Get Ahead in Geometric Probability: Expert Guidance and Real-World Applications

An interesting identity about the expectation $\mathbb{E}(X{\mathbb{I}}_{\{X<a\}}{\mathbb{I}}_{\{Y<b\}})$
Given (X,Y) follows a bivariate normal distribution $\mathcal{N}(\left(\begin{array}{c}0\\ 0\end{array}\right);\left(\begin{array}{cc}1& r\\ r& 1\end{array}\right))$ with $-<!--\; -\; -->1<r<1$

Geometric probability question, find the probability that the area of the triagle is more than twelve.
$|AB|=8,|CD|=6$. We choose the point P by chance on AB and again a point Q by chance on CD find the probability so the the triangle whose height is AP, and base is CQ is larger than 12. The answer is in the picture in it's condesed form, can anyone provide an explanation?

$y=\frac{24}{x},p={\int <!--\; \int \; -->}_4^8\frac{(6-<!--\; -\; -->\frac{24}{x})dx}{48}$

How to recognize what type of probability distribution to use in solving probability problems?
Bernoulli's, Binomial, Geometric, Hypergeometric, Negative binomial, Poisson's, Uniform, Exponential, Normal, Gamma, Beta, Chi square, Student's distribution.
I would like to know how and when to use each of these distributions when solving problems in probability. If possible, make analogy with combinatorics (when we use permutations, variations and combinations).

A certain discrete random variable has probability generating function: ${\mathrm{\pi <!--\; \pi \; -->}}_x(q)={\displaystyle \frac{1}{3}}{\displaystyle \frac{2+q}{2-<!--\; -\; -->q}}$
Compute $p(x)\text{for}x=0,1,2,3,4,5$. (Hint: the formula for summing a geometric series will help you expand the denominator)."

Geometric probability
Inside a square of side 2 units , five points are marked at random. What is the probability that there are at least two points such that the distance between them is at most $\sqrt{2}$ units?

Finding probability distribution of quantity depending on other distributions
I have a vector that depends on the coordinates of randomly drawn unit vectors in ${\mathbb{R}}^2$:
$\mathrm{\sigma <!--\; \sigma \; -->}=\left(\begin{array}{c}\frac{\cos <!--\; \; -->({\mathrm{\theta <!--\; \theta \; -->}}_a)+\cos <!--\; \; -->({\mathrm{\theta <!--\; \theta \; -->}}_b)}{1+\cos <!--\; \; -->({\mathrm{\theta <!--\; \theta \; -->}}_a)\cos <!--\; \; -->({\mathrm{\theta <!--\; \theta \; -->}}_b)}\\ \frac{\sin <!--\; \; -->({\mathrm{\theta <!--\; \theta \; -->}}_a)+\sin <!--\; \; -->({\mathrm{\theta <!--\; \theta \; -->}}_b)}{1+\sin <!--\; \; -->({\mathrm{\theta <!--\; \theta \; -->}}_a)\sin <!--\; \; -->({\mathrm{\theta <!--\; \theta \; -->}}_b)}\end{array}\right)$
Here, ${\mathrm{\theta <!--\; \theta \; -->}}_a,{\mathrm{\theta <!--\; \theta \; -->}}_b\in <!--\; \in \; -->[0,2\mathrm{\pi <!--\; \pi \; -->})$, are drawn uniformly random from the unit circle, and are thus angles that parametrize unit vectors in ${\mathbb{R}}^2$. I am interested in figuring out a probability distribution for the coordinates of the $\mathrm{\sigma <!--\; \sigma \; -->}$-vector, but I am not well-versed in probability theory. I have been told that it is possible to somehow find a probability distribution for a quantity that depends on other distributions by somehow using the Jacobian matrix, but I have been unable to figure out how on my own.
My questions are thus:
Is there a method for finding a probability distribution for a quantity that depends on other distributions? If so, how? Does it have a name so I can learn about form other sources?
If the method exists, how would I use it on my concrete example?

