This tag is for basic questions about probability and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using measure theory), please ask under ...

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3
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0answers
6 views

What are some examples of isotrophic sets?

What are some examples of isotrophic sets? and is there a "good" way to describe them?
1
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0answers
12 views

Strong law of large numbers for square-integrable random variables with bounded variance

Let $(\Omega,\mathcal{A},P)$ be a probability space and $(X_n)_{n\in\mathbb{N}}$ be a sequence of square-integrable random variables $\Omega\to [0,\infty]$ with ...
0
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1answer
22 views

Football pool question

I'm in a football pool where before the game is played, we pick numbers from 0 to 9 from a bag. The winner is whoever picks the number that is the sum of the last number of the two scores. Eg. 32-27, ...
1
vote
2answers
25 views

Let $X$ be Hypergeometric, Find $E\left(\binom{X}{2}\right)$

Let X be Hypergeometric: $X \sim \operatorname{HGeom}(w,b,n)$, so that $X$ is the number of white balls in a sample of size $n$ out of a population of $w+b$ white and black balls. Find ...
0
votes
2answers
64 views

Using probability to detect exam cheating (identical wrong answers)

Hypothetical: What’s the probability that two people taking a test with 10 questions get the identical wrong answers? (Let's say there are 4 choices per problem) Should we first break this down ...
0
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0answers
11 views

Card Matching: expected value of correctly predicted cards with partial feedback

A standard deck of cards is shuffled, and the cards are dealt face down one by one. Just after each card is dealt, you name any card (as your prediction). Let X be the number of cards you predict ...
0
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3answers
43 views

Monty Hall problem (shifting probabilities)

I was explaining the Monty Hall problem to someone, and I explained it in this way: You have three doors, and you pick one, giving you a $1/3$ chance of being right. The presenter opens one of the ...
0
votes
1answer
26 views

Getting P-value of test; statistics

In order to test $H_0 : \mu = 50$ vs $H_{\text{a}} : \mu \neq 50$, a random sample of 9 observations (from a normally distributed population) is obtained, yielding $\bar{x} = 61$ and $s = 21$. What is ...
1
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1answer
26 views

Probability and range partition

in this question we have a fixed partition and we want to partition the range to obtain a three subsets with the condition below.
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0answers
16 views

Evaluate the sum $n$ of geometric random variables

Let $X_i\sim G\left (1-\frac{1-i}{n}\right)$. Evaluate $ \sum_{n=1}^n X_i$ My Try: $$ \sum_{i=1}^n X_i = \sum_{i=1}^n \sum_{k=1}^\infty \left(\frac {i-1}{n}\right)^{k-1}\left( 1 - ...
1
vote
2answers
23 views

Show that $\left| \mathbb{P}(X=m)-\mathbb{P}(Y=m) \right| \le \mathbb{P}(Y\ne X)$

Let $X,Y$ two random variables of the same probability space. Show that $$\left| \mathbb{P}(X=m)-\mathbb{P}(Y=m) \right| \le \mathbb{P}(Y\ne X)$$ I think I need to start from LHS and split it ...
0
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0answers
23 views

Three state probability where one state has yet to factor results.

I'm currently trying to explain something to someone else using probability to make it simpler to understand. As I have it now there have been 5 examples that have happened. In one case there are 3 ...
-3
votes
1answer
70 views

I do not understand the last step of this proof. [on hold]

1. PLEASE LOOK THE FOLLOWING PROOF FIRST. 2. Suzu explained the fist several steps to me in this page :Explanation of an integral formula for the expectation of $(X_1-X_2)(Y_1-Y_2)$ . But I still ...
0
votes
0answers
35 views

Truncation of partitions generating function question

$A (x)$ is the generating function for partitions. $B(x)=\sum_{n=0}^{\infty}b_nx^n $ $$b_n =\binom{\text{number of partitions of }n}{\text{into an even number of parts}}-\binom{\text{number of ...
1
vote
3answers
90 views

Series expansion of $\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$?

How would I find the series expansion $\displaystyle\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$ so that it will turn into an infinite power series again??
4
votes
1answer
90 views

How to write $1-x-x^3+x^4+x^5+x^6-x^7 \cdots$ as a power series representation

How can I write $1-x-x^3+x^4+x^5+x^6-x^7 ....$ as a power series representation (i.e., a neat fraction such as $\frac{1}{1-x}$. This stems from $\binom{\text{number of partitions of }n}{\text{into an ...
0
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0answers
19 views

An example of memoryless yet non-independent random process?

