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

The number of Bernoulli trials required to produce exactly 1 success and at least 1 failure.
Let X be the number of Bernoulli (p) trials required to produce exactly 1 success and at least 1 failure. Find the distribution of X.
How can I answer this question if I don't know how many trials have taken place?
For context, the section this comes from was about Poisson distribution.
$(\genfrac{}{}{0ex}{}{n}{k})p^k(1-<!--\; -\; -->p{)}^{(n-<!--\; -\; -->k)}\to <!--\; \to \; -->\frac{e^{-<!--\; -\; -->\mathrm{\mu <!--\; \mu \; -->}}{\mathrm{\mu <!--\; \mu \; -->}}^k}{k!}$
$\text{as }n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}\text{ and }p\to <!--\; \to \; -->0\text{ with }np=\mathrm{\mu <!--\; \mu \; -->}$
However I read in a different book that you can use the Geometric probability model for Bernoulli trials: Geom(p)
$P(X=x)=q^{X-<!--\; -\; -->1}P$
where p is prob. of success, X is number of trials, and q is prob. of failure

Why is probability on Lie groups nice?
The question may sound weird. I am a probabilist who has heard of works connecting probability with (compact) Lie groups. What is the motivation? Is it generalisation just for the sake of generalisation?
Applebaum’s book is the classic but I am firstly interested to know why doing probability on Lie groups is a worthwhile effort. I know little lie theory from a differential geometric point of view. Is it enough for reading about connections between Lie groups and probability?

An experiment is conducted until it results in success: the first step has probability $\frac{1}{2}$ to be successful, the second step (only conducted if the first step had no success) has probability $\frac{1}{3}$ to be successful, the third step (only conducted if the first two steps had no success) has probability $\frac{1}{4}$ to be successful : if none of the steps were successful, we repeat the experiment until success is achieved. Assuming the first step has cost 2, and the second step as well as the third step have cost 1, what is the overall cost until success ?

Probability that $xy\le <!--\; \le \; -->1$ and $x/y\ge <!--\; \ge \; -->2$
Let x,y be any real numbers in the interval [0,2]. What is the probability that $xy\le <!--\; \le \; -->1$ and $x/y\ge <!--\; \ge \; -->2$?

Two points are chosen at random on a line segment of length 9 cm. The probability that the distance between these 2 points is less than 3 cm is?

Expected value for geometric probability
Points $P=(X_P,Y_P)$ and $Q=(X_Q,Y_Q)$ were independently chosen from square (−1,0),(0,-1),(1,0),(0,1) with geometric probability.
How does one find $\mathbb{E}|X_P-<!--\; -\; -->X_Q{|}^2?$
How do you even define expected value here?

A problem on geometrical probability
Two points P and Q are taken at random on a straight line OA of length a,show that the chance that $PQ><!--\; >\; -->b$, where $b<<!--\; <\; -->a$ is $(\frac{a-<!--\; -\; -->b}{a}{)}^2$

Geometric distribution expected number of rolls
A pair of dice are repeatedly rolled until the two sum to $\ge <!--\; \ge \; -->10$. The expected number of times the pair is rolled is?
I understand how to apply the geometric to get the probability of an event, but I do not understand how to use it to get the expected number of rolls (in this case) to achieve a certain result.

Relation via derivative of two angle-based probabilities for n points on a circle.
Suppose we sample n points uniformly (and independently) on the unit circle in the sense that the probability that a point lies within some circular arc is proportional to its length. For this question, we'll choose the convention to measure angles from 0 to 1 (rather than from 0 to $2\mathrm{\pi <!--\; \pi \; -->}$ when we use radians) to make the formulae come out nicer.
Then if we are given a fixed circular arc subtending an angle $0\le <!--\; \le \; -->\mathrm{\alpha <!--\; \alpha \; -->}\le <!--\; \le \; -->\frac{1}{2}$ (the bounds chosen due to technicalities from reflex angles), we know that the probability that all n points lie in this arc is ${\mathrm{\alpha <!--\; \alpha \; -->}}^n$. It is also a known result/elementary exercise that the probability that n such points lie within some arc which subtends an angle of $\mathrm{\alpha <!--\; \alpha \; -->}$ is $n{\mathrm{\alpha <!--\; \alpha \; -->}}^{n-<!--\; -\; -->1}$.
Is there some probabilistic reason that these formulae are related by differentiation with respect to $\mathrm{\alpha <!--\; \alpha \; -->}$?

Suppose a certain exam is classified as either difficult (with probability 90/92) or fair (with probability 2/92). Exams are taken one after the other. What is the probability that at least 4 difficult exams will occur before the first fair one?

