things a little bit concrete but you can see this is a generalizable thing if we have the probability of one condition or another condition so let me rewrite it the probability and I'll just write a little bit more generally here this gives us an interesting idea the probability of getting one condition of an object being a member of set a or or a member of set B is equal to the probability that it is a member of set a plus the probability that it is a member of set B minus the probability that it is a member of both mine is a probability member of both this is a really useful result I think sometimes it's called like the addition rule of probability but I wanted to show you it's a completely common-sense thing the reason why you can't just add these two probabilities is because they might have some overlap there's a probability of getting both and if you just added both of these you would be double counting that overlap which we've already seen earlier in this video so you have to subtract one version of the overlap out so you are not double counting it and I'll throw other one other idea out sometimes you have possibilities that have no overlap so let's say this is a set of all possibilities this is the set of all possibilities and let's say this is the set that meets condition a let's say this is set the Meuse condition a and let this is let me do this in a different color and let's say that this is the set that meets condition B so in this situation there is no overlap there's no way nothing is a member of both set a and B so in this situation the probability of a and B is zero there is no overlap and these type these type of conditions or these two events are called mutually exclusive mutually exclusive so if events are mutually exclusive that means that they both cannot happen at the same time there's no there's no event that meets both of these conditions and if things are mutually exclusive then you can say the probability of A or B is the probability of a plus B because this thing is 0 but if things are not mutually exclusive you would have to subtract out the overlap and these the probably the best way to think about it is to just always realize that you have to subtract out the overlap and obviously if something is mutually exclusive the probability of getting a and B is going to be 0