High school probability Uncovered: Discover the Secrets with Plainmath's Expert Help

Probability of first increase in ordering of iid random variables
What is the probability that the first $n-<!--\; -\; -->1$ terms of iid Unif(0,1) random draws are in decreasing order, but the first n terms are not?
I know that due to exchangability, $\mathbb{P}(X_1<X_2<\cdots <!--\; \cdots \; --><X_{n-<!--\; -\; -->1})=\frac{1}{n-<!--\; -\; -->1!}$. Why can't a similar exchangability argument hold to show that $\mathbb{P}(X_1>X_2>\cdots <!--\; \cdots \; -->>X_{n-<!--\; -\; -->1}<X_n)=\frac{1}{n!}$? My reasoning is that the distribution of $(X_1,X_2,\dots <!--\; \dots \; -->,X_n)$ is symmetric, so no matter which permutation of the random variables exist in the random vector, they will all have the same probability.

Derivatives and Integrals of Stochastic Processes?
I am trying to wrap my head around this idea and am trying to understand why this might be useful. For instance, suppose I have some stochastic process like the Brownian Motion. Why might I be interested in knowing "how quickly the Brownian Motion might change" at some point (i.e. the derivative) and the "area that the Brownian Motion might cover over two time intervals" (i.e. the integral)?
I understand this is more complicated than evaluating derivatives and integrals on deterministic functions. In a deterministic function, you only need to take one derivative and one integral to answer your question. On the other hand, each time I simulate a Stochastic Process on a computer, each realization of this Stochastic Process will look different. Thus, it is likely more challenging to take the integral and derivative of a function that can take many forms, compared to a function that can only take a single form.
But this point aside - why is it useful to know the derivative and the integral of a stochastic process? What do I gain from knowing this - how might I be able to apply this information to some real world application? For example, in financial models such as the Black-Scholes Model, are we using Stochastic Calculus to infer the amount of "risk" or "volatility" (i.e. the area under the stochastic process) that the stochastic process corresponds to over some period of time?

For a binomial distribution, which probability is not equal to the probability of 1 success in 7 trials where the probability of success is 0.277?
Select one:
a) Probability of 6.000 failures in 7 trials where the probability of success is 0.723
b) Probability of 1 success in 7 trials where the probability of failure is 0.723
c) Probability of 6.000 failures in 7 trials where the probability of failure is 0.723
d) Probability of 6.000 failures in 7 trials where the probability of success is 0.277

this is probably going to be a very dumb question, so my apologies in advance for that.
besides, i'm seeking to decide if there may be a large difference inside the quantity of testimonies that mention sweet, that point out fruit, or that mention neither candy nor fruit, among (a) the instances and the put up, (b) the times and the herald, and, (c) the publish and the usher in.
I ran this Chi-Square test with all three newspapers (from socscistatistics.com):
Chi-Square contingency table
However, is that a correct approach to make these three separate conclusions?
Or do I need to - instead - run three separate Chi-Square tests: (1) the Times and the Post (2) the Times and the Herald (3) the Post and the Herald

If ${\mathrm{\chi <!--\; \chi \; -->}}^2$=0 for a dataset, are the frequencies of the values in the contingency table all the same?
Could I say that if the ${\mathrm{\chi <!--\; \chi \; -->}}^2$ value of a dataset is 0, then the frequencies of the values of the cells in the contingency table are all the same? I have noticed that if I change the frequency of any of these values to be more than the other, that ${\mathrm{\chi <!--\; \chi \; -->}}^2$ stops being 0.
$$\begin{array}{lcc}& 1& 2\\ 1& 8& 8\\ 2& 8& 8\end{array}$$
${\mathrm{\chi <!--\; \chi \; -->}}^2=0$

Income percentile ranks are numbers from 0 to 100 that indicate where you sit in the income distribution - for example, 75 would mean you earn more income than three quarters of people, but less than the top quarter. Let's assume a child's expected income percentile is a linear function of their parents' income percentile. It turns out the intercept is enough to pin down the slope. Do you see why? If the intercept is 20 - which means the lowest-income parents have children who end up at the 20th percentile, on average - what is the slope?
This problem is from an app called probability puzzles.
My approach: I realised $y=ax+b$ could be the line for child but we don't have any information about the slope. We know that percentile is uniformly distributed between 0 and 100 so expected value is 50 but I don't see how to proceed from here.

