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# Intro to theoretical probability

AP.STATS:
UNC‑2 (EU)
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UNC‑2.A (LO)
,
UNC‑2.A.1 (EK)
,
UNC‑2.A.2 (EK)
,
UNC‑2.A.3 (EK)
,
VAR‑4 (EU)
,
VAR‑4.A (LO)
,
VAR‑4.A.1 (EK)
,
VAR‑4.A.2 (EK)
,
VAR‑4.A.3 (EK)
CCSS.Math:

## Video transcript

what I want to do in this video is give you at least a basic overview of probability probability a word that you've probably heard a lot of and you are probably a little bit familiar with it but hopefully this will give you a little deeper understanding so let's say that I have let's say that I have a fair coin over here and so when I talk about a fair coin I mean that it has an equal chance of landing on one side or another so you can maybe view it as the sides are equal the weight is the same on either side it's not if I flip it in the air it's not more likely to land on one side or the other it's equally likely and so you have one side of this coin so this would be the heads I guess try to draw try to draw George Washington I'll assume it's a quarter of some kind and then the other side of course is the tails so that is heads the other side right over there is tails and so if I were to ask you what is the probability I'm going to flip a coin I'm going to flip a coin and I want to know what is the probability of getting heads and I could write that like this the probability of getting heads and you probably just based on that question have a sense of what probability is asking it's asking for some type of some type of way of getting your hands around an event that's fundamentally random we don't know whether it's heads or tails but we can start to describe the chances of it being heads or tails and we'll talk about different ways of describing that so one way to think about it and this is the way the probability tends to be introduced in textbooks as you say well look how many different equally likely possibilities are there so how many equally likely possibilities so number of equally let me write equally of equally likely equally likely possibilities possibilities and of the number of equally possibilities I care about the number that have that contain my event right here so the number of possibilities possibilities that meet my constraint that meet my conditions that meet my conditions so in the case of the probability of figuring out heads what is the number of equally likely possibilities well there's only two possibilities we're just not we're assuming that the coin can't land on its corner and just stand straight up we're assuming that it lands flat so there's two possibilities here two equally likely possibilities you could either get heads or you could get tails and what's the number of possibilities that meet my conditions well there's only one the condition of heads so it'll be one over two so the one way to think about it is the probability of getting heads is equal to one over two is equal to one-half if I wanted to write that as a percentage we know that 1/2 is the same thing as 50% now another way to think about or conceptualize probability that will give you this exact same answer is to say well if I were to run the experiment of flipping a coin so this flip you view this as an experiment I know this is a kind of experiment that you're used to you know you'd only think an experiment is doing something with in chemistry or physics or all the rest but an experiment is every time you do you run this random event so one way to think about probability is if I were to do this experiment an experiment many many many times if I do it a thousand times or a million times or a billion times or trillion times and the more the better what percentage of those would give me what I care about what percentage of those would give me heads and so another way to think about this 50 percent probability of getting heads is if I were to run this experiment tons of times if I were to run this forever though and closer or an infinite number of times what percentage of those would be heads you would get this 50 percent and you can run that simulation you can flip a coin and it's actually a fun thing to do I encourage you to do it if you put take a hundred or two hundred quarters or penny stick them in a big box shake the box so you're kind of simultaneously flipping all of the quote all of the coins and then count how many of those are going to be heads and you're going to see that the larger the number that you are you are you are doing the more likely you're going to get something really close to 50% there's always some chance even if you flip the coin a million times there's some SuperDuper small chance that you get all tails but the more you do the more likely that you're going to get you're going to that things are going to trend towards 50% of them are going to be heads now let's just apply these same ideas and write well we're starting with probability cut at least kind of the basic this is probably an easier thing to conceptualize but a lot of times this is actually a helpful one to this idea that if you run the experiment many many many many times what percentage of those trials are going to give you what you're asking for in this case it was heads now let's do another very typical example when you first learn probability and this is the idea of rolling a die so here's my die right over here and of course you have you know you have the different sides of the die so that's the one that's the two that's the three and what I want to do and we know of course that there are and I'm assuming this is a fair die and so there are six equally likely possibilities you could when you roll this you could get a 1 a 2 a 3 a 4 a 5 or 6 and they are all equally likely so if I were to ask you what is the probability what is the probability given that I'm rolling a fair die so the experiment is rolling this fair die what is the probability of getting a 1 well what are the number of equally likely possibilities well I have six equally likely possibilities and how many of those meet my conditions well only one of them meets my condition that right there so there is a 1/6 probability of rolling a 1 what is the probability what is the probability of rolling a 1 or a 6 well once again there are six equal equally there are six equally likely possibilities for what I can get and there's two there are now two possibilities that meet my conditions I could roll a 1 or I could roll a 6 so now there's two equally likely possibilities that meet my constraints my conditions so this is there's a 1/3 probability of rolling a 1 or a 6 now what is what is the probability this might seem a little silly to even ask this question but I'll ask it just to make it clear what is the probability of rolling a 2 & 2 and a 3 and I'm just talking about one roll of the die well in any roll of the die I can only get a 2 or a 3 I'm not talking about taking two rolls of this die so in this situation it is there's six there are six possibilities but none of these possibilities are two and a three none of these are two and a three two and three cannot exist on one trial you cannot get a two and a three in the same in the same experiment these are getting a 2 and a 3 are mutually exclusive events they cannot happen at the same time so the probability of this is actually 0 there's no way to roll this normal die and all of a sudden you get a 2 and a 3 in fact and I don't want to confuse you with that because it's just you know it's kind of abstracted impossible so let's cross this out right over here now what is the probability what is the probability of getting an even number of getting an even number so once again you have six equally likely possibilities when I roll that die and which of these which of these possibilities meet my conditions the condition of being even well 2 is even 4 is even and 6 is even so three of the possibilities meet my conditions meet my constraints so this is one half if I roll a die I have a 1/2 chance of getting an even number