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Questions tagged [probability]

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 (probability-theory) instead. For questions about specific probability distributions, please use (probability-distributions).

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Probability - Too simple but confusing - Why two different approaches for similar problems?

There are two questions and the approach seems to be different. a) Probability of choosing 3 queens from a deck of cards? Ans: 4/52 * 3/51 * 2/50 However, if the question is how many diamonds ...
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1answer
8 views

What's the probability that picking $3$ coins from a bag containing all coins from $1$ cent through $2$ euro will yield a total of $80$ cents?

I recently ran into this interview question and am wondering if my solution is correct. The setting is We have a bag containing one coin of each type, i.e. we have one $1$ euro cent coin, one $2$ ...
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0answers
5 views

How to estimate the parameters of a censored exponential mixture process.

I have a real world scenario that involves a process behaving as follows: there are two kinds of machines and when these machines fail, their recoveries follow exponential distributions. Let's say the ...
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2answers
12 views

What is the probability of right and left handedness given these probabilities?

Me and some friends were having a debate over one of the questions presented during a lecture. According to my one of friend, who attended the lecture, he said that the professor said the correct ...
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0answers
10 views

A poker hand contains five cards. Find the probability that a poker hand can be…

d) one pair Since there are 13 ranks, we pick 1 rank. $\binom{13}{1}$, for each of these ways we want to pick 2 cards out of 4 that belong in this rank. So $\binom{13}{1} \binom{4}{2}$. Now we want 3 ...
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2answers
25 views

Trying to understand a probability question/concept

Discrete Probability: Four dice are thrown, what's the probability that... So I made a post and there were a lot of incorrect answers, and two people were able to provide the correct answers, but ...
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1answer
32 views

Monty Hall problem: If contestant knows the door with goat…

I need to know the probability of this variation of the problem. Suppose that the contestant is a psychic and in some way he/she knows one and only one door with a goat. *The psychic can't know the ...
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5answers
36 views

Discrete Probability: Four dice are thrown, what's the probability that…

Four dice are thrown, what's the probability that: a) None of them fall higher than three? b) None of them fall higher than four? c) That four is the highest number thrown? So for a, I wanna think ...
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2answers
27 views

Calculate $E(X|X+Y)$ where $X,Y$ have expotential distribution with parameter 1

Calculate $E(X|X+Y)$ where $X,Y$ have expotential distribution with parameter 1. I calculated distribution of $X+Y$, which if I am not mistaken is $\frac{e^{-x-y}}{x+y}$, where $x+y>=0$, but I am ...
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0answers
8 views

Infinitesimal generator/intensity matrix

I am trying to understand the following example, but I do not understand how they get $P(t)$. What are they doing with the diagonal matrix to get the three matrices with $e^{-t}$, $e^{-3t}$ and $e^{-...
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0answers
14 views

PDF of conditional uniformly distributed random variable

Given two barrels of water, $A,B$, with 1 liter each. We pour an $X\sim U[0,1]$ amount of water from $A$ to $B$ and then $Y$ amount of water randomly from $B$ to $A$ $(Y|X=x\sim U[0,1+x])$. ...
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0answers
9 views

Deriving AR(1) Time Series Model with repeated substitutions

I am simply trying to go from formula (2.10) to (2.11) in Analysis of Financial Time Series from Ruey Tsay. What I am not understanding is how the substitution results in 2.11 do not include the mean,...
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1answer
18 views

Trying to find the CDF of $X+Y$ when $X\sim exp(\alpha)$ and $Y \sim exp(\beta)$ (independent) without convolution, but it doesn't seem to work

The textbook I am using, using convolution in order to find the CDF of the $X+Y$ when $X\sim exp(\alpha)$ $Y\sim exp(\beta)$, and X and Y are are independent. However, I have no background with ...
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2answers
124 views

Cube and unit cubes

Consider a $3\times 3\times 3$ cube consisting of smaller $1\times 1\times 1$ unit cubes. The big cube is painted black on the outside. Suppose we disassemble the cube and pick a random unit cube, ...
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2answers
11 views

Odds of drawing any 3 identical cards from 27-card deck (9 unique x 3 copies each) in 9 draws

I have a 27-card deck with 3 copies of each of 9 distinct cards. If I'm drawing 9 cards, what are the odds that I draw all 3 copies of any one of the 9 possible options? [Also known as "What are the ...
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1answer
20 views

Convergence of the sum of iid scaled by $n^\alpha$

I am interested in the convergence of the sequence $\mathbb{P}(|X_1+...+X_n|/n^\alpha<z)$ where $z>0$, $\{X_n\}_n$ is an iid sequence with mean zero and finite variance. I can easily prove that ...
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1answer
9 views

