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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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Genius prediction

It's a long story, but trust me, it's worth it. Matt is a detective and there's news of 3 murders about to happen in the upcoming week. Matt's usual success rate is 60%, i.e. He'll be successful in a ...
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Bayesian network problem: third day rainy, given first day is

I've tried searching for this problem online but could not find a solution, hopefully you can help me. I have three random variables [r1,r2,r3], these three variables shows the probabilities of it ...
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How $\mathbb{P} (A | B \cup B^c)$ is $\mathbb{P} (A | B) \cdot P(B) + \mathbb{P} (A | B^c) \cdot \mathbb{P} (B^c)$?

P(A) = $P(A|\Omega)$         = $ P(A|B \cup B^c)$ But how to reach P(A | B) * P(B) + P(A | B$^c$) * P(B$^c$) from P(A | B U B$^c$) ?
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How is P(N|$H_3$) derived?

An exercise with a solution attached below. I do not understand how is, in ii), $P(N\mid H_3$) derived. Could somebody please explain me?
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Question about how to obtain the value of f X|Y(x|y)

In the following example question (from Bertsekas, edition 1), i have one question: Why the value of fY|X(y|x) is 1/2? Is it because Y is Y|X is either 0 or 1/6 (50% probability), or because some ...
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Probability for sampling at least one member of a set from a larger set

Let's assume the following: Population size: N Individuals with a particular feature: x% of N Sample size: y% of N What's the chance that we get at least one ...
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Law of total variance and covariance given X and Y are normal

I have a problem which asks me to find $\Bbb E[Y]$ and $Var(Y)$ given that $Y\text{~}Normal(x,1)$ conditional on $X=x$. $X$ is standard normal. So I have worked out that $\Bbb E[Y]=0$ using the law of ...
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How to use the law of total variance

I know that the law of total variance states $$Var(X)=\Bbb E[Var(X|Y)]+Var(\Bbb E[X|Y])$$ But how does one treat $Var(X|Y)$ and $\Bbb E[X|Y]$ as random variables? For example, say we know that $$\Bbb ...
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Conditional Joint probability of three random variables

I have encountered a problem in my work where x,y,z are my independent exponential random variables. I need to find out the probability given I have the limit for random variable z which is $(z \lt \...
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Can Conditional Expected Value be negative in normal distribution?

So, the problem gives me this facts (for a Normal bivariate distribution X,Y) $$Var(Y|X=x) = 5$$ $$E(Y|X=x) = 2 + x$$ It asks me to find $$E[Y^2|X=7]$$ I tried this: using the conditional variance ...
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Expectation of Gaussian r.v. conditioned on positive r.v.s with positive covariances is positive

Suppose that $(X_1,\dotsc,X_K)^T \sim \mathcal{N}(0, \Sigma)$, with $\mathrm{cov}(X_i, X_j) > 0$ for all $i,j$. Prove that $$ \mathbb{E}[X_K 1\{ X_1> 0, \dotsc, X_{K-1}> 0 \} ] > 0$$ ...
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Conditional Interpretations of Linear Regression

We estimate a linear regressor in the 1 dimensional with x and y random variables with zero mean: y/x = $\alpha$ x We can rewrite this using the variance of the variables as: y/x = $\rho \frac{\...
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combine several dependent variables

I have 3 boolean variables X,Y,Z (indicating if the same event has happened). Let's assume the error of each of them is 0.3. This means, for example, that if X is true then the probability of an ...
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conditional probability about gambler winning x amount of coins

A gambler plays seven games one after the other and the chance to win each of them is $\frac{1}{3}$, independently of the others. For $k = 1, ..., 7$, if the gambler wins game number $k$, then the ...
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Intuition of TailVaR

As per the actuarial guide I have called the CMP - from Acted - tailVaR is the expected loss in excess of the benchmark value L. I don't really get that, so I tried splitting the equation into: $...
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Finding the conditional probability of an event without indication if the events are independent.

I'm studying an introductory statistics textbook and, unfortunately, it doesn't come with an answer key. The textbook gave this problem, which I've spending hours trying to figure out: "Roll two fair ...
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How can calculate conditional pdf of Y when you dont know about f(y)

X is a uniform distribution on the interval (0,1). Y is a also uniform distribution on the interval (0,x). Its the only information that I could know. Then how can I calculate p(Y|x)? If you teach me, ...
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Actual meaning of the formula $E[\phi(X,Y)|\mathcal{G}]= E[\phi(x, \cdot)]|_{x=X}$

Let's suppose we have a probability space $(\Omega,\mathcal{F},P)$ and a sub sigma algebra $\mathcal{G} \subset \mathcal{F}$. Let's say we have $X$ that is $\mathcal{G}$-measurable and $Y$ that is $\...
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Conditional Independence and product of random variables

I am stuck at the following situation: Let random variables $Y, X, W_1, W_2$. I know that $W_1$ and $W_2$ are each independent from $Y$ conditional on $X$: $$p\left(Y\mid \{X,W_1\}\right) = p\left(...
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Conditional distribution for two random variables

I recently came across and exercise from a past exam and I was wondering how to solve it. Two independent random variables $A$ and $B$ are given and they both follow the exponential distribution but ...
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If $X\sim \mathrm{lognormal}$ then $Y:=(X-d|x\geq d)$ has approximately a Generalized Pareto distribution.

Let $X$ be a random variable with lognormal distribution. Show that when sufficiently large then $Y:=(X-d|x\geq d)$ is approximately a random variable with generalized Pareto distribution. Hint: Use ...
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Conditional probability with exponential distribution [closed]

Suppose that $X \sim \mathcal{E}(1.3)$ and $Y \sim \mathcal{E}(1.7)$ are two exponential random variables and define $U := \min\{X, Y\}$. How do I calculate following values? $\mathbb{P}[U > 0.32 ...
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Calculating the probability of conditional expectation

I'm absolutely baffled by this question: Given X~Geo(0.75), Y|X = x~Geo(1/(x+1) What is P(E(Y|X) = 2) ? I'm very obviously missing something - I know how to calculate a conditional expectation, ...
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Finding Conditional Expectation and Distribution of Shuffled Songs

You have a playlist of $N$ songs. You listen to $K$ songs on shuffle (as the playlist shuffles, it selects a song out of the $N$ with replacement). While listening, you observe that you listened to ...
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Proof for Void probability of a Binomial Point Process

We need to prove: $$P[N(B)=0|N(A)=n]=\left(1-\frac{|B|}{|A|}\right)^n$$ The attempt: Let $\bar{B}=A\text\B$ \begin{align}P[N(B)=0|N(A)=n]&=\frac{P[N(B)=0\bigcap N(A)=n]}{P[N(A)=n]} \\\\ & =...
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Independent of two Random Variables

Let $X, Y$ and $Z$ are all binary random variables, and $Y = X + Z \mod 2.$ $X$ and $Z$ are independent. a) Suppose $X$ and $Z$ are both uniformly distributed. Are $X$ and $Y$ independent? Why? (...
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Deriving distribution from conditional distribution

Hi guys I am having problems deriving $P(X = k)$ if $P(X = k|X+Y = n)$ = ${n}\choose{k}$ $\times$ $2^{-n} $ X and Y are i.i.d. random variables with values in $\mathbb{N_0}$. After playing a bit ...
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Finding conditional probabilities [duplicate]

A bag contains 25 coins. 24 are fair coins (50% chance of heads and 50% chance of tails). 1 Coin is bias (100% chance of heads) One coin is taken from the bag and is flipped n times to test if it is ...
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conditonal distribution question

For conditional distribution $$f_{X|Y}(x|y) = \frac{f(x,y