The probability of winning in a shootout, using a geometric random variable
Sir Lancelot and Sir Galahad are doing a shoot out, in which they try to shoot each other while shooting in the same time at each other. The probability of Sir Lancelot to hit Sir Galahad is 0.5 and the probability of Sir Galahad to hit Sir Lancelot is 0.25. All the shots are independent.
A. What is the probability that the shoot out will end in the n's round ?
B. If it is known that after k rounds the shoot out did not end, what is the chance that is will end within two rounds ?
C. What is the chance of Sir Lancelot to win ?
D. What is the chance of Sir Galahad to win ?

Discrete Probability - geometric and uniform
A fair 20-sided die is rolled repeatedly, until a gambler decides to stop. The gambler receives the amount shown on the die when the gambler stops. The gambler decides in advance to roll the die until a value of m or greater is obtained, and then stop (where m is a fixed integer with $1<m<20$).
(a) What is the expected number of rolls (simplify)?

Probability distribution of a random variable
We roll a dice until we get 6. Knowing that we have rolled 10 times, evaluate the probability that in the next 20 rolls there will be no 6. So in this question are we supposed to use binomial distribution or geometric distribution?

Geometric distribution with given probability value.
The probability of a man hitting a target is 2/3. If he doesn't stop shooting until he hits the target for the first time, a) What is the probability of taking 5 shots to hit the target? b) Which is the least number of shots needed in order to have a probability (of hitting the target) greater than 0.95?

Geometric Probability. Sphere
Suppose a sphere with radius R. Find the probability of the event such that n selected points of the sphere are within the distance of $r=\frac{R}{2}$ from the center of the sphere. The points are uniformly distributed and selected independently.

Geometric distribution and Conditional probability problem
I got this problem in my book.. I will directly go the last part which I could not solve.
$P(goodcondition)=0.91854$
$P(badcondition)=0.08146$
a random package check is checked by a company till there is a package in a bad condition .
1. What is the probability that the company will check exactly 4 packages?
2. It is known that the first 3 packages that were checked are in good condition , what is the probability that more than 8 packages will be checked(edit: the answer should be 0.654).

Geometric Interpretation of Independence of Random Variables on the Standard Probability Space
Let $([0,1],\mathcal{R}\cap <!--\; \cap \; -->[0,1],\mathrm{\lambda <!--\; \lambda \; -->})$ be the standard probability space on [0,1] with $\mathrm{\lambda <!--\; \lambda \; -->}$, the Lebesgue measure. What do independent random variables "look like" in this space? Is there an easy geometric approach to creating independent functions on this space?

Finding the probability distribution of the sum of geometric distributions?
X and Y are both geometric distributions with success p. What it the $Pr(X+Y=n)$
Would I use a convolution with a sum for this? Do I need to define a third random variable?

Continuous/probability version of arithmetic/geometric mean inequality
If one has a random variable X, described by a finite probability distribution with equally likely possible values $x_1,\dots <!--\; \dots \; -->,x_n$, then the inequality: $E[\log <!--\; \; -->X]\le <!--\; \le \; -->\log <!--\; \; -->(E[X])$ is a reformulation of the arithmetic/geometric mean inequality.
And a necessary and sufficient condition for equality (maybe only if the distribution is concentrated at one point, for example)?

Alice throws a die 12 times then gives it to Bob, Bob throws the die 6 times. They then honestly report that the die landed 3 times on 1. What is the probability that Alice will get 1 on exactly 2 times rolling on 12 times rolling?

Find the probability that k is between 16 and 19, inclusive, for $p=0.15$. Using Geometric Distribution.
Can you give a formula for calculation?

In Geometric distribution what does it mean to include first success
I was watching the harvard stat110 course on Geometric distribution and the PMF is $P(X=k)=p{q}^k$ where p is the probability of success and $q=1-<!--\; -\; -->p$. later he mentions another r.v Y which he defined as the 1st success distribution and he mentions that this distribution, unlike the Geometric Distribution, includes the first success.
I got a bit confused as to the meaning of including the first success, in the PMF of Geo(p) there is a p, isn't this including the first success already?

Geometric Distribution Probability Problem
Suppose that X is a geometric random variable with parameter (probability of success) p.
Show that $Pr(X>a+b\mid <!--\; \mid \; -->X>a)=Pr(X>b)$.

Tossing a coin. X is the number tosses. What is the probability that X is an odd number?
A "fair" coin is tossed until the first time we get tails. Let X be the number tosses. What is the probability that X is an odd number?