Probability of Binomial twice of Geometric
I've come up with an interesting result:
Let X be the amount of failures of Bernoulli(p) until we get (p).
$X=Geo(p)$
$B=Bin(2X,\frac{1}{2})$
$P(B=X)=?$
Turns out: $P(B=X)=\sqrt{p}$
I found it using the Taylor expansion of $\frac{1}{\sqrt{1-<!--\; -\; -->p}}$, where the coefficient of $p^i$ turns out to be $P(Bin(2i,\frac{1}{2})=i)$.
I would like to see a probabilistic proof of this result.
Explanation of the process in words:
Roll a die with probability p of getting "X". Each time that we don't get "X", toss 2 balanced coins and accumulate the number of heads and tails. When you get "X", check if you got the same amount of heads and tails.
Programmatic explanation:
$i=0$
$while(!bernuli(p))\text{i++;}$
$Bin(2i,\frac{1}{2})\stackrel{?}{=}i$
Example:
If we succeed immediately (with probability p), $X=0$, and $P(Bin(2\cdot <!--\; \cdot \; -->0,\frac{1}{2})=0)=1$, thus $P(B=0)=1$, thus contributing p to the conditional sum, $p<\sqrt{p}$, and everything is alright.

What is the probability that the determinant of a matrix of order 2 is positive ,whose elements are i.u. ran.var.in the interval(0,1)?
I have been thinking about the problem for quite a while and it seems to me a problem on geometric probability albeit in four dimensions.For if the elements of the matrix are a,b,c,d(row-wise) the value of the determinant is ad-bc.the problem reduces to finding the probability P(ad-bc>0) subject to given domain and distributation of a,b,c,d.In our problem the sample space is the 4D unit cube.If we take the axes as X,Y,Z,U THE REQUISITE PROBABILITY IS GIVEN BY THE FRACTION OF 4D VOLUME OF THE UNIT CUBE FOR WHICH XY-ZU>0. I would love to see a solution along these goemetrical lines.

Finding the Probability function of the remainder.
Let X have a geometric distribution with $f(x)=p(1-p)x$, $x=0,1,2,\dots <!--\; \dots \; -->$. Find the probability function of R, the remainder when X is divided by 4.
I am stuck on the above practice question: If I say $R=0,1/4,2/4,\dots <!--\; \dots \; -->$… and then use the continuous case equivalent of geometric distribution to find the probability function of R, which is exponential distribution, will I be correct?

Proof of geometric probability mass function
Show that $P(X=n+x\mid <!--\; \mid \; -->X>n)=P(X=x)$ for $x=1,2,3\dots <!--\; \dots \; -->$ and $n=1,2,3\dots <!--\; \dots \; -->$

What is the surface area of a cylindrical cylindrical ring, or toru, ifits out side diameter is 16mm and its inside diameter is 10mm?round off your answer to the nearnest whle number wich one is correct is it A 92mm or B 118mm or C 123mm or D 385 mm wich one is correct

What is this geometric probability
In a circle of radius R two points are chosen at random(the points can be anywhere, either within the circle or on the boundary). For a fixed number c, lying between 0 and R, what is the probability that the distance between the two points will not exceed c?

How to find sum of infinite geometric series with coefficient?
Given this series, $p+p(1-<!--\; -\; -->p{)}^3+p(1-<!--\; -\; -->p{)}^6+p(1-<!--\; -\; -->p{)}^9+...$
This is an infinite geometric series with ratio less than 1 since it's probability.
$\underset{n=0}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}p(1-<!--\; -\; -->p{)}^{3n}$
Can you use geometric series sum formula? Is it $p/(1-<!--\; -\; -->(1-<!--\; -\; -->p{)}^3)$?
How do you deal with 3 that's in front of n?

100 Gambles of Roulette
I am stuck on the following problem: You go to Las Vegas with $1000 and play roulette 100 times by betting $10 on red each time. Compute the probability of losing more than $100. Hint: Each bet you have chance 18/38 of winning $10 and chance 20/38 of losing $10.
So essentially this is recurring 100 times. Now each time you do it you have an 18/38 chance of winning and 20/38 chance of losing. You have a greater chance of losing than winning, so based on speculation you need to lose 56/100 in order to lose more than $100, to counterbalance your winnings . How would I set up an equation to solve for this?

Conditional Probability in Geometric Distribution
Ms. A is taking driving test to get her driving license. The probability of passing a test is 0.3 which remains same no matter how many times she takes the test. Let X is a random variable which is the number of tests she takes to get her license.
1) Use conditional probability rule to determine her chance of getting the license on the 6th test starting from her first attempt considering it is known that she has already failed on the first 4 tests.
2) Though one can take the driving test 6 times within a six-month period, he/she can take tests until getting licensed. How many tests is Ms. A expected to take to get the license? What is the standard deviation of the number of tests she takes to get licensed?

Suppose 10 patients are to be tested for a blood disease and that the test in guaranteed to detect the disease. Furthermore, suppose that the probability that a patient has the disease is 0.02 and that presence of the disease in any patient is independent of all other patients. One option is to test all patients individually and that would require 10 tests. Another option is to combine blood samples from the 10 patients and test the combined sample. If the test is negative, then all patients are declared disease free and only 1 test is required. If, however, the combined sample tests positive, then all 10 patients are tested individually, requiring a total of 11 tests. In the latter option derive the probability distribution of , the number of tests, and determine Expected Value?
Among all the distribution models, it seems that geometric distribution seems best fit. But, when I use geometric distribution pmf and its' $EV=1/p$ , I get unreasonable answers.