00:01
So we have a dart board here, it's a square dart board, and we're throwing a dart at it randomly on this board so it'll equally likely to land at any point in the square, under a certain reasonable definition of equally likely.
00:15
We want to know basically the probability that it lands closer to the edge of the board than to the center of the board.
00:25
We'll look at this in two parts.
00:30
Part a, we want to argue by symmetry is necessary only to consider one quarter of the board.
00:34
They've actually drawn on that quarter here, up top, is like the region r.
00:40
And the reason basically is it says by symmetry, the board is rotationally symmetric.
00:45
Wherever the dart lands, it's going to be in one of these four regions, which i will actually draw.
00:53
See either like this upper green region they've indicated, or this one, or either of these red ones.
01:03
And by rotating the board by 90 or 180 or maybe 270 degrees, you'll be able to land it in that region.
01:12
And as such, you only need to do analysis in that region.
01:16
That's what's meant by symmetry.
01:18
So then the idea basically here is, the closest argument to that is going to be b.
01:25
A given region is a square, and it's rotationally symmetric.
01:29
Everything here is rotationally symmetric.
01:31
So solving it for one quarter, you can extend that into the entire region.
01:38
That's good, that's nice.
01:40
Part b, find the curve in this region that is equidistant from the center of the board and the top edge of the board.
01:46
Well, let's see.
01:46
If we have a point here, x, y, in the region r, which is to say that the absolute value of x is less than or equal to the value of y, which is less than or equal to 1...