You've just found a neat way to place points uniformly randomly inside a unit circle: simply place points at random inside a circumscribed square -- x and y uniformly chosen on [-1, 1] -- and ignore the points near the square's corners that are outside the circle. There are other ways, but this works, and it's simple to do.

And why are you excited about this? The reason is that you've often wondered about the expected size of randomly drawn triangles inside a unit circle. And now you can find out. You sequentially place a million sets of three random points in the circle, calculating (and then averaging) the areas of the million triangles they define. And you find something pretty amazing: the million triangles had an average area that is only ~ 7.4% of the circle's area. You also note that the median area was ~ 5.4%.

OK, so that's a fairly long set-up for a pretty short puzzle. Read on.

You tell a friend about how amazingly small random triangles constrained by a circle are, and he replies with a question of his own: "That's cool," he says, "but I wonder what fraction of those triangles cover the circle's center?" You admit that was a piece of information that you did not take note of. "Oh, that's OK," your friend replies, "I think I can tell you."

What answer did your friend (correctly) come up with?

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## bonanova 85

You've just found a neat way to place points uniformly randomly inside a unit circle: simply place points at random inside a circumscribed square -- x and y uniformly chosen on [-1, 1] -- and ignore the points near the square's corners that are outside the circle. There are other ways, but this works, and it's simple to do.

And why are you excited about this? The reason is that you've often wondered about the expected size of randomly drawn triangles inside a unit circle. And now you can find out. You sequentially place a million sets of three random points in the circle, calculating (and then averaging) the areas of the million triangles they define. And you find something pretty amazing: the million triangles had an average area that is only ~ 7.4% of the circle's area. You also note that the median area was ~ 5.4%.

OK, so that's a fairly long set-up for a pretty short puzzle. Read on.

You tell a friend about how amazingly small random triangles constrained by a circle are, and he replies with a question of his own: "That's cool," he says, "but I wonder what fraction of those triangles cover the circle's center?" You admit that was a piece of information that you did not take note of. "Oh, that's OK," your friend replies, "I think I can tell you."

What answer did your friend (correctly) come up with?

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