What is another name for mutually exclusive events?

Why is Binomial Probability used here?
A test consists of 10 multiple choice questions with five choices for each question. As an experiment, you GUESS on each and every answer without even reading the questions.What is the probability of getting exactly 6 questions correct on this test?
The answer says:
$$P=(\genfrac{}{}{0ex}{}{10}{6})(0.2{)}^6(0.8{)}^4$$
I was thinking though:
$$P(A\text{and}B)=P(A)\cdot <!--\; \cdot \; -->P(B)$$
Probability of getting 6 right and 4 wrong would be just:
$$(0.2{)}^6(0.8{)}^4$$
Why is the binomial coefficient used there?

What is Cumulative Binomial probabilities?
I am new to this so don't know if I am asking the right question as I just read about its usage but didn't know what exactly a Cumulative Binomial probability is.
So my question is, What is Cumulative Binomial probabilities ? any example will be of great help.

If $\frac{1+3p}{3},\frac{1-<!--\; -\; -->p}{4},\frac{1-<!--\; -\; -->2p}{2}$ are the probabilities of mutually exclusive events, then the set of all values of $p$ is?

The standard telephone number in the US has 10 digits: for example, NOVA's telephone number is (703) 323-3000, How many phone numbers are possible for the 703 area code it the first digit that follows the 703 cannot be a or a 1?

Calculate Binomial Probability for inequality.
How could the binomial probability be calculated for a case of $P(X\ge <!--\; \ge \; -->2)$ for a given value of n and p ?

How do you add probabilities?

Find the number of Sylow 2-subgroups of the special linear group of order 2 on $\mathbb{Z}$ (modulo 3). I think it will be 1. But I failed to prove it using the counting principle. It has 4 sylow 3-subgroups.

Classify the following as an example of classical probability, empirical probability or subjective probability. Based on a sample of 200 cats, the probability that a randomly chosen cat has blue eyes is 0.76
a) empirical probability
b) subjective probability
c) classical probability

Statistics Binomial Probability
Suppose that in a group of 10 people, each person has a 0.1 probability of having a certain disease, independently from person to person. Now consider a testing plan where the group is split into two groups of 5 people where in each group blood samples are taken from each person, then their blood samples are combined and tested together (so there is one test per group of 5). If a group tests positive for the disease then each person in that group is tested individually for the disease, if a group does not test positive, then the testing is concluded for that group. Let the random variable X represent the total number of tests required with this plan.
Find the probability distribution of X.
I know that there are 2, 7, 12 possible values of X but I don't know what formula to use to find P(X). I'm assuming its the binomial probability formula with $nCk(p^k)(q^n-<!--\; -\; -->k)$
but how do you use 2, 7 and 12 in that? what is n and k?

There are ${\displaystyle \frac{m}{gcd(m,x)}}$ distinct elements in the set $\{ax(\mathrm{mod}m):a\in <!--\; \in \; -->\{0,...,m-<!--\; -\; -->1\}\}$
I have only known these by using a computer to generate the number of distinct elements. But I am not sure how to prove this conjecture.
And is there any way that we can connect this problem to Euler's phi function so that we can simply use properties of $\mathrm{\varphi <!--\; \varphi \; -->}$ function to prove it?
And can we also use some counting principle here to give an exact answer?

List contains events $A_1$, $A_2$, …, $A_5$ which are mutually exclusive and collectively exhaustive.
Compute the following:
$\underset{i=1}{\overset{k}{\sum <!--\; \sum \; -->}}P(A_i^c)$