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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8 views

Concentration inequality for sum of squares of independent and identically distributed sub-exponential random variables?

Suppose $X_1, X_2, \ldots, X_n$ are independent and each has the same distribution with a sub-exponential random variable $X$ (for example, $X$ is the square of a standard normal Gaussian variable). ...
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0answers
13 views

Find the minimum number of tickets to guarantee the win of a n-bit binary lottery?

Here's the problem. I just don't know how to approach it. If the 'one error tolerance' were removed, then this would be a simple binomial distribution problem. But now I can't figure it out. In ...
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1answer
22 views

probability of not getting same number twice in a row after n die rolls

Having rolled a die $n$ times, I want to determine the probability of not getting any number twice in a row. If I wanted the probability of not getting any number three times in a row, I could use the ...
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0answers
21 views

When is the conditional expectation function equal to a continuous function a.e.?

We are given a random variable $Y$ and a $d$-dimensional random vector $X$. Suppose $Y$ is $L_1$ (has first moment). Then $f(x)=\mathbb E[Y\mid X=x]$ is a Borel function. Lusin's theorem says that for ...
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1answer
11 views

Is relative entropy with respect to a pmf a continuous function?

Is the relative entropy $D(p || q)$ with a fixed pmf $q$, continuous over $p$, where $p \in \{x \in \mathbb{R}^n: \sum_{i=1}^n x_i = 1 , x_i \geq 0 \}$?
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1answer
21 views

Suppose you draw a five-card hand randomly from the deck and get four cards that that would make a straight if you could replace the fifth card…

Suppose you draw a five-card hand randomly from the deck and get four cards that that would make a straight if you could replace the fifth card. (e.g. J 10 9 8 3 or K 7 6 4 3). If you are allowed to ...
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1answer
21 views

Suppose that after a five-card hand is drawn, the cards in it are put back in the deck and another five-card hand is drawn.

Suppose that after a five-card hand is drawn, the cards in it are put back in the deck and another five-card hand is drawn. a) What is the probability that the two hands have no card in common? b) ...
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2answers
70 views

What does $-p \ln p$ mean if p is probability?

In statistical mechanics entropy is defined with the following relation: $$S=-k_B\sum_{i=1}^N p_i\ln p_i,$$ where $p_i$ is probability of occupying $i$th state, and $N$ is number of accessible ...
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1answer
24 views

sum of two Dice game

The question is: You have 2 fairly weighted dice. You and an opponent pick any integer one after the other. If your number is closer to the sum of the faces on the rolled dice, you win. Do you want ...
0
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0answers
33 views

How does a loaded die affect this probability

Suppose I own five different six-sided dice. Four of the dice are fair dice and they are equally likely show the values $1, 2, 3, 4, 5,$ and $6$. One of the dice is loaded and never shows ...
0
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1answer
50 views

Probability of experiencing rain

The question is: You are going camping over the weekend, and there is $50\%$ chance of rain on Saturday and $60\%$ on Sunday (independent). What is the probability that you will not experience rain? ...
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22 views

conditional probability (question)

Let $X,Y$ be random variables with $f$ the density of $Y$ and $x \geq t$ \begin{align} & P(X \leq u \mid Y=x)=E(P(X \leq u\mid Y=x,Y \geq t)) \\[10pt] = {} & \int P(X \leq u\mid Y=x,Y \geq ...
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1answer
40 views

Is there an equation to find out how after $\frac{6!}{6}$ to locate clockwise increase in numbers in sets of 2

So I asked this question last night what is the max possible combinations of 1 2 3 4 5 6 without repeating And as stated I don't know what symbols mean, but I learned what $!$ is and how it works ...
0
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1answer
23 views

Number of Unique Ranks of High Card in Three Card Brag

Well the game is called Teen Patti in India. Almost similar to Three Card Brag a British game. There are total $16440$ Unique High Card hands are present. (Considering the suit.) Hand $1 = 5$ Heart, ...
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0answers
16 views

Equivalent definition of singular random variable

I'm taking an intermediate course in probability theory (that is without measure theory) and when defining singular random variables (after showing the devil's function), the book defines: $X$ is a ...
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3answers
28 views

What is the probability that none of the cans of soup are next to each other? [on hold]

On a empty shelf you have to arrange $3$ cans of soup, $4$ cans of beans, and $5$ cans of tomato sauce. What is the probability that none of the cans of soup are next to each other? I tried working ...
2
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1answer
14 views

Maximizing the probability of a poll prediction

Using the central limit theorem, I was able to find out the first part of this question. However, part b is eluding me. How do I, in general, find a value for $n$ such that we can ensure the ...
0
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1answer
22 views

