I have this task to solve:
A father suggests two algorithms to divide a round pie between his sons: A) The elder son gets 2/3, and the younger son gets 1/3; B) The pie is cut along the line passing through two points chosen at random at its circumference, and the younger son gets the smaller piece. Which algorithm gives a larger mathematical expectation of the younger son's part?
EDIT: I have a brutal error below. My only question is:
What is the name of this class of problems? What topic should I read to be able to solve these?
To model algorithm B):
Let's take just half a circle and put it on a graph in such a way that the first randomly chosen point lies exactly at (0,0). Let radius $r = 1$
Then let's choose another random point and define the equations for these functions:
$y_l = kx$ for the green line, $b = 0$ because $y_l(0) = 0$
$y_c = \sqrt{1 - (x - 1)^2}, y > 0, x > 0$ for the red line-circumference,
where the first 1 is radius
The $S$ area on the graph is equal to the part which will be given to the younger brother.
Then the area of $S$ must be equal
EDIT: the error is in subtracting $\int_0^2 y_l \mathrm{d}x$ as this subtracts extra area in the top-right corner
\begin{align*} S &= \int_0^2 y_c \mathrm{d}x - \int_0^2 y_l \mathrm{d}x \\ & = \int_0^2 \sqrt{1 - (x - 1)^2} \mathrm{d}x - \int_0^2 kx \mathrm{d}x \\ & \text{let's substitute} \, x - 1 = sin(u) \:\text{and}\: \mathrm{d}x = cos(u)\mathrm{d}u \: \text{for 1st eq. and keep int. intervals as a and b for simplicity} \\ & = \int_a^b \sqrt{1 - sin^2(u)}\:cos(u)\mathrm{d}u - (\frac{kx^2}{2})|_0^2 \\ & = \int_a^b cos^2(u)\mathrm{d}u - (\frac{kx^2}{2})|_0^2 \\ & = \int_b^b (\frac{1}{2} + \frac{cos(2u)}{2}) \mathrm{d}u - (\frac{kx^2}{2})|_0^2 \\ & = (\frac{1}{2}u - \frac{sin(2u)}{4})|_a^b - (\frac{kx^2}{2})|_0^2 \\ & \mid\:sin(2u) = 2sin(u)cos(u) = 2sin(u)\sqrt{1 - sin^2(u)}\:\mid \\ & = (\frac{arcsin(x-1)}{2} - \frac{(x - 1)\sqrt{1-(x-1)^2}}{2})|_0^2 - (\frac{kx^2}{2})|_0^2 \\ & = [(\frac{\pi}{4} - 0) - (\frac{-\pi}{4} - 0)] - (2k -0) \\ & = \frac{\pi}{2} - 2k \end{align*}
So I have that in the algorithm A) the younger son gets 1/3 of a circle, $S_c = \pi \times 1^2 = \pi$, so it's $\frac{\pi}{3}$.
In the algorithm B) the younger son gets $S(k) = \frac{\pi}{2} - 2k$.
And the question is: Which algorithm gives a larger mathematical expectation of the younger son's part?
I have 2 questions:
Where do I go from there?
What is the name of this class of problems? What topic should I read to be able to solve these?
This is a problem of geometric probability (not to be confused with the geometric distribution). There's a geometric-probability tag here on math.SE with $348$ questions that you can look at to get ideas for such problems.