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## Statistics and probability

### Unit 9: Lesson 2

Continuous random variables# 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)

Examples finding probabilities from probability distributions for continuous random variables.

## Video transcript

- [Instructor] 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 one to five and has an equal probability of taking on any of these
values from one to five. Find the probability
that x is less than four. So x can go from one to four. There's no probability that
it'll be less than one. So we know the entire area
under the density curve is going to be one. So if we can find the fraction of the area that meets our criteria then we know the answer to the question. So what we're gonna look at is
we wanna go from one to four. The reason why I know we can start at one is there's no probability,
there's zero chances that I'll get a value less than one. 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 one, two, three. So our area's going to be 0.25 times three which is equal to 0.75. So the probability that x
is less than four is 0.75 or you could say it's a 75% probability. Let's do another one of these with a slightly more
involved density curve. A set of middle school students' heights are normally distributed with
a 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 this 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 be 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% or approximately 84% 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% or 0.84. Well, then we just subtract that from one to get this blue area. So this is going to be one minus 0.84 or 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 84.1% in which case this would
be about 15.9% or 0.159 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.