than the average? x minus mu is the this is the probability that you're less than that, And that should happen if we put to be really close to the mean than further away. So there's 1 minus equal to 1 if we're heads or 0 if we're tails. between whatever point we want to find. the central limit theorem. histogram or that bar chart, and you say, oh, dice-- or let's say, since it's faster to draw, the coin-- the And we'll say there's only a What is the variance of sum of the squared distances from the mean. videos if you want to learn a little bit more about point, and I subtract from that the cumulative out the mean, variance and standard deviation for our Screen Draw tool. And I really encourage What is the probability So we could be talking about www.khanacademy.org/downloads/-- and if you just type that little bit what's going on here a little bit better, right around there. And the difference arguably the most important concept in statistics. People might talk about You have to say between a range. At some point, just the way we inches of rain, then of course the probability is much higher. This is going to be, this term everything out. deviation below, and one standard deviation do down here-- let me move this down a little bit. And just so you know how to use They do tend to be are going to be to the mean. above the mean, which-- it's a number get the probability that you land between those two. written this way, but it gives me a intuitive sense. And then you say if you were to look at this and I were to ask you, what is the of all of your flips-- if you were to give yourself you just go right over here, this essentially And I cover that number is right here. from the larger thing, you're just left with what's just so you know, is downloadable at exists, and all of that. see the-- if you go to 20, you just go right to negative 2p squared you're left with negative p under the curve just under 0-- there's no area. It'd be given by this area. you'd calculate the area in the curve all the way to infinity, you, the point minus 20 is 25 less than the mean, right? sum of the squared distances from the mean. And now you know. is exactly 2 inches long. There you go. 1 or lower, right? I think this is so That makes sense, because most So let me draw a probability this-- you're like, oh, wow, there's so many from minus infinity to x of our probability The whole bell curve just shifts Now unfortunately, it at minus 15 down here, it tells you the probability I mean, one way you to an exponent, you can just multiply And it should make is 1 inch, this is 2 inches, this is 3 inches, 4 inches. to my Pen tool-- what I do is, I figure out, amount of rain tomorrow. seat I'm sitting on. Let me get out of the Pen tool. So this just tells molecular interactions, and every time compound x manufacture things, there's going to be an extra atom For those of you who haven't, Greek letters here. Is equal to the exact what the actual probability distribution function for this And if you take its square people will say that's 2. for normal distribution-- let me actually get my just another word. 1 and 1-- and I'll show you the behind The mean was minus 5. interacts with compound y, what might result doesn't have There is some to become as familiar with this formula as possible. discrete random variable, they all have to add up to 1. Let's say we're here. me put that in. deviation is 10 times the square root of 2 pi, Now we have an interval here. essentially the probability of success times the probability This might be taxing my so half of your results should be above that, And we know that our variance is green-blue color. value of Y minus is 2 is less than some tolerance? from 0 to our mean? or almost everything we do in inferential is already going on here. that you land at some value less than your x value? So if we said that the absolute the time for now. the random variable. probability that we get failure, that we get 0. you make that Excel call. So if we're doing know, is it a 0.5 chance? I talk about it in Now what's the probability anything being exactly a certain measurement to the It's not a continuous curve. Now what is going to maybe 90%, roughly. functions for it, and you can even approximate it. is p to the third. and it probably won't happen in the life of the bit with in this chart, and see what that means. standard deviation squared. distance from the mean. a suitably small number here, and a suitably probability isn't given by just reading this graph. large number here. And the these, I was going Let me do it-- between minus Then I did the And then you'll get this I don't know what phenomenon I mean, there's not a single of 10, the probability of getting 1,000 here you have this p right over here, so this is equal to p. And then when you add p squared It calculated at 7%, of that point and that point, and multiplying it by the base. Let me change colors, So once again it's a value that 1 minus p squared? You evaluate it at plus 5, a normal distribution. probability of getting a tails as well. it the cumulative distribution function-- it's a function of x. So this is 1, 0, where x is And obviously, if you add these Our mission is to provide a free, world-class education to anyone, anywhere. 1 and 1, what I do is-- let me get back tomorrow we have exactly 2 inches of rain? a probability density function. know, how far north they live from my house, is exactly 2 inches. under the curve up to x. Donate or volunteer today! And area is key. Practice: Probability in normal density curves. in the right standard. capital letters. point right here, you can see that this And then we have the It might not be obvious to you, Well, let me rewrite that. so that's a good thing. don't know, roughly 15% or so? to email me if you see some insight on why this part in, you'll see everything that's downloadable-- but I don't know. It's the probability weighted because what's this? And then this is a We get 100%, although And get an intuitive under the curve right there. And you can watch the calculus what's the probability that I land within one It's not 0. that's only because I round it. this in a second. So when the standard it's pretty high. But in a continuous So you could figure based on data points, is to some degree based on rounded, so it says there, close to 0. So that cancels out. Because when a random variable look at a population where the probability of success-- we'll two values, they are going to add to 1. plus the probability that we get a 1, which is just p-- this you're always, if someone says, oh, we're assuming a normal p times 1 is p. p times negative 2p is There's some probability the height of the function. So let's say, at minus 20, to play with the spreadsheet, and to even make a spreadsheet I've never seen it That's our x. So this is a cumulative product of this. What's the probability that than the average right here, what you deviation is a measure of-- the variance is the average around with the formula, because I really want to see don't know calculus yet, I encourage you to And then you know the Whoops. So it's just e to the this what's the probability of getting less than minus 5? This 7%, or actually 0.07, is the height is very low. in the last video, I now want to calculate the expected 0.24, which is exactly what we got in the last example. go onto Wikipedia, and if you were to type Then your probability 2 inches, that's the case this area under the curve. is the probability that I get a value less than 20, right? won't make any sense. in this example was 0.6, probability of failure times e to the minus 1/2, times x minus our mean. Here's a cumulative these exponents. And the way you get it is with

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