Predictive Distribution with Normal Prior

Given $\Theta = \theta$, let $X_1, X_2, \dots, X_n, X_{n+1} \sim \mathcal{N}(\theta, \sigma^2)$ be independent. $\Theta \sim \mathcal{N}(\theta_0, \tau^2)$. What is the easiest way to find the ...
1
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2answers
12 views

how many trials of independent event with probability p needed to reach chance q of at least one success

Given an independent event with probability $p$ and a number of trials $k$, if I want there to be a probability of at least $q$ that the event has occurred at least once, how big does $k$ have to be ...
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0answers
12 views

Independence of time intervals between visits of a state $x$ on a Markov chain

The question is like the following, Let $X_0,X_1,...,X_n,...$ be a Markov chain with finite state space. Define $τ_{x,0}^+=0$, and $τ_{x,k}^+=\min\{t:t>τ_{x,k-1}^+,X_t=x\}$. In plain words, ...
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0answers
8 views

Rigorous Derivation of Metropolis-Hastings Transition Density

The Metropolis-Hastings MCMC algorithm is as follows. Set $X_0$ to some initial value in the support of the target density $f$ and choose a proposal density $q(y \mid x)$; a density in $y$ for each ...
3
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2answers
39 views

If a fair six-sided die is rolled four times, in how many outcomes is the value of each roll at least as large as the value of the previous roll?

Suppose you roll a fair 6-sided die four times. Let C be the event that the value of each roll is at least as large as the value of the previous roll. What is the probability of C? I know that ...
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2answers
39 views

Is the condition $\;P(X^2>1|X>0)\;$ the same as $\;P(X>1)\;$?

I saw two examples for this question the condition $\;X>0\;$ means that $\;X^2>1\;$ is true only when $\;X>1\;$, and the probability is $\;P(X^2>1|X>0)=P(X>1)\;$ But I also saw the ...
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1answer
24 views

conditional expectation and equality (question) [on hold]

Let X,Y random variable and f is the density of Y. $P(X<u)=E(P(X<u|Y))=\int P(X<u|Y=x)f(x)dx$ Is it true? Thank you
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1answer
32 views

Probability of balls of colors in two urns

In one urn A there is $2$ red balls and $3$ white balls. In another urn B there is $3$ red balls and $1$ white ball. $4$ balls are taken out and returned from urn A and $5$ balls from urn B. I could ...
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1answer
36 views

Number of solutions of $N_9 + N_8 + N_7 + N_6 + N_5 + N_4 + N_3 + N_2 + N_1 = 82$ in the positive odd integers with $N_i \leq N_{i - 1}$

Given $N_{tot}=82$ where $N = [N_9 \: N_8 \:N_7 \:N_6 \:N_5 \:N_4 \:N_3 \:N_2 \:N_1 \:N_0]$, how many possible combinations are there if each $N_i$ must be odd and $N_i \leq N_{i-1}$, i.e. one ...
2
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2answers
23 views

Find the new variance

In a sample of size $21$ the sample mean is $58$ and the sample variance is $10.7$. If an observation of value $52$ is added to the sample, what now is the sample variance of the observations? I ...
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0answers
20 views

Probability of winning at least once in four trials [on hold]

If you have an 8% chance of winning over 4 mutually exclusive draws. So 8/100, 8/100 8/100, 8/100 over each draw. You can win once, twice, three times or all four. What are the odds of winning at ...
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1answer
29 views

A box with $3$ types of colored balls.

In a box there are $15$ white balls, $8$ black balls, and $12$ red balls. We extract $6$ balls, without putting them back. $(a)$ What is the probability that the first ball is red, the second and ...
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3answers
43 views

An ice-cream shop sells $11$ kinds of ice-cream, including mango and lemon.

An ice-cream shop sells $11$ kinds of ice-cream, including mango and lemon. For a bowl, one chooses at random five kinds (not necessarily different). $(a)$ How many different bowls can be made? ...
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1answer
26 views

Calculate the density of $X=X_1*X_2$ using dirac function.

Let $X_1$ have p.d.f $$p_1(x_1)=\gamma^2x_1 \cdot \text{exp} \left( \frac{-x_1^2}{2} \right),$$ and $X_2$ have p.d.f $$p_2(x_2) = \frac{1}{2 \pi} \text{exp} \left( \frac{-x_2^2}{2} \right). $$ ...
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0answers
13 views

process and renewal equation [on hold]

The renewal equation is: $Z=z+F*Z$ and $Z(t)=z(t)+\int_0^t Z(t-u)F(du)$ Let $A(t)=\sum_0^{\infty} F^{*n}(t)$ the renewal function How to show $A(t)<\infty$? Thank you
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1answer
28 views

Expected number of red balls in an urn | a specific ball being in it

This is a follow-up on this question. We toss balls into urns. Denote with $x$ the number of balls in an urn. And $x_r$ denotes the number of red balls. The share of red balls among the balls is ...
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0answers
26 views

How to find out the following probability?

