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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0
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1answer
53 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? ...
4
votes
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 ...
1
vote
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 ...
1
vote
0answers
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). ...
1
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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 ...
3
votes
0answers
23 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 ...
5
votes
0answers
100 views
+50

Multiplicative version of Mcdiarmid's inequality?

Suppose you have $n$ i.i.d. random variables taking values in $\{0,1\}$, and $X$ represents their sum. Then you can use a Chernoff bound to control the deviation of $X$ from its expectation. The ...
0
votes
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 $p \in \{x \in \mathbb{R}^n: \sum_{i=1}^n x_i = 1 , x_i \geq 0 \}$?
-3
votes
1answer
22 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 ...
0
votes
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) ...
4
votes
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 ...
0
votes
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 ...
2
votes
1answer
50 views
+200

The sums over RVs between two return times are independent for a Markov chain

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, $τ_{x,k}^+$ is the time of the $k$th return ...
1
vote
0answers
27 views

What is the likelyhood that they will see all the 5 posters? in 5 different elevators? [on hold]

We are putting up 5 different posters in 5 different elevators for 5 days. There is 300 staff accessing the elevators. The elevators cover 6 floors. Staff normally go up to their office 4 times a day ...
0
votes
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 ...
0
votes
0answers
34 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 ...
-4
votes
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 ...
0
votes
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 ...
1
vote
0answers
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 ...
0
votes
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 ...
1
vote
1answer
14 views

Distribution function derivative bounds give bounds on associated measures? Billingsley theorem 31.4 proof.

I am working through Billingsley, Probability & Measure. Struggling with the proof of theorem 31.4: Suppose $u(a,b) = F(b) - F(a)$ and that $F'$ exists throughout a Borel set $A$. If $F' ≤ c$ ...
2
votes
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
votes
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, ...
0
votes
2answers
41 views

Root mean square distance explanation

We know that $D_{rms}=\sqrt N$ where $N$ is the number of steps taken by the random walker. Now,consider a situation where a random walker walks $2$ steps in positive direction in the first two ...
0
votes
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 ...
0
votes
1answer
940 views

PDF of the ratio of two independent Gamma random variables

Let $X \sim \operatorname{Gamma}(a,\lambda)$ and $Y \sim \operatorname{Gamma}(b,\lambda)$ being independent. Find the PDF of the ratio $W=X/Y$. I found $$ f_W(w) = \frac{\Gamma(a+b)}{\Gamma(a) + ...
1
vote
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 ...
0
votes
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 ...
3
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
1answer
16k views

Finding the median value on a probability density function

Quick question here that I cannot find in my textbook or online. I have a probability density function as follows: $\begin{cases} 0.04x & 0 \le x < 5 \\ 0.4 - 0.04x & 5 \le x < 10 \\ ...
1
vote
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, ...
0
votes
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
1
vote
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 ...
0
votes
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 ...
2
votes
2answers
24 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 ...
1
vote
1answer
35 views

Symmetric random walk: mean duration given absorption occurs at 0

This is exercise 2 from Section 3.9 of Probability and Random Processes by Grimmett and Stirzaker: For a simple random walk $S$ with absorbing barriers at 0 and $N$, let $W$ be the event that the ...
1
vote
1answer
42 views

The sum of all combinations greater than $x$

Suppose I choose $3$ integers at random from $\{1,…, 100\}$. What is the chance that the sum of those integers exceeds some number $x$? I know the probability that the numbers will sum to a ...
1
vote
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? ...
0
votes
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 ...
0
votes
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 $$p_2(x_2) = \frac{1}{2 \pi} \text{exp} \left( \frac{-x_2^2}{2} \right). $$ ...
1
vote
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 ...
-1
votes
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
0
votes
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
votes
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 ...
0
votes
5answers
42 views

Probability derivation using axioms

$$P((A \cap B^c) \cup (A^c \cap B))=P(A) + P(B) -2P(A \cap B).$$ I need to show this holds. I see it with Venn diagrams but I need to show it using only the axiom, for the union of two disjoint sets: ...
-1
votes
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? ...
0
votes
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) ...