Expert Help with High School Geometry

Probability of ending with 2 blue balls in a row after n draws
The scenario is, we're playing a game where we have in a box 9 balls: 4 red, 3 green, 2 blue.
- We draw one ball at a time, and note down its color and then return it for the next draw, the game stops only when we have drawn the same color twice in a row.
- What is the probability of the game ending after n draws? $P(X=n)$?
- What is the probability of the last two balls being blue?

How to use congruence and angle bisector theorem for this geometry question involving isosceles triangle?
Here is the diagram for the question:

In triangle $\mathrm{△<!--\; △\; -->}ADC$, $|AB|=|AC|=8$ and $|NC|=6$. AN is the bisector of $\mathrm{\measuredangle <!--\; \measuredangle \; -->}BAC$. $\mathrm{\measuredangle <!--\; \measuredangle \; -->}ADC=2\cdot <!--\; \cdot \; -->\mathrm{\measuredangle <!--\; \measuredangle \; -->}BCD$. $|DN|=x$. Find x.

What is the distance between (-1,4,1) and (6,-7,-2)?

Rectangle inscribed on two parabolas but not parallel to the axis
Let $f(x)=2x^2$ and $g(x)=36-<!--\; -\; -->x^2$. Let R be the closed region between them. Draw a rectangle ABCD such that A and B are points on f(x), and C and D are points on g(x), and the rectangle is not parallel to the axis, or prove that to be impossible.

The Probability the number of flips of a fair coin to achieve 3 heads is mod 3
Compute the probability that the number of flips of a fair coin needed in order to achieve a total of exactly 3 Heads will be a multiple of 3.

Finding volume of a frustum of a pyramid
I need to find the volume of a frustum of a pyramid with square base of side b, square top of side a, and height h(using integrals). I have no idea how to do questions like these, I only know to use the disc/washer/cylindrical shells methods and rotate a region around any line and find its volume. For these type of question, I find myself at a loss as to where to even start. Any hints on general about starting these kind of problems are also appreciated!

Given a circle with radius 1. Take an arbitrary starting point. Then go around the circle an infinite number of times, always drawing a point when you are one unit further. So the next point is always an arc length 1 step further. Easy to prove that all points will be different! But my question is: is each point randomly close approached? I believe this is true, but I can’t prove it!

Finding the volume of a rectangular-based pyramid using calculus?
Find the volume of a pyramid with height h and rectangular base with dimensions b and 2b.

CDF and Survival Function of Geometric Distribution
I am having trouble understanding the intuition behind the CDF and survival probability of a geometric distribution on both {0,1,…} and on {1,2,3,…}.
I know that a geometric starting from 0 is the number of failures before the first success and the geometric starting at 1 is the number of failures including the first success.
Can someone please explain the intuition behind the CDF and survival functions and explain what the formula is?
I think that the part confusing me is: Why is the following not true?
$p(X>n)={(1-<!--\; -\; -->p)}^{n+1}+{(1-<!--\; -\; -->p)}^{n+2}+...$ for a geometric starting at 0.

Geometric interpretation of multiplication of probabilities?
When dealing with abstract probability space $\mathrm{\Omega <!--\; \Omega \; -->}$ which consists of atomic events with measure ($P:\mathrm{\Omega <!--\; \Omega \; -->}\to <!--\; \to \; -->\mathbb{R}$) defined for them, it seems natural to start immediately imagining simple cases like this: $\mathrm{\Omega <!--\; \Omega \; -->}$ is some closed area in 2-D space partitioned into subareas, representing atomic events. Measure is the area. The total area of $\mathrm{\Omega <!--\; \Omega \; -->}$ is 1.
In fact this is what they sometimes picture in textbooks with the help of vienne diagrams or whatever.
This intuition works fine for simple cases, especially with adding probabilities.
But then it comes to probability multiplication: I don't understand how to interpret it within this simple model. Is there a way?.. Is there at all some mental model to think about probability multiplication in simple geometric terms, besides Lebesgue terms?

Please, help. People refer to the dimensions of things as ''2 by 4'' etc. I can't tell if the length dimension is vertical (up and down) or horizontal (side to side).

Distance between (−3,−48) and (-17,−42)?

Help with finding the volume of a solid of revolution
Revolve $x^2+4y^2=4$.
a. About $y=2$.
b. About $x=2$.
The answer at the back of the book is the same for both
$a.2\mathrm{\pi <!--\; \pi \; -->}{\int <!--\; \int \; -->}_0^{2}8\sqrt{1-<!--\; -\; -->\frac{x^2}{4}}dx=78.95684$
$b.2\mathrm{\pi <!--\; \pi \; -->}{\int <!--\; \int \; -->}_0^{1}8\sqrt{4-<!--\; -\; -->4y^2}dy=78.95684$
I tried splitting the ellipse into its upper and lower half but I still couldn't get the answer. I get that its symmetrical that's why it's multiplied to two but I don't know how you get the equation inside the integral.

Midpoint polygons?
A midpoint polygon is formed by taking the midpoints of each side of a polygon, and making a new polygon out of those points. The end result is the midpoint polygon inscribed in the polygon you started off with.
Given an arbitrary polygon, is it always the midpoint polygon of some other polygon? And if so, is this other polygon unique? Is there a simple construction to find it? And also, are there any special cases of the original question which are easier to answer?

If X is a nonnegative $\mathrm{\sigma <!--\; \sigma \; -->}$-subGaussian random variable with $P(X=0)\ge <!--\; \ge \; -->p$, what is a good upper bound for $P(X\ge <!--\; \ge \; -->h)$?

In right $\mathrm{\Delta <!--\; \Delta \; -->}ABC$, $\mathrm{\angle <!--\; \angle \; -->}C={90}^{\circ <!--\; \circ \; -->}$. E is on BC such that $AC=BE$. D is on AB such that $DE\perp <!--\; \perp \; -->BC$ . Given that $DE+BC=1$ and $BD=\frac{1}{2}$, Find $\mathrm{\angle <!--\; \angle \; -->}B$.

Given an item with a value of X, it has a 50% chance to split/double, so now we have two items, each with value X. They each have separate, and half the chance to split as their parent object, so each one has in this case 25% chance to split. How can I calculate the average number of items, given that the initial chance to split is instead Y, and assuming it can theoretically split forever, and if the splits is also capped to some number N. Maybe format from 1 to N, where N could be infinity?

Decide whether each of these integers is congruent to 3 modulo 7.
-17

Decide whether each of these integers is congruent to 3 modulo 7.
37

Suppose that a and b are integers, ${\displaystyle a\equiv 11\left(b\text{mod}19\right),\text{and}b\equiv 3\left(b\text{mod}19\right)}$. Find the integer c with ${\displaystyle 0\le c\le 18}$ such that ${\displaystyle c\equiv 13a\left(b\text{mod}19\right)}$