Finding area-based probabilities

0:00- [Instructor] So what we have drawn here or depicted here are two dartboards that are each perfect squares. They're 10 centimeters by 10 centimeters. Now what we're going to do is think about throwing darts at these dartboards. And let's say when we throw darts, we're equally likely to land anywhere on this dartboard.

0:20What I wanna do is think about in which dartboard do I have a higher probability of landing in the gray? So pause this video and have a go at that before we do this together. Okay, so how do we figure out the probability that we're landing in the gray? Sometimes people say, "Oh, well, lemme just look at how many sections are gray compared to how many are white.

0:43That won't quite work, because they don't all have the same area. So what we really just wanna do is figure out what is the gray area over the total area. And we can already see the total area is gonna be 100 square centimeters, 10 by 10. So let's compare the gray areas of the left one and the right one, and whichever one has more gray area, that one has a higher probability of throwing a dart there.

1:08So let's look at this first example. How do we figure out the gray area? Well, there's two ways of thinking about this. What we have embedded here, and I should have said this, these are squares embedded in each other. And here, these are quarter circles that are embedded, or at least part of this picture, I guess, is how you could talk about it.

1:27But if we look at this one, you might be able to recognize that these lengths are going to be all the same. Another way to think about it is the corners of these inner squares are at the midpoint. And so you could do something like use the Pythagorean theorem to figure out what this length right over here is going to be.

1:51We know that this is 5 cm, we know that this is 5 cm. You can then figure that out, then we could figure out the area of this square, and then you could use the Pythagorean theorem again, taking this part and this part, and then figuring out that, and then subtracting that out.

2:07I propose doing it another way, a little bit more visually, so to speak. If you were to imagine, let's imagine, let me draw a little bit smaller right over, let me do it over here, I'm running outta space. So if you have a square like this, and I'm gonna cut it up into four equal quadrants.

2:30You have your inner square that is like that. Your inner square is like that. And if we think about the area of the inner square, notice, we'd be shading in half of those quadrants. So one way to think about it is when you have a square embedded in a square like this, the embedded square has half the area of the outer square.

2:52So, for example, we know that the larger square has an area, this larger one out here, that has an area of 100 square centimeters. And then if we wanted to figure out, or actually, let me just start with the gray square. The gray square is going to have half that area, so it's going to have an area of 50 square centimeters.

3:16And then from that we are going to wanna subtract out this white square. Well, that's going to have half the area of the 50. So we're gonna subtract out the 25. And so we are going to have a gray area of 25 square centimeters on this left-hand side. That's this plus this plus this plus this.

3:37Now let's think about the gray area on the right-hand side. So the key realization is, is it's really just the area of the whole thing minus the area of these two quarter circles. And we know how to find the area of a circle or a quarter circle if we know its radius.

3:51So how do we figure out the radius of either of these quarter circles? Well, a key insight might be, let's draw another diagonal over here. I'm just hand-drawing it, so it's not perfectly, shaky hand there. So these diagonals are gonna intersect at a right angle.

4:06So this is gonna be a right angle. So we know that this length and this length are going to be the same, so we can use the Pythagorean theorem again, because the hypotenuse, if this is the right angle, the hypotenuse is 10 centimeters. So we could say, let's call each of those lengths X.

4:26So if this is X, and if this is X, we could say that X squared + X squared is = 10 squared, is = 100. Or we could say that just adding these two X squareds, we could say that 2X squared is = 100. Divide both sides by 2, we get X squared is = 50, or that X is = to the square root of 50.

4:57I could keep simplifying it, but that's going to serve us well. The reason why this is useful is because X is the radius of either of these quarter circles. Let me do that. X is the radius. So if we wanted to find the area of the entire circle, if I were to just keep drawing this and going all the way around, we know that area is equal to pi r2.

5:24In this case, it would be equal to pi x the square root of 50 squared, which is going to be equal to 50 pi. Now we have to be careful. 50 pi would be if we had the whole circle. But we have two quarter circles. So two quarter circles is equivalent to the half circle.

5:43And so this white area of these two are going to be 25 pi. So if we were to add these two areas, they're going to be 25 pi. They make a half circle of area, which is half of the entire circle's area. So what is the area in gray? It's not gonna be the 25 pi. It's going to be the bigger square minus the 25 pi.

6:07So it is going to be the area of, the gray area, I should say, is going to be 100, the area of the whole thing, minus 25 pi, and that's going to be in square centimeters. Now which one has a bigger gray area? Well, one way to think about it. We already know this one on the left is 25 square centimeters.

6:31Here on the right, if pi were equal to 3, then 100 - 3 x 25 would get us to 25 square centimeters. But pi is greater than 3. So this right over here, this part, is going to be greater than 75. So if you subtract something larger than 75 from 100, then this whole thing is going to be less than 25 square centimeters, which is all we need to know in order to compare them.

7:07If this gray area is less than 25 square centimeters, and the one on the left is 25 square centimeters, then you have a higher probability of hitting the gray area on the left one. Now, if you wanna know what the actual probabilities are, well, then you just would put it over the total area.

7:23So in this case, it would be 25 over 100, which is a 1/4 probability of hitting the gray area. While in this one, it's a little bit more complicated of an expression, it would be 100 - 25 pi. Lemme make sure I have enough space to write all of that. It would be 100 - 25 pi over 100, which we were able to already reason is going to be less than 1/4.

7:55And we are done.