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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finding the limit for a martingale

I have trouble finding the exact limit for a martingale: Let $\{\xi_n\}_{n\in\mathbb{N}}$ be a Markov chain with $\xi_0$ uniformly distributed in $[0,1]$ and $$ ...
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2answers
16 views

Calculate dependent probabilities

Imagine that two raffles will happen, every raffle will reward 1 person. 10 people will participate. The first raffle will reward 1 person. The second raffle will reward another person, but the ...
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0answers
12 views

Bayes theorem with multiple conditions

how to calculate P(a|b,c,d). knowing that b, c and d are NOT independent from each others ? i know how to solve it if there is independency assumption. however, i am just wondering if there is any ...
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1answer
14 views

Show that $|F_{X,Y}(x,y)|^2\leq F_X(x)F_Y(y)$

Consider the random variables $X$ and $Y$ defined in the same space $\Omega$. Show that $$|F_{X,Y}(x,y)|^2\leq F_X(x)F_Y(y)$$ This question comes from an old test, I know that ...
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2answers
23 views

Find the density of their average

If $f_{X,Y,Z}(x,y,z)=e^{-(x+y+z)}I_{[0,\infty]}(x)I_{[0,\infty]}(y)I_{[0,\infty]}(z)$ find the density of their average $\frac{X+Y+Z}{3}$ I'm a little lost on how to solve this exercise, ...
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1answer
18 views

The number of distinct values taken by a sequence of partial sums of iid

I'm working on an old exam problem as follows: Let $\{X_n\}_{n\in\mathbb{N}}$ be a sequence of i.i.d taking values in $\mathbb{Z}$. Define $S_0 :=0$ and $S_n:=X_1+X_2+...+X_n,$ and $\theta_n$ to be ...
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1answer
27 views

Find the density

Suppose that radius $R$ of one sphere is a continuous random variable with density $$f_R(r)=6r(1-r) I_{[0,1]}(r)$$ Find $f_V(v)$ and $f_S(s)$ the densities of volume and surface area I did ...
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2answers
24 views

Probability of the simultaneous failure of two components where one component would take the full load if only one failed instead.

Working through an example question in Applied Probability for Engineers and Scientists, 1st Ed., by Ephraim Suhir. Example 1.9, beginning on p. 5, reads as follows: A heavy, nondeformable beam ...
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1answer
26 views

Expectation of a random variable in terms of its distribution function

Here is a theorem on expectation of a random variable in terms of its distribution function Theroem: Let $X$ be a (continuous or discrete) non-negative random variable with distribution function ...
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2answers
22 views

A confusing probability question..

A magician holds one six-sided die in his left hand and two in his right. What is the probability the number on the dice in his left hand is greater than the sum of the dice in his right?
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3answers
50 views

Probability that an integer is divisible by $8$

If $n$ is an integer from $1$ to $96$ (inclusive), what is the probability that $n(n+1)(n+2)$ is divisible by 8?
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1answer
25 views

How do I determine the weight to assign to each bucket?

Someone will answer a series of questions and will mark each important (I), very important (V), or extremely important (E). I'll then match their answers with answers given by everyone else, compute ...
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1answer
19 views

Poisson Distribution practice problem

I have just started to learn Poisson Distribution and I have no idea how to deal with the following practice from my textbook: Suppose the average amount of cars passing on a street per minute is ...
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1answer
30 views

Proof the statements

Proof the statements below i)If $P(A)=0$ and $B$ is any event, then $A$ and $B$ are independents ii)If $P(A)=1$ and $B$ is any event, then $A$ and $B$ are independents iii)The events ...
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0answers
18 views

Probability of selecting a sequence in order

If "X" number of attempts are made by "Z" number of persons to select a random number from a range "r", where "X <= r". Then I am interested in finding the probability that a particular sequence in ...
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1answer
28 views

Does this game have infinite expected payout?

Consider the following game: Suppose the initial value of the pot is $ S $. Our player Josephine then rolls a fair $n$-sided die. If the roll is not $1$, then the pot is multiplied by that roll, and ...
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1answer
31 views

Dice probability

You roll a fair dice twice A. What is the probability that the first roll is odd and the second roll is even? B. What is the probability that one roll will be odd and the other roll will be even?
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2answers
50 views

Disjoint events

Let $A$ and $B$ two disjoint events such that $P(A)=0.3$ and $P(B)=0.5$. Find the probability that i)$A$ or $B$ ocurrs ii)$A$ occur but not $B$ iii)repeat $i)$ and $ii)$ with $A$ ...
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2answers
14 views

Understanding different definitions of bayes theorem

I am taking course on probability and reading about bayes theorem. In Sheldon Ross' book, it given as $$P(E) = P(E|F)P(F) + P(E|F^C)P(F^C)$$ with note: Equation above states that the probability of ...
2
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1answer
25 views

Inequality for the derivative of a density of a random variable convolved with a normal r.v.

