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

How can you tell when you need to use the binomial coefficient for probabilities?
For example, the probability of being dealt a hand with 3 spades and 2 hearts from a standard deck of cards. I would think the geometric probability is:
${\displaystyle \frac{13\ast <!--\; \ast \; -->12\ast <!--\; \ast \; -->11\ast <!--\; \ast \; -->13\ast <!--\; \ast \; -->12}{52\ast <!--\; \ast \; -->51\ast <!--\; \ast \; -->50\ast <!--\; \ast \; -->49\ast <!--\; \ast \; -->48}}$ because I usually don't use the binomial coefficient with geometric probabilities, but the answer is incorrect unless I multiply by the binomial coefficient (which is a factor of 10 here).

Suppose we roll a fair dice repeatedly. Find the probability that we get our 4th prime number on the 25th roll. I think it should be 4 successes and 21 failures but I feel it incorrect.

Geometric distribution with unequal probabilities for trials
I am researching an engineering problem in which I want to model the probability distribution of the number X of independent trials needed to get one success. If the probability of success at each trial is a constant p, the solution clearly reduces to the simple geometric distribution. However, in this particular case, the probability varies with each trial in a way that I can predict.
I feel sure there must already be a body of work on this type of problem that I can refer to, but I have been unable to find it. I have come across the Poisson Binomial distribution which models the probability distribution of the number of successes in a sequence of X independent yes/no experiments with different success probabilities. So I naturally searched for the Poisson Geometric or Geometric Poisson distribution, but this in fact refers to something quite different (a variation on the Poisson rather than Geometric distribution).
Can anyone help with the name of the distribution I am looking for?

Evaluating min/max probability with Geometric random variables
Suppose that $X_1,\dots <!--\; \dots \; -->,X_n$ are independent, ${\text{geometric}}_1(p)$ random variables. Evaluate $P(\text{min}(X_1,\dots <!--\; \dots \; -->,X_n)><!--\; >\; -->l)$ and $P(\text{max}(X_1,\dots <!--\; \dots \; -->,X_n)\le <!--\; \le \; -->l)$
The textbook solution is as follows:
$\begin{array}{rl}P(\text{min}(X_1,\dots <!--\; \dots \; -->,X_n)><!--\; >\; -->l)& =P(\underset{k=1}{\overset{n}{\bigcap <!--\; \bigcap \; -->}}\{X_k><!--\; >\; -->l\})\\ & =\underset{k=1}{\overset{n}{\prod <!--\; \prod \; -->}}P(X_k><!--\; >\; -->l)\\ & =\underset{k=1}{\overset{n}{\prod <!--\; \prod \; -->}}{p}^l\\ & =p^{ln}\end{array}$
and $\begin{array}{rl}P(\text{max}(X_1,\dots <!--\; \dots \; -->,X_n)\le <!--\; \le \; -->l)& =P(\underset{k=1}{\overset{n}{\bigcap <!--\; \bigcap \; -->}}\{X_k\le <!--\; \le \; -->l\})\\ & =\underset{k=1}{\overset{n}{\prod <!--\; \prod \; -->}}P(X_k\le <!--\; \le \; -->l)\\ & =\underset{k=1}{\overset{n}{\prod <!--\; \prod \; -->}}(1-<!--\; -\; -->p{)}^l\\ & =(1-<!--\; -\; -->p^l{)}^n\end{array}$
I'm having trouble understanding pretty much the entire solution. If I were to summarize what's particularly puzzling me:
1) How was the first line of each solution derived (i.e. $P(\text{min}(X_1,\dots <!--\; \dots \; -->,X_n><!--\; >\; -->l)=P({\cap <!--\; \cap \; -->}_{k=1}^n\{X_k><!--\; >\; -->l\})$ and the corresponding equation for max)?
2) The PMF for ${\text{Geometric}}_1(p)$ is given as $p_X(k)=(1-<!--\; -\; -->p)p^{k-<!--\; -\; -->1}$. How is $P(X_k><!--\; >\; -->l)=p^l$? The same applies for the max case as well.

Probability question about geometric distribution
Let X have a geometric distribution. Show that $P(X>k+j|X>k)=P(X>j)$, where k and j are nonnegative integers.

Geometric probability - line segment
On the segment [0,1] two numbers X, Y are being randomly selected. Find the probability that $X+Y\le <!--\; \le \; -->1$ and $X-Y>0.1$.

Joint probability of geometric random variables
X and Y are geometric, independent random variables with parameters x and y respectively. Use their joint P.M.F. to compute an approximate summation for $P(X<Y)$.