I need to find $\mathbb{E}_d[\mathbb{P}\left\{X\le\mu\right\}|\hspace{1mm}d]$ with \begin{equation} ...
0
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1answer
28 views

Probability of infected but does not show symptoms of disease?

A person moving through a tuberculosis prone zone has a $50\%$ probability of becoming infected. However, only $30\%$ of infected people develop the disease. What percentage of people moving through a ...
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1answer
41 views

what is the max possible combinations of 1 2 3 4 5 6 without repeating

Each number has to be used and only once in each set. I don't know how to put it but it can't cycle . here is my example 123456 Is the same as 234561 Same as 345612 This isn't for any homework or ...
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2answers
40 views

drawing cards in the deck [on hold]

Suppose 3 cards are drawn from a shuffled 52 card deck. The face cards are the Jacks, Queens, and Kings. Let A = {all diamonds} and B = {All face cards} Are the events A and B independent? ...
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1answer
15 views

Question about flat prior

Suppose we have a bent coin with unknown probability θ of heads. We toss it 12 times and get 8 heads and 4 tails. Starting with a flat prior, I want to show that the posterior pdf is a beta(9, 5) ...
2
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1answer
22 views

Measurablity of functions defined over sections of product measures

I have to solve the following exercise but I am unable to proceed. Could you please give me some hints to how to solve it? Let $(\Omega_1, \mathcal{F}_1)$ and $(\Omega_2, \mathcal{F}_2)$ be ...
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2answers
23 views

Distribtution of the maximum of three uniform random variables.

How do I get the cumulative density function of $Y$? $X$ is a continuous random variable with pdf $$f(x) = 1,\quad 0 < x < 1. $$ Three independent observations of $X$ are made. Find the pdf ...
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1answer
36 views

How to compute the $p$ value? and the correct explanation of the overall experiment.(Is my answer correct?)

Hello community first of all thanks for helping me with my math problems. Here I'm again with hypothesis test exercise. I want to know if I made some mistake in my answer and if someone can help me ...
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1answer
18 views

How to derive mean and variance for a Bayes estimator?

Let $X_1,...,X_n \sim$ iid $\mathcal{N}\left(\theta , \sigma ^2\right)$, where the variance is known. Also, suppose the prior distribution $\theta \sim ...
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2answers
36 views

How to find the expected value of the radius of the meat ball? Assuming its shape is a perfect sphere? [on hold]

To make a meatball, you choose beef with probability $2/3$ and turkey with probability $1/3$. If you choose beef, the number of ounces you take is uniformly distributed between $2$ and $4$. If you ...
0
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1answer
19 views

Probability of rolling higher than $N$ by summing the highest $X$ number of dice out of a set $Y$ number of dice, each with $Z$ sides. [on hold]

I'd like help finding a formula for the probability of rolling higher than a target number, $N$, by summing the highest $X$ number of dice out of a set $Y$ number of dice, each with $Z$ sides, ...
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0answers
18 views

conditional probability, logical product

I was working my way through Kruschke's textbook and got to Chapter 9 and the result on factoring out conditional probabilities for hierarchical models, seemed similar to something in Feller Vol1 ...
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2answers
19 views

The covariance between $X$ and $Y$.

Suppose that $X$ and $Y$ are both continuous random variables that have a joint probability density that is uniform over the rectangle given by the four $(x,y)$ coordinates $(0,0)$ , $(2.46,0)$ , ...
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1answer
30 views

Suppose there are 5 dollar bills in a box…find the PMF

Suppose that there are $5$ dollar bills in a box: three $1$ dollar bills, one $5$ dollar bill and one $10$ dollar bill. You are allowed to pick up two bills at the same time from the box randomly. Let ...
1
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0answers
15 views

Deriving sample size using Hoeffding's Inequality

I want to use Hoeffding's Inequality to determine the necessary sample size $n$ to construct a confidence interval of $\epsilon$ and $\alpha$. I've consulted the Wikipedia article and am confused as ...
1
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1answer
21 views

Breaking sides of equation to prove a probability.

I am trying to prove that $$P(B \cap C \mid A) = P(B \mid A) P(C \mid A \cap B)$$ So far I have been trying to break the equation down LHS and RHS, but I am having trouble figuring out the right ...
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0answers
8 views

renewal function and constant

Let F a density and $U(t)=\sum_{n=0}^{\infty}F^{*n}(t)$ with $F^{*n}(t)=F*...*F$ Show if for all $a<\infty$, $F(a)<\infty$ then for all $t<\infty$ and $\delta<1$, there exist ...