I am new to random process. I know that independence indicates memorylessness yet the memorylessness is not necessarily independence. There are abundant examples of independent random process (like ...
1
vote
1answer
26 views

If $X\ge 0$ and $a\ge E[X]$, then $P(X\gt a)\ge (E(X)-a)^2/ E(X^2)$ [on hold]

I need help with this problem. Prove that if $X\ge0$ and $E[X^2]<\infty$ then for all $a\neq0$, $E[X]\le a$, we have $$P(X\gt a)\ge\frac {(E(X)-a)^2}{E(X^2)}$$ Progress I have my doubts if ...
1
vote
0answers
22 views

Pollaczek-Khinchin formula for ruin probability - proof

I got stuck in a specific part of proof of the Pollaczek-Khinchin formula (in book "Stochastic Processes for Insurance and Finance", T. Rolski et al., section 5.3.3, theorem 5.3.4). Namely, why the ...
2
votes
1answer
29 views

Adjacent dominos in a train

Definition of a domino -- a domino contains two squares separated by a line. In both of the squares, there are some numbers of dots (can be 0). Definition of "double-n" domino set: It contains one of ...
3
votes
1answer
29 views

Identifiying Biased and Unbiased Samples

My little nephew asked me a question about biased/unbiased samples in which is teachers answer is something I disagree with to say the least (I don't agree with the assumption made by the teacher nor ...
1
vote
1answer
28 views

Probability of drawing an element from a countably infinite sequence

Consider a sequence containing $A$ and $B$ where, starting at $n=0$, there are $2^n A$'s followed by $2^{n+1} B \ $'s, so the sequence begins $$A, B, B, A, A, B, B, B, B, A, A, A, A, B, B, B, B, B, ...
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votes
1answer
29 views

What is the probability of that event? [on hold]

A fair coin is tossed repeatedly. What is the probability of the event "Three consecutive heads occur before two consecutive tails"?
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1answer
36 views

Expected number of matching “cards”. Why is $\sum_{m=0}^n D_{n,m} = \sum_{m=0}^n m \cdot D_{n,m}$?

Each of n ≥ 2 people puts his or her name on a slip of paper (no two have the same name). The slips of paper are shuffled in a hat, and then each person draws one (uni- formly at random at each ...
5
votes
4answers
286 views

Find the fraction where the decimal expansion is infinite?

Find the fraction with integers for the numerator and denominator, where the decimal expansion is $0.11235.....$ The numerator and denominator must be less than $100$. Find the fraction. I ...
1
vote
1answer
16 views

Random Walk With Absorbing Barrier [on hold]

Consider a random walk $S_{t}$ with a lower absorbing barrier at $0$, and no upper absorbing barrier. $$ {\mathbb P}\left(\, S_{t + 1} - S_{t} = 2.5\,\right) =0.5\,,\quad\mbox{and}\quad{\mathbb ...
0
votes
1answer
24 views

$P(X=c)=0$ for normally distributed $X$?

Let $X$ be norm $(a, b)$-distributed and let $c$ be some real number. Does this imply $P(X=c)=0$? What if $b=0$?
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0answers
26 views

A treatise on Probabilistic arguments and Laplace/Fourier transforms to solve limits/integrals from basic calculus.

I've seen in some answers in Brilliant.org to some very complicated limits and integrals that uses probabilistic arguments (Let $X$ be a random variable from $[0,1]$... some examples are in those ...
2
votes
0answers
36 views

Calculate expected values of the lengths of line segments

There is a line segment of the length of $1$. $N-1$ points are randomly chosen in it, so it is divided by $N$ parts. The question is to calculate expected values of all these parts, from the shortest ...
0
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1answer
12 views

definition of Cumulative distribution function

let X be RV, and his Cumulative distribution function: there is a difference if in my case if $X<x$ ? the definition is the same?
4
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2answers
93 views

Probabilistic techniques, methods, and ideas in (“undergraduate”) real analysis

As the book Probabilistic Techniques in Analysis by Richard Bass shows, nowadays techniques drawn from probability are used to tackle problems in analysis. The mentioned book presents a survey of ...
2
votes
1answer
36 views

Prove that $\int k(w)o(h^2w^2)dw=o(h^2)$ for $\int k(w)dw=1$

Suppose that $k$ is nonnegative real-valued function satisfying $$ \int k(w)dw=1,\quad\int wk(w)dw=0,\quad\int w^2k(w)dw=\kappa_2<\infty.\tag{$\star$} $$ (The limits of the integrals are all ...
-1
votes
3answers
50 views

