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Current time:0:00Total duration:9:10

Video transcript

I'm hoping to explore in this video is the notion of a sample space and it's a pretty hopefully you'll find straightforward idea if you're having a if you're doing a trial a rant something that is probabilistic a trial or an experiment a sample space is just all of the is the set of the possible outcomes so a very simple trial might be a coin flip so if you're talking about a coin a coin flip well then the sample space is going to be the set of all the possible outcomes so you could get a heads or you could you could get a tails that right over here is the sample space for the coin flip so and it's very useful because in for example if these are equally likely outcomes then and you say well I want what's the probability of the event of a heads you say okay that's one out of the two equally likely outcomes or you can even construct it once you know all of the possible outcomes even if they aren't equally likely you could say well let's create a probability distribution we at least know what the sample space is we know what the possible outcomes are now let's think about the probability of each of those outcomes but a lot of times when people talk about sample spaces they're often they're there they tend to be most useful I would say when you have equally likely outcomes I'd like in the in the case of a fair coin flip because then from the sample space it's fairly straightforward to think about the probability of various events but this is a simple sample space right over here but let's make things a little bit more interesting let's imagine a world so let's just put this aside a little bit let's imagine a world where there's a bakery and at that bakery there are three types three flavors of cupcakes but there's also three different sizes of cupcakes so now we're we're essentially looking at two different ways in which that the thing that we're we're going to be sampling can vary so what we're doing or do we write this down so we have our flavors flavors of cupcakes at this bakery and let's say that you have chocolate chocolate you have let's say there's strawberry strawberry and let's say that there is vanilla there is vanilla and it comes in they come in three different sizes so those sizes sizes could be small so I'll just write it out small medium or large so if you were and let's say you know each of these flavors come in each of these sizes or you could view the other way around each of these sizes come in all three flavors so now how do you construct the sample space if you say look I'm gonna go you know I'm gonna blindfold myself and walk into this bakery and and randomly you know somehow pick up a cupcake and you know my fingers can't tell the flavor or the size of the cupcake what are the possible what are the possible outcomes for the cupcake I'll pick and the up outcome would be both the flavor and the size of the cupcake well there's a bunch of ways to think about this one way is you could draw a tree you could say okay well I'm going to pick three different flavors I could either pick chocolate chocolate I'm going to pick strawberry strawberry or I'm going to pick vanilla vanilla and then for each of those flavors I'm going to pick a small medium or large so you could say small medium large small and then so this is a small chocolate this is a medium chocolate this is a large chocolate this is a small strawberry medium strawberry large strawberry this is a small vanilla medium vanilla large vanilla and so you see there's nine possible outcomes once again this is a medium chocolate you picked up chocolate and then it was a medium one this is a large vanilla you picked a vanilla and it is a large one and you could have done it the other way around you could have said well okay I'm going to either pick a small medium or large and then for each of those I'm going to pick either a chocolate strawberry or vanilla and I'll just use the first letters two so I'm either going to pick a chocolate a strawberry or vanilla when I write the s over here in this magenta color I'm talking about the flavor and if I write the s in green I'm talking about small so here you you can have if you have a medium cupcake it could be chocolate it could be strawberry or it could be vanilla you have a large cupcake it could be chocolate strawberry or or vanilla so for example this was a medium chocolate cupcake over here a medium chocolate cupcake is this one it's medium chocolate it would be that one over here so you could use these I have a tree diagram like this to think about the sample space to think about the nine possible outcomes here but you could also you could also do a I guess you could say grid or you could write the flavor so you can have chocolate chocolate actually let me just write that well let me write them out all right so let me just let me just write the letters it's going to take a long time to do so you can have the flavors chocolate strawberry vanilla so that's along that axis and then you have your sizes you can have a small a medium or large and you can set up a grid here so this is another way to do it and notice this grid has nine boxes so let's look at it so set up the grid set up the grid and so what is this one going to be this is a small chocolate small small chocolate small chocolate what is this one this is going to be a small a small strawberry a small strawberry and you could just keep constructing like this where everything in this row there's all about small whoops we do the this is small having trouble changing colors all right there is a small and then this is a small this is a small vanilla this color-changing is really that's a difficult thing some small vanilla and all of these these are all this would be a medium chocolate medium strawberry medium vanilla large chocolate large strawberry large vanilla and once again you have nine outcomes this is another way to think about all of the possible outcomes when you're looking at these two ways in which are my cupcakes could vary another way a third way that you could do it is you can literally just construct a table well you could say okay I can have a chocolate actually I'm going to use the letters again so let's say we make this is the flavor column and then this is the size column size column and so you could say I could have a chocolate that is so let's see there's three types of chocolate that I can have there and they could be they could be small medium or large you could say there is three types of three types of strawberry to be small medium or large so let me write that in small medium large or you could say well there's three types of vanilla there's three types of color-changing again three types of vanilla once again it could be small medium or large so you have these nine possibilities now the sample space the sample space isn't telling you if they're equally likely or not it's just telling you if you're going to do an experiment what are all the different possibilities the possible outcomes for that experiment now in the case where they are equally likely it can be very very useful because you could say you could do something like if you said that okay it's equally likely to pick any one of these nine outcomes you could say well what's the probability of what's the probability of getting a something that is either small or chocolate and so you can see well how many of those events out of the total actually meet that constraint but we'll do more of that in future videos that's just a little bit of a clue of why we we even care about things like sample spaces especially sample spaces like this where we're looking along two ways or multiple ways that something can vary and these types of sample spaces in particular are called compound sample spaces so these right over this is a compound sample space because we're looking at two different ways that it can vary not just two heads or tails can vary by size or by flavor and you can even have compound the sample spaces that vary in more than two ways