I have a question about the following proof. The statement is: Let $X$ be a random variable and $Z_\tau \sim N(0,\tau)$ be an independent random variable. Then $Y_\tau := X + Z_\tau$ has a ...
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1answer
27 views

Random variable with all higher order moments zero?

Is there a random variable with finite first and second moment but all higher order (non-central) moments are zero?
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1answer
53 views

Why does this expectation integrate to 1

Let $p(y|\theta )$ be our likelihood, and $p_{N}(\hat{y}|\theta)$ be an unbiased estimator of our likelihood. Let $z=\ln p_{N}(\hat{y}|\theta) - \ln p(y|\theta )$, and $g_{N}(z|\theta)$ be the ...
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1answer
12 views

Distribution of the summation of k random variables and k is also variable

We have a set of positive random variables $\boldsymbol X=\{X_1,X_2,\ldots\}$, where $X_1,X_2,\ldots,$ are independent and identically distributed (i.i.d.). The CDF $F(x)$ and PDF $f(x)$ for Xi are ...
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1answer
46 views

expected number of steps for chossing randomly each number between 1 to $n$ at least $k$ times

Assume the following game: Every step choose a number between 1 to $n$ randomly i.e. every integer between 1 to $n$ is chosen with probability $\frac{1}{n}$. Success is when every number has been ...
2
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1answer
29 views

the probability density function (PDF) of concatenation of two Gaussian variables

Gaussian variable $x$ follows from $N(u_x,\sigma_x^2)$ and $y$ follows from $N(u_y,\sigma_y^2)$. Assume we have the vector $\bf{z}=[x,y]^T\in R^2$, then it seems that no matter whether $x$ and $y$ are ...
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1answer
30 views

Is this a binomial or multinomial question?

You can donate to a company: $10$ dollars , $20$ dollars or nothing. In a mall there are $70$% young people and $30$ % old people. $50$% from the old people aren't donating anything. ...
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2answers
26 views

Understanding Conditional Probability Basics

In many online sources I've read a statement similar to: Probability of B happening given A is equal to the probability of A and B both happening divided by B happening or $p(A | B) = p(A \cap ...
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1answer
32 views

Is this a misuse of the term “probability space”?

Let me first state the definitions as I am using them. Do correct me if I am wrong here! A "probability space" is a triple $(\Omega, F \subseteq 2^{\Omega}, \mu : F \rightarrow [0,1])$. The ...
3
votes
1answer
27 views

Probability of drawing >18 when drawing 3 cards

I am trying to calculate some probabilities for a card game. Players have to draw 3 cards each time and the cards must add up to a certain value for them to win - the value changes depending on the ...
2
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2answers
37 views

Erin rolls 4 four-sided dice all at once, then can roll a subset of her choosing a 2nd time. What is the probability of getting all the same number?

Here's what I have so far: All 4 same on first try = (1/4)^4 * 4 3 same, then get 4th on 2nd roll = 4 * (1/4)^3 * (3/4) * (4!/3!) Here's where I'm confused: 2 same = 4 * (1/4)^2 * (3/4)(2/4 :to ...
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1answer
23 views

Prove that $\tilde{X}_{\tilde{\theta}}(t)$ is a martingale

Let me introduce the objects: 0) $(\Omega, \mathcal{F},\Bbb{P})$ is a probability space 1)$S_N $ is the set of symmetric, non-negative definite $N\times N$ matrices 2)$a:[0, \infty) \times \Omega ...
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0answers
28 views

Probability of Probabilities :)

So here is a tough one (or so i think). i have 15 games (30 teams). and only 2 options i can chose from (even / odd number of goals). I want to bet a ticket with each possible combination. How many ...
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1answer
29 views

Conditional Probability for Exponential Random Variables

I'm working through a practice problem for an exam and I would like to verify that I've done it correctly. Additionally I'd like some insight on the intuition behind the numbers I'm getting. Problem ...
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1answer
18 views

Probability of picking all white marbles?

