# Thread: The very first probability exercise, should be easy

1. ## The very first probability exercise, should be easy

Hi,

I just began a probability course and I am already in problems. This should be extremely easy for anyone ever done these, but I'm just lost with these notations. So I'm first and foremost looking to understand that what the actual question is, not so much of the answer (because I'm sure once I get the idea I'll be able to solve these by myself).

So here's the exercise:
Perform the following set operation:

What am I supposed to do with it? k goes from 1 to infinity, making the first part of the comparison one, half, third, quarter etc. and the last part of comparison is 3, 3½, 3.67, 3.75... but what else I'm supposed to do? What kind (what form) of answer I should get? Since this is union, is the answer a 2d-area where lower left corner (so minimum of x and y) is almost in 0,0 and upper right corner (maximum of x and y) is almost in 4,4?

So step-by-step, what am I supposed to do with this exercise?

Thanks for anyone even trying to help!

2. Start by drawing some pictures so that you understand what is going on.

$\displaystyle x^2+ y^2= \frac{1}{k}$ is a circle with center at (0, 0) and radius $\displaystyle \frac{1}{\sqrt{k}}$. $\displaystyle x^2+ y^2= 4- \frac{1}{k}$ is a circle with center at (0, 0) and radius $\displaystyle \sqrt{4- \frac{1}{k}}$.

$\displaystyle \{(x, y)| \frac{1}{k}\le x^2+ y^2\le 4- \frac{1}{k}\}$ is the region on and between those two circles. When k= 1, that is the "washer" between radius 1 and radius $\displaystyle \sqrt{3}= 1.73$. When k= 2, that is the "washer" between radius $\displaystyle \frac{1}{\sqrt{2}}= .7071$ and $\displaystyle \sqrt{3.5}= 1.87$. Notice that the first set is completely contained in the second so the union of the two sets is just the second set. What does $\displaystyle \frac{1}{k}$ go to as k goes to infinity? What does $\displaystyle 4- \frac{1}{k}$ go to as k goes to infinity?

3. Ok, I think I got this. For those two questions first one goes to almost 0 (never reaching it) and similarly the second one goes to almost 4.