Expressing conditional independence in terms of third conditionally independent variable

So we have three variables $A$, $B$, and $C$. We know that $A$ and $B$ are conditionally independent given $C$ (ie $P(A,B|C)=P(A|C)*P(B|C)$) How would I prove $P(C|A)=\frac{P(B|A)}{P(B|C)}$. My ...
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0answers
29 views

probability one man with one coin and another with n-1 coins

I can't find the probability of the following problem: There are two men, the first one with one coin and the second one with n-1 coins. Every mini-game each of them put one coin on the table, and ...
0
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1answer
16 views

Odds of having at least one pair in a 4-card hand

I'm trying the calculate the odds of having at least one pair in a 4-card hand. I made the calculations in 2 different ways but the math doesn't add up. odds of having at least 1 pair in a starting ...
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1answer
20 views

Calculate $EX_{\tau}$ where $\tau=[inf\space n: \space X_n=2 \space or\space X_n=3]$

I came up with this task myself so it might be blurry, actually I changed a bit another exercise which was easy, but I'd like to know the way of coming up to an answer if it was like that. Let $X_n$ ...
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0answers
19 views

Conditional Independence, Decomposition

Is there some set of independence relations between three random variables $X$, $Y$ and $Z$ such that $P(Z \mid X, Y)$ = $P(Z \mid X) \cdot P(Z \mid Y)$? (I feel like there should be, but I can't find ...
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0answers
9 views

Markov process intensity matrix

$X$ is a Markov process with state space $(1,2,3)$. How can I find the matrices of transition probabilities $P(t)$ if the generator is \begin{bmatrix}-2&2&0\\2&-4&2\\0&2&-2\end{...
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1answer
20 views

Optimal Bet Game Strategy

I start with a certain amount of cash $C$ and I play a game where on each win, I double my money. The game is structured such that the first time I play, I have a 50% chance of winning. Let's say I ...
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0answers
30 views

Which of the two conditional probabilities is larger?

I have two events at my disposal where the objective is to find out which of the two is more likely to occur. After some manipulation I get: $$ \Pr\left( A\geq \theta \mid A\leq B \right) - \Pr\left( ...
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0answers
24 views

Finding a constant in a probability function

A basket contains one blue ball. Firstly, a random number of white balls, X, is added to the basket. The probability function of X is: $P(x=k) = \frac{c}{k!}$ ($k = 0,1,2,...$) ($c$ is a constant). ...
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0answers
12 views

Consistency for maximum likelihood estimator with a single sample

Suppose you have a finite family of probability measures $\{\mu_\theta: \theta \in S\}$ on a finite space $\Omega$ (with respect to the discrete sigma algebra). Let $X$ be a random element of $\Omega$ ...
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1answer
23 views

Is my computation correct? distribution of an inverse function

Let $ X \sim U[-1,2]$ and let Y: $$ Y=\begin{cases} \frac{1}{X^2} && X \ne 0 \\ 0 && X=0 \end{cases} $$ Find the distribution of Y. What I did was: $F_Y(t)=P(Y\leq t)=P(\...
2
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2answers
25 views

Distributing 5 distinct balls into 3 distinct boxes

Suppose $5$ distinct balls are distributed into $3$ distinct boxes such that each of the $5$ balls can get into any of the $3$ boxes. What is the Probability that exactly one box is empty. Also What ...
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0answers
7 views

Average Number of users based on some condition and error calculation

I want to Calculate average No. of users selecting any particular No from random number range (2 to 6) and I have defined error condition as when more than one user selects same number. Now based on ...
0
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1answer
15 views

Notation Question: $X \sim \mu$ in the definition of Coupling

My professor defined a coupling as the joint distribution of $(X, Y)$ such that $X \sim \mu$ and $Y \sim \nu$ for probability measures $\mu$ and $\nu$. I was not quite sure whether $X \sim \mu$ ...
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1answer
55 views

Monty Hall Problem. If contestant choose door with goat and host reveal it…

I need to confirm this scenario. 1 - The contestant picks a door with a goat behind it. 2 - The host opens this door and reveals the goat. 3 - The host gives the contestant the chance to pick a new ...
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3answers
34 views

2 dice are thrown once Find the probability of getting either 1 or 5

Two dice are thrown once find the probability of getting either 1 or 5 ? Here I get into a situation that I don't understand If the answer is 20 /36 oR it is 2/36 I m bit confuse between the ...
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2answers
45 views

Pdf of $X+Y+Z$ where $X,Y,Z$ are independent $U(0,1)$

This is my working out of the problem so far, I want to know if there is a more simpler way to solve this, or I would just be interested in other methods that one could use to solve a similar problem
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0answers
18 views

Setting up expression for distance of random point to origin

Problem Statement: Suppose that the coordinates of a point $(X, Y)$ are such that $X ∼ U[0, 1)$ and $Y ∼ U[0, 1)$ Compute the probability that the distance from this point to the origin is ...
3
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1answer
41 views

Should bridge/hearts/etc players sort their hands behind a screen?