Probability when n balls put randomly in n boxes such that each box contain 1 ball [on hold]

There are 100 boxes in front of you. You have 100 balls in your pocket which you throw one by one towards the boxes in front of you. Each ball will definitely end up in a box and has equal probability ...
2
votes
2answers
31 views

Probability in dice, Feller exercise

I am stuck with exercise 2 of Chapter 4 Feller vol 1 "an introduction to probability theory and its application". Here I report the exercise text: Five dice are thrown. Find the probability that at ...
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0answers
18 views

Probability (Dependent Events) [on hold]

The M.com class consists of 60 students, 12 of them are girls and 48 boys, 10 of them are rich and 50 not, 15 of them are fair complexioned. What is the probability of selecting a fair complexioned ...
2
votes
1answer
21 views

Derivation of the third moment of Poisson distribution using Stein-Chen identity

(a) Use LOTUS to show that for $X \sim \operatorname{Pois}(\lambda)$ and any function g, $E(Xg(X)) = λE(g(X + 1))$. This is called the Stein-Chen identity for the Poisson. (b) Find the third ...
1
vote
3answers
43 views

Where does this conditional probability law come from?

I was trying to follow a computation done in my class notes, and was having difficulty seeing the inspiration for a part of the manipulation in a question regarding probability. I did some Googling, ...
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0answers
12 views

Deviation of number of cycles of length 4 in Erdős–Rényi random graphs.

I'm working on my homework and can't find any relevant information for this problem. Problem: Let $G(n, p)$ be Erdős–Rényi random graph. I need to find deviation of number of cycles of length 4 in ...
0
votes
1answer
23 views

Two series of independent Bernoulli trials. Find distributions of being simultaneously successful and of first success being simultaneous.

Nick and Penny are independently performing independent Bernoulli trials. For concreteness, assume that Nick is flipping a nickel with probability p1 of Heads and Penny is flipping a penny with ...
0
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0answers
33 views

Can somebody help to understand the last step of this proof?

PLEASE look this proof first. Suzu Hirose helped me a lot! But I still do not understand the last step:(Suzu explained the fist several steps to me in this page :Explanation of an integral formula ...
5
votes
2answers
41 views

Distributing candies

Suppose ther are B boys and G girls in a classroom.Teacher wants to distribute candies among B boys and G girls such that: 1.Each student gets atleast one candy and atmost N candies. 2.sum of ...
1
vote
2answers
38 views

problem related with probability [on hold]

Three shots are fired in succession. The probability of a hit in the first shot is 0.3, in the second is 0.6, in the third is 0.8. In the case of one hit, the probability of destroying the target is ...
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0answers
31 views

Who can find mistakes in this calculation process?

I compute the conditional Separman's rho using the following method, but I do not know whether it is right? Who can tell me whether it is right? Thanks.
1
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1answer
71 views

Explanation of an integral formula for the expectation of $(X_1-X_2)(Y_1-Y_2)$

I do not understand the proof of this expression. Who can explain it to me using simpler words? I do not understand the following black part:
4
votes
1answer
45 views

probability circle determined by chord determined by two random points is enclosed in bigger circle

Two points $A$ and $B$ are chosen uniformly at random from the interior of a circle $X_1$. Let $X_2$ be the circle whose diameter is the segment $AB$. What is the probability that $X_2$ is contained ...
1
vote
2answers
42 views

Health Risk Probability

Question: An actuary is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of women. For each of the three factors, the probability is 0.1 that a woman ...
2
votes
1answer
34 views

Chance of winning a game of hearts with four players

I play hearts with a computer game program. The game is set up so that four people are playing the game. The question is: What are the mistakes, if any, with assuming that the probability of winning a ...
0
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0answers
26 views

How to calculate the following conditonal expectation? am I right?

How to calculate the following conditonal expectation? Is the following calculation process right?
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0answers
21 views

How to calculate the following conditional expectation? Is my calculation process right?

I want to calculate the conditional person's correlation coefficient. But I don't know how to calculate the following expressions,especially the conditional expectation of ...
9
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2answers
83 views

true story about probability? [duplicate]

A women's organization was contemplating suing a famous American university when it learned that the percentage of women who received tenure in the university was smaller than the percentage of men. ...