EX of area of a circle
In a circle k(O,r) we have fixed point A, point B lands randomly on the circle. Find EX of the area of the triangle AOB
What I have tried: since the area of the triangle will be $\frac{1}{2}\ast <!--\; \ast \; -->r^2(sin(\mathrm{\alpha <!--\; \alpha \; -->}))$ where $\mathrm{\alpha <!--\; \alpha \; -->}$ is from 0 to 360 degrees then we have to find the integral ${\displaystyle {\int <!--\; \int \; -->}_0^{2\mathrm{\pi <!--\; \pi \; -->}}\frac{1}{2}\ast <!--\; \ast \; -->\mathrm{\alpha <!--\; \alpha \; -->}\ast <!--\; \ast \; -->r^2(sin(\mathrm{\alpha <!--\; \alpha \; -->}))d\mathrm{\alpha <!--\; \alpha \; -->}}$.

On the interval [0, 5] we randomly and independently choose two numbers that divide this interval into three sections. The smaller number is denoted by a and the larger number by b. What is the probability event that section [a, b] will be the shortest and section [0, a] the longest?

Median from Probability distribution formula
How i can get formula for median of geometric distribution with this probability mass function formula $P_r=\frac{(1-<!--\; -\; -->\mathrm{\alpha <!--\; \alpha \; -->}){\mathrm{\alpha <!--\; \alpha \; -->}}^{CW}}{1-<!--\; -\; -->{\mathrm{\alpha <!--\; \alpha \; -->}}^{CW}}.{\mathrm{\alpha <!--\; \alpha \; -->}}^{-<!--\; -\; -->r}$

Probability of rolling a 1 before a 6 on a dice
what is the probability that I roll a "1" on a dice before rolling a "6". I do not know how to tackle this problem. I was thinking that this is a Geometric random variable but I do not know how to solve it.

Apparent paradox in Geometric probability problem
Suppose we have N particles in a box. We can consider these particles to be distinguishable, and move about the box randomly, like molecules of gas in a box. Suppose, the volume of this box is V.
I've been asked to find the probability that at a certain time, a small sub-volume $V_0$ does not have a single particle inside it.

Given two iid geometric random variables with $p=\frac{1}{5}$, what is the probability that variates from them equate?

Is the sum of two independent geometric random variables with the same success probability a geometric random variable?

Expected area of a triangle where 1 point is within another triangle
Suppose we have triangle ABC with area k with a point P chosen inside ABC. What is the expected area of triangle PBC?
I'm pretty sure if we let P be the centroid we get k/3. Also, how would you solve this question for an n-sided polygon?

Finding the Probability of a multiple using geometric distribution
Let X be a geometric distribution with parameter $p=\frac{1}{9}$ i.e. $P(X=k)=\frac{{\frac{8}{9}}^{(k-1)}}{9}$ for $k\in <!--\; \in \; -->\{1,2,3,...\}$.
Let A be the event that X is a multiple of four. What is the value of P(A)? How do I find k in this instance?

Entropy of geometric random variable?
I am wondering how to derive the entropy of a geometric random variable? Or where I can find some proof/derivation? I tried to search online, but seems not much resources is available.
Here is the probability density function of geometric distribution: $(1-<!--\; -\; -->p{)}^{k-<!--\; -\; -->1}p$
Here is the entropy of a geometric distribution: $\frac{-<!--\; -\; -->(1-<!--\; -\; -->p){\log }_2<!--\; \; -->(1-<!--\; -\; -->p)-<!--\; -\; -->p{\log }_2<!--\; \; -->p}{p}$
Where p is the probability for the event to occur during each single experiment.

Probability of 3 numbers chosen between 0 and 10 being within 1 of each other
I was trying to find the probability that 3 real numbers uniformly chosen from 0 to 10 are within 1 of each other (the largest number minus the smallest number is at most 1).
I tried using geometric probability, and I got a region that looks like what you would get if you put one vertex of a unit cube at the origin (and orient the edges such that they line up with the positive axes), and move the corner at the origin from (0,0,0) to (9,9,9) (is this region correct?). I calculated the volume by finding the area of the one of the faces of the cube (which is 1), and multiplying it by the distance it moved (which is 9). Since there are three of these squares, the volume is 27. I then add this to the volume of the cube, which results in 28. The total volume is ${10}^3=1000$, so the probability is $28/1000=7/250$.

Recursive proof of expectation of geometric distribution?
If $T\sim <!--\; \sim \; -->\mathsf{G}\mathsf{e}\mathsf{o}(p)$ ($0<p<1$) then $\mathbf{E}T=1/p$ is well known.
Suppose that I toss a sequence of coins that come up heads with probability p. On average, how many coins do I have to toss before I see a heads?