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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What are some examples of isotrophic sets?

What are some examples of isotrophic sets? and is there a "good" way to describe them?
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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 ...
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1answer
21 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, ...
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1answer
372 views

Time based probability question

A restaurant sees 70 customers in a 24 period based on historical patterns. However, 40 customers came in today between 9AM-11AM. What is the probability that in the next 22 hours that there will be ...
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1answer
21 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 ...
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1answer
286 views

Probability - rolling a fair die 10 times, what is the probability you would match a separate set of 10 numbers?

Having some trouble with this problem... Say someone is rolling a fair die 10 times, and using that roll as an attempt to guess what number (1-6) someone else has written down on a piece of paper for ...
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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 ...
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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 ...
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2answers
158 views
+100

3-dimensional light up cube, # of rows/cols/diags in/on a 4 × 4 × 4 cube

Imagine a 3-dimensional cube (much like a 4 × 4 × 4 Rubik's cube) except the planes of the cube cannot be twisted individually and instead of faces with different colors, it is clear (see ...
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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 ...
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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 ...
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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 ...
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2answers
425 views

Expectation score of multiple choice test

A multiple choice exam has 100 questions, each with 5 possible answers. One mark is awarded for a correct answer and 1/4 mark is deducted for an incorrect answer. A particular student has probability ...
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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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1answer
3k views

Expected number of calls for bingo win

Before I begin, I did a search through math.stackexchange and came across two previous attempts to get people to solve probability problems involving bingo. Neither produced a response. So what ...
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1answer
39 views

Conditional expectation of an uniformly distributed random variable

Suppose $U_1, \ldots, U_n$ are i.i.d. random variables with $U_1$ distributed uniformly on the interval $(-1, 1)$. Compute $\mathbb{E}(U_1 + \ldots + U_n |\max(U_1, \ldots, U_n) = t)$ for $t \in (-1, ...
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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 ...
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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 - ...
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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 ...
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1answer
327 views

Distributions of $X^2$ and $X-1$ when $X$ is geometric

Let$ X$ be a discrete random variable with the probability mass function given by $p_x(x)= 2^{-x}$ for $x=1,2,3,\ldots$ and $0$ otherwise. a) Let $Y=X^2$, find the probability mass function of ...
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3answers
89 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??
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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 ...
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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 ...
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1answer
141 views
+100

Number of flips to get to a Set of Positive Lebesgue Measure

A consequence of Exercise 1.1.19 on page 13 of Stroock's "Probability Theory: An Analytic View" is that if a set $E\subset[0,1)$ has positive (Lebesgue) measure, then for almost every $x\in[0,1)$, a ...
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1answer
69 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 ...
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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 ...
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1answer
43 views
+100

Probability that there is sub-sequence of exact length

Can you help me to solve the following: Find probability that in sequence of N random uniformly distributed numbers there is increasing sub-sequence of exact length L.
3
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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 ...
4
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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 ...
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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 ...
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1answer
30 views

The odds of winning two lotteries back to back

Not one with a math background I ask the following question; what are the odds of winning both the Mega Millions and PowerBall back to back? A question that arose out of my statement to a friend’s ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
28 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
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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1answer
24 views

Tail bounds for maximum of chi-squares

Suppose one has $n$ i.i.d. chi-square variables $X_i$ with degrees of freedom $k$. Is there any literature on the distribution of $\max(X_i)$? In particular are there any good tail-bounds for the ...
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0answers
20 views
+50

Largest magnitude of off-diagonals of Wishart matrix

Let $X_1,\dots,X_n\sim N(0,\Sigma)$ be a multivariate normal, with sample covariance $\hat\Sigma$. Of course the diagonals of this matrix are chi-square distributed and there exist tail bounds for how ...
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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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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 ...
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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?
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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 ...
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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 ...
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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
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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 ...
3
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1answer
34 views

Possibilities with unit digits and numbers

$x$ is a three digit number greater than $700$. If $x$ is an odd number and each digit is not equal to zero, what is the possible number of $x$? (Replacement is not allowed) Answer: $91$ Can ...
2
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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 ...