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Questions tagged [conditional-probability]

In probability, conditional probability, is the probability that an event occurs given something else has already occurred.

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Probability of two dice against one

If you roll two dice, what are the odds that at least one has a higher value than a third die you roll?" And for (B), it's the same as (A) except you roll three dice before checking against a fourth?
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Probably of winning with 2 dice (maximum of them) against another one

could some of you help me to find out what is the probability of A) obtain with two dice a greather number than another die? B) and if the dice are 3 how can I do? Not the sum of the 2 dice, but the ...
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Probably of winning with 2 dice against 1

could some of you help me to find out what is the probability of A) obtain with two dice a greather number than another die? B) and if the dice are 3 how can I do? Not the sum of the 2 dice, but the ...
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Let X and Y iid be ~ N(0,1)[Gaussian]

Given Z=XY.Find the conditional pdf Z given X=x. The way I calculated assumed W=X, then proceeded with Jacobian and replaced W=X in the final answer. Not sure whether it is right or not. Answer i got ...
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How to find conditional probability, given parent node and child node

Currently I am working on a sample question for my course: Calculate P(Sprinkler | Cloudy=True, WetGrass=True) based on this simple Bayesian Network diagram. My process is as follows: Given the ...
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Conditional expectation of number of trials

Consider $n$ independent trials, each of which results in one of the outcomes $\{1, ..., k\}$, with respective probabilities $p_1, p_2, ...,p_k$ where those probabilites sum to $1$. Let $N_i$ denote ...
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Find the conditional probability of this event.

Suppose a lab test has the following statistics for detecting a certain disease. $A$ is the event that the test result is positive, and $B$ is the event the tested person has the disease. $P(A \mid ...
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Deriving marginal likelihood formula

The formula for marginal likelihood is the following: $ p(D | m) = \int P(D | \theta)p(\theta | m)d\theta $ But if I try to simplify the right-hand-side, how would I prove this equality $ = \int \...
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How many different possible sequences of 6 marks can Ben achieve?

Ben attempts all 6 questions of a maths test in order. Each question is marked out of 1-10. Ben never scores more in a later question than in an earlier question. How many different possible sequences ...
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Expectation, variance and conditional probability of combined discrete and continuous random variables

Category: Introductory Probability I have seen many of the other questions with similar titles (there are quite a few!), but unfortunately I am struggling to apply the concepts and knowledge I have ...
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Separating conditional joint probability

I am trying to separate the probability expression $P(S,\vec \theta_a, \vec \theta_b, \vec \theta_c | \vec \alpha)$. It describes the probability of a sequence S, which is itself based on the three ...
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I think there is a little mistake in this exercise about the memoryless of Geometric Distribution

An exercise of Jacod and Protter: Let $X$ be Geometric. Show that for $i, j > 0$, $$P(X > i + j | X > i) = P(X > j)$$ I did it and I got a different asnwer: $$P(X > i + j | X > i) = ...
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Conditional probability of four events

Suppose I have a group of people where $\ A_1 $ willing to recycle papers, $\ A_2 $ willing to recycle plastic $\ A_3 $ willing to recycle glass and $\ A_4 $ willing to recycle batteries. and I'm ...
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Example of E(E(X|F)|G) \neq E(E(X|G)|F) [duplicate]

Can you find an example where E(E(X|F)|G) $\neq$ E(E(X|G)|F) (F and G is $\sigma$-field in probability theory)
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Bayes Theorem to find probability of getting a sum given a coin toss

Suppose you flip a fair coin, and if the result is heads you will roll a pair of fair dice, and if the result is tails you will roll the biased dice. Using Bayes’ Theorem: a. derive the probability ...
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Let us have 10 balls in a urn, probability that all the balls in the urn are white if…

Let us have a urn with 10 balls, the balls are either white or black ( we don't know in which proportions ).We extract 4 balls without reintroducing them back and after each extraction the ball is ...
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is this P(X<Y | Y = y) = P(X<y) true