In many card games, players are dealt a hand and then sort it for their own convenience. It occurred to me that this might be a way of leaking information about your hand, although probably not in a ...
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0answers
26 views

Help understanding the Binomial Distribution for nonindependent events?

Apologies if this seems like a basic question, but I haven't been able to find a clear answer in my textbook or online. The original problem; Please determine the probability of finding exactly 3 ...
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0answers
18 views

Expected distance between leaf nodes in a binary tree

Let T be a full binary tree with 8 leaves. (A full binary tree has every level full). Suppose that two leaves a and b of T are chosen uniformly and independently at random. The expected value of the ...
0
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1answer
31 views

If A is an event with probability 1, then A is the sample space: true or false

If the statement is true, prove it. If it is false, give a counterexample. If $A$ is an event with probability 1, then $A$ is the sample space. I know the statement is false. I just can't think of ...
2
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0answers
27 views

Maximum (expectation of) volume of a regular tetrahedron whose angles are i.i.d. random variables

I am proposing here a multi-d version of the question I proposed here and from which I have really learnt a lot. Problem. Given three points "on the Earth", i.e. three couples of longitude/latitude $...
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1answer
55 views

Kelly bet question

I stumbled upon Kelly bet during studies and decided to try simulating it. I chose a simple example where $b = 1$ and $p = 0.6$, which gave me the Kelly bet equal to $0.2$ of the wealth. I took $25$ ...
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1answer
15 views

Probability Mass Function, Cumulative density function, probability distribution function [on hold]

Can someone help with these few problems and help explain how I can answer them.
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2answers
15 views

Suppose we have 2 sets of billiard/pool balls, what is the probability of drawing 4 unique balls?

The Cue ball is not included in either set, so we just have 1-15 in each set (30 total balls).
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2answers
35 views

Compare two normal probabilities

Assume $X_1,X_2,X_3$ are IID $N(0,1)$ random variables. I want to show that $$P(X_1>0,X_1+X_2+X_3<0)>P(X_1+X_2>0,X_1+X_2+X_3<0)$$ I know I can show this by calcultating these two ...
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1answer
41 views

Can $\min_{a,b} E[(V -(aU+b))^2]$ be found without differentiation?

Let $(U,V)$ be joint random variables (assume zero means for simplicity). It is well known that \begin{align} \min_{a,b} E[(V -(aU+b))^2] \end{align} is minimized by $a= \frac{E[VU]}{E[U^2]}$ and $b=...
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0answers
34 views

Odds of winning a video game combat

We have a combat between 2 characters in a video game. Character 1 has starting hit points = 'c1_hit_points' Character 2 has starting hit points = 'c2_hit_points' Every turn character 1 deals a ...
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2answers
67 views

Show that $\mathbb E[X_{n}]\xrightarrow{n \to \infty} \infty$ while $X_{n} \xrightarrow{n \to \infty} 0$ a.s.

Say I have a biased coin that shows heads with probability $p \in ]1/3,1/2[$ and I initially have capital of $100 $EUR. Every time heads is shown, my capital is doubled, in the other case I pay half ...
5
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3answers
54 views

Sum of three independent random variables

Let $X$, $Y$, and $Z$ be independently and identically distributed variables, each uniformly distributed between $0$ and $2$. What is the probability that $X+Y+Z\leq2$? All the other answers on ...
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0answers
9 views

MATLAB: color scale with probplot

I have two 1D vectors of the same length, data and my_parameter. I use probplot to see the ...
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0answers
21 views

Finding the expected value of a negative binomial distribution with two success indications

Flip a coin until a head appears or until the fourth trial. Let $X$ be the number of coin tosses. What is $E(X)$? ...
3
votes
2answers
30 views

$X \sim \exp(1)$, $Y \sim \exp(1)$ Independent what is the CDF of $Z=X-Y$?

Let $X \sim \exp(1)$, $Y \sim \exp(1)$ $Z=X-Y$. $X,Y$ are independent. What is the distribution of Z? For $t\geq0$, I simply calculated that the old fashioned way. $F_Y(t)=P(Z\leq t)=P(X-Y\leq t)=\...