Probabilities from density curves | Random variables | AP Statistics | Khan Academy

Consider the density curve below. 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 four. So, X can go from one to four. There's no probability that it'll be less than one, so you know the entire area under the density curve is going to be one. If we can find the fraction of the area that meets our criteria, then we know the answer to the question.

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 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. 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 to 3.

So our area is going to be 0.25 * 3, 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 cm and a standard deviation of 20 cm. Let H be the height of a randomly selected student from this set. Find and interpret the probability that H, that is, the height of a randomly selected student from the set, is greater than 170 cm.

So, let's first visualize the density curve. It is a normal distribution. They tell us that the mean is 150 cm, so let me draw that. The mean is 150, and they also say that we have a standard deviation of 20 cm. So, 20 cm above the mean, one standard deviation above the mean is 170, and one standard deviation below the mean is 130.

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.

How can we figure that out? Well, there are 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. 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.

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%.

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. So, if 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 1 - 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.