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# Probabilities from density curves

AP.STATS:
VAR‑6.A (LO)
,
VAR‑6.A.2 (EK)
,
VAR‑6.A.3 (EK)
,
VAR‑6.B (LO)
,
VAR‑6.B.1 (EK)
,
VAR‑6.B.2 (EK)

## Video transcript

consider the density curve below and so we have a density curve that describes the probability distribution for a continuous random variable this random variable can take on values from 1 to 5 and has an equal probability of taking on any of these values from 1 to 5 find the probability that X is less than 4 so X can go from 1 to 4 there's no probability that will be less than 1 so we know the entire area under the density curve is going to be 1 so if we can find a fraction of the area that meets our criteria then we know the answer to the question so what we're going to look at is we want to go from 1 to 4 the reason why I know we can start at 1 is there's no probability there's zero chances that I'll get a value less than 1 we see that from the density curve and so we just need to think about what is the area here what is this area right over here well this is just a rectangle where the height is 0.25 and the width is 1 2 3 so our area is going to be 0.25 times 3 which is equal to 0.75 so the probability that X is less than 4 is 0.75 or you could say it's a 75% probability let's do another one of these are slightly more involved in a density curve a set of middle school students Heights are normally distributed with the mean of 150 centimeters and a standard deviation of 20 centimeters let H be the height of a randomly selected student from this set find and interpret the probability that H that the height of a randomly selected student from the set is greater than 170 centimeters so let's first visualize the density curve it is a normal distribution they tell us that the mean is 150 centimeters so let me draw that so the mean that is 150 and they also say that we have a standard deviation of 20 centimeters so 20 centimeters above the mean one standard deviation above the mean is 170 one standard deviation below the mean is 130 and we want the probability of if we randomly select from these middle school students what's the probability that the height is greater than 170 so that's going to be this area under this normal distribution curve it's going to beat that area so how can we figure that out well there's several ways to do it we know that this is the area above one standard deviation above the mean you could use a Z table or you could use some generally useful knowledge about normal distributions and that's that the area between one standard deviation below the mean and one standard deviation above the mean this area right over here is roughly 68% it's closer to 68.2% for our purposes 68 will work fine and so if we're looking at just from the mean to one standard deviation above the mean it would be half of that so this is going to be approximately 34% now we also know that for a normal distribution the area below the mean is going to be 50% so we know all of that is 50% and so the combined area below 170 below one standard deviation above the mean is going to be 84 percent or approximately 84 percent and so that helps us figure out what is the area above one standard deviation above the mean which will answer our question the entire area under this density curve under any density curve is going to be equal to one and so the entire area is one this green area is 84 percent or 0.8 for well then we just subtract that from 1 to get this blue area so this is going to be 1 minus 0.8 4 I'll say approximately and so that's going to be approximately 0.16 if you want a slightly more precise value you could use a Z table the area below one standard deviation above the mean will be closer to about eighty four point one percent in which case this would be about fifteen point nine percent or 0.15 nine but you can see that we got pretty close by knowing the general rule that it's roughly 68% between one standard deviation below the mean and one standard deviation above the mean for a normal distribution
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