Is this formula true? $P(X<Y | Y = y) = P(X<y)$ I am trying to solve this with conditional formula, but don't know how $P(X<Y | Y= y) = \frac{P(X<Y, Y= y)}{P(Y=y)} = \frac{P(X<y)}{P(...
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Understand the probability formula of a random walk

I have the following problem: Let's assume $G$ is a graph with vertices in red or blue colour. There is no limitation on how we connect the vertices, i.e., a red vertex can be connected with either ...
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Conditional probability - formula validity over different sample spaces

Is the conditional probability formula $P(B\mid A) =\frac{ P(B\cap A) }{ P(A)}$ always true? What if the sample space is non-uniform?
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How to calculate probably when the odds change over time

Sorry for the dumb wording, or asking a question that may have been answered before, I'm not familiar with the vocabulary so I don't really know how to ask the question or what to search for. I can ...
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What is the meaning, both intuitively and mathematically, behind the probability of a set?

I'm currently studying a course on applied probability and I keep running into a notation that I can't quite understand. Let's say we are working in a progrability space $(\Omega,\mathcal{F},P)$ and ...
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$P[X+Y \in \Gamma \mid Y=y]=P[X+y \in \Gamma] \text{ for } PY^{-1} \text{-a.s.} y \in \mathbb{R}^d$

Let $X,Y$ be two random variables on $(\Omega,\mathcal{F},P)$ taking values in $\mathbb{R}^d, \mathcal{B}(\mathbb{R}^d)$ such that $X$ is independent of $\mathcal{G} \subset \mathcal{F}$ and $Y$ is $\...
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Approximate $p(X_{1000000}=2|X_1=1)$

Let the distribution on variables $(X_t)$ satisfy a Markov Chain, where each variable can take values {1,2}.And we are given that$ p(X_1)=0.5$ given that the transition matrix P is: $P $= $$ \...
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Cheat Card Game Probability [on hold]

I'm a college student working on a Computer Science project where I am going to make a playable GUI for the cheat card game (rules and explanation: https://www.theguardian.com/lifeandstyle/2008/nov/22/...
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Binomial Distribution Application in determining conditional probability

Teams A and B play a series of games, with the first winning 3 games being declared the winner. Suppose that A independently wins each game with a probability 'p'. Find the conditional probability ...
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Probability of an event with multiple conditions

Could you inform me please, how can I calculate conditioned probability of several events? I have 3 events A, B, C; I know P(B|C) and I want calculate P(A|B,C). Is it possible? In the special case B,...
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Probability theory - Logic, Notation, simulation

i need some help in probability theory. The thing is im not sure if im thinking about this correctly and if i express my thoughts correctly. I really got lost in all the dice examples of the internet. ...
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Conditional Probability Drawing Candy

I figured out the first one and need help with the second, with the second one could you provide solutions for both with and without replacement? I have two bowls of candy. One is supposed to be ...
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calculating variance and expectation of unknown binomial variables over a window

I have $2^m$ independent random variables. All have binomial distributions, each with $m$ samples. The probability of success for each binomial distribution is somewhere in the range $[0,p]$ (...
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Game theory - a shooter problem

I just met an interesting question but did not know how to approach it... Suppose two gunmen (A and B) are moving in a straight line towards each other in a fixed speed. Each of them only have one ...
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Book on probability theory which is comprehensive and covers covariance, conditional probability and probability distributions, all with proofs?

My background is in Computer Science and I'd like to establish a strong foundation in probability theory. I was reading the GraphSLAM paper to get a sense of the algorithms used for SLAM purposes in ...
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Finding the Conditional Distribution of a Random Variable Given Another Random Variable of a Different Distribution

This is the problem i'm currently tackling from my textbook: Let $P$ have a uniform distribution on $[0,1]$, and, conditional on $P=p$, let $X$ have a Bernoulli distribution with parameter $p$. ...
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Is this probability function monotone? [on hold]

If two random variables $\tilde{x}$ and $\tilde{y}$ are mutually independent, $\tilde{x}$ has pdf $f(x)$ and CDF $F(x)$, $\tilde{y}$ has pdf $g(y)$ and CDF $G(y)$. Given a constant $R$, $\Delta(R)\...
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Marginal pdf from conditional pdf

Let the conditional pdf of $X$,given $Y=y$ be given by $f(x|y)=e^{y-x} , x>y$ and let $Y$ have the pdf $g(y)=\lambda{e^{-\lambda y}},y>0,\lambda>0,\lambda \neq 1$ We need to find the ...
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Conditional probability between two random variables

A conditional probability $P(X |Y)$ where $X$ and $Y$ are two random variables can be represented in a graphical way as: Now, my question is: does $P(X=x|Y)$ make sense? If the answer is affirmative,...
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Question using Bayes' theorem with marbles and rolling a die

A urn contains 10 white marbles and 20 blue marbles. You roll a die and pick the amount of marbles without replacement out depending on the number you roll on the die. Find the probability you roll a ...
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Is this step in computing the conditional expectation of a product correct?

Given a sequence of independent variables {$X_i, i \in \mathbb{N}$} with $X_i$ ~ $N(0,1)$ for all $i$ and a Poisson distributed random variable $Y$ with parameter $1$ independent of the aforementioned ...
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Conditional probability for Bayse rule.

So I have the following problem. There are 300 million people and 2 million of them are green. Say there are 10 people who are terrorists. 9 out of 10 of these terrorists are green. What is the ...
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Conditional probability of intersection

Let $A,B R$ events such that the probabilities $P(R)$, $P(A|R)$, $P(A|R^c)$, $P(B|R)$ and $P(B|R^c)$ are known. Assume $A$ and $B$ are independent. I would like to compute $P(R|A\cap B)$. With total ...
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Is $P(X=x|Y=y)\le P(X=x)$?

I know that $$P(X\,|\,Y)\le P(X)\quad\Leftrightarrow\quad \frac{P(X , Y)}{P(Y)}\le P(X)\quad\Leftrightarrow\quad P(X , Y)\le P(Y)P(X)$$ is a proof for $P(X=x|Y=y)\le P(X=x)$ but i don't get the ...
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Don't understand the answer to a uniform distribution question with conditional probability for probability and statistics.

The question is this: You arrive at a bus stop at 10 A.M., knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30. (a) What is the probability that you will have ...
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figuring out a conditional probability question

here is the question: a contractor is hired for the job of paving streets and roads, in the end if the quality of pavement is approved by quality assessments, contractor's work is accepted. we know ...
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If $P \ll Q$, are the regular conditional probabilities a.s. absolutely continuous?

Let $P$ and $Q$ be probabilities on $(\Omega, \mathcal{A})$, and let $\mathcal{F}$ be a sigma-subalgebra of $\mathcal{A}$. Assume $P \ll Q$. Assume that $P(\cdot \mid \mathcal{F})$ and $Q(\cdot \mid \...
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Probability exercise - system reliability

Probability of failure of each component in subsystem A & C is 0.1. The failure of each component is independent of each other component and both components of subsystem B have the same unknown ...
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conditional probability problem anomaly

This link: https://www.youtube.com/watch?v=5UQU1oBpAic, points to a youtube video that presents the following problem: suppose you have a chess match against an inferior opponent where 3/4 of the ...
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Probability of Drawing the nth Ball from urn A, and then figure out the values of certain constants

"Consider two urns (A and B), each containing both white and black balls. The probabilities of drawing white balls from the first and second urns are pA and pB, respectively. Balls are sequentially ...
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Conditional distribution of mean parameters in a Normal mixture

I'm solving an exercise from Casella's book, Introduction to Monte Carlo with R and he askes us to build a Gibbs sampler for the following mixture of Normals \begin{align} p\mathcal{N}(\mu_{1},\sigma^...
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Number of parameters for conditional probabilities

Let's assume we have P(A|B) with A and B beeing two binary random variables. Why do we only need 2 parameters to specify the distribution? Namely P(A=true | B = true) and P(A = true | B = false).
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Bayes rule for multiple conditions

Let us consider following table: Let us suppose that those are conditional probabilities, and we are asked to predict ...