Consider that you have a drawer with n marbles of various colors. There are 5 white colored marbles. You grab k marbles from the drawer, where k <= n. What is the probability you find all 5 white ...
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1answer
18 views

Probability and Stats (loaded coin)

Smith is offered the following gamble: he is to choose a coin at random from a large collection of coins and toss it randomly.The proportion of the coins in the collection that are loaded towards a ...
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2answers
33 views

Distribution of Summation of two discrete random variables

Here, $\tilde{x}_1$ and $\tilde{x}_2$ are two non-negative independent discrete integer-valued random variable and the support set of $\tilde{x}_1$ and $\tilde{x}_2$ are below: $$ X_{1} = \{ ...
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2answers
44 views

Dice Roll Probabilities

I'm trying to figure out the probabilities for the following casino game: You and the dealer each roll a pair of dice and the person with the highest individual die roll wins. If its a tie, you win. ...
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0answers
44 views

Strategy for choosing lottery numbers when buying many tickets

In a given lottery a user must choose 5 out of 50 numbers, without replacement. Prizes are offered for matching at least 2 of the winning numbers. If a user can purchase multiple tickets (let's say ...
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2answers
25 views

Probability of Winning a Toss

I have an unfair coin with two sides 1 and 2. I have a problem and its constraints. The coin has to be tossed until I win; which happens when 1 shows up in a toss. Constraints: Since the coin in ...
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1answer
21 views

Confidence interval of a uniform distribution

I need some help with the following problem: I want to estimate $n$ of $X_i \sim U(1, n)$ with a $90\%$ confidence level. What is given is the sample size with $10$ and the maximum of the sample with ...
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1answer
19 views

Proving Properties of Discrete Time Markov Chain mathematically

I want to prove that the queue length at a store is not a Discrete Parameter Markov Chain (DPMC). Now I have the equation: $$Q_k = (Q_{k-1} - 1) + V_k$$ $Q_k$ is the queue length at time instant ...
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2answers
40 views

12 six-sided dice are thrown. What is the probability of getting each number twice?

I got this: $\frac{6!12!}{6^{12}2!^6}$ but the answer is this: $\frac{{12!}}{6^{12}2!^6}$ Im not sure I understand why you wouldn't write the $6!$ because if the first die's value is #3 then you have ...
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1answer
37 views

Asymptotic Behavior of Binomial Distribution

I am considering the following problem: Given the following equation: \begin{equation*} c = \sum_{k=n}^{2n-1} \binom{2n-1}{k} p(c)^k (1-p(c))^{2n-k-1} \end{equation*} Which is the probability that ...
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1answer
18 views

probability of a proportion point estimate

I've got a problem where I'm supposed to find the probability of a point estimate but cannot see how my answer is differing from the given one. The problem is: Unknown to an experimenter, the ...
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1answer
15 views

Illegal lottery problem (Merging dependent bernoulli trials)

Suppose I am in a town that playing lottery is illegal. If I buy a ticket for 1 dollar, I will win the lottery with probability $p$. Each time I buy a ticket, the police may catch me and confiscate ...
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2answers
32 views

One-One Correspondences

Adam the ant starts at $(0,0)$. Each minute, he flips a fair coin. If he flips heads, he moves $1$ unit up; if he flips tails, he moves $1$ unit right. Betty the beetle starts at $(2,4)$. Each ...
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1answer
31 views

On a 50 question multiple choice exam with 5 choices per questions, What are the odds that I get 100% if I were to Guess every answer? [on hold]

What would the odds be to get 100% on a multiple choice exam where I guessed the answer to all 50 of the multiple choice questions (5 choices per questions)? A 1 in how many chance?
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1answer
12 views

Density of a distribution function at upper bound [on hold]

Consider a strictly increasing continuously differentiable distribution F with support on $[a,b]$. Let $f$ be the pdf of $F$. What can we say about $f(b)$? Under what conditions is $f(b)>0$? ...
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2answers
35 views

Tail probability of a random variable

Here are two theorems about the "tail probability" of a random variable X Thm1: The expectation $E(|X|^\alpha) < \infty$ for some positive $\alpha$ if and only if $$\sum_{n=1}^\infty ...
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1answer
34 views

A detail on a proof of the strong Law of Large Numbers.

In the following blog post https://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/ one is presented with a nice account of the LLN. Suppose that I have shown that if $(n_j)$ is a ...