I would like to share the solution of a very simple problem. I tried solving it during an exam. It looked really complex (to me) at first but then I finally did it!

Problem: There is a square of side . There are two circles drawn with radius equal to the side of the square. What is the probability of a particle to be found in the shaded (yellow) region?

Usually when we solve the probability problems (not that I am an expert), the formula that we use is, (number of desired items e.g. red balls) divided by (total number of items e.g. red and yellow balls). Similarly, here we can use, (area of shaded region) divided by (total area). To calculate it, we need to figure out the area of the yellow region.

It can be seen, that there are two circles here. And the area of circle is, , but the circle inside the square is just , so area of first circle inside the square is, . The area of square itself will be . Now if we subtract the area of the circle from the square, we get the area of one purple region. . The area of second purple region will also be the same, and hence the total area is, . In the end what we need to do is, subtract this from the area of square to get the area of yellow region. Which is going to be, .

Finally, we can get the probability of the particle to be found in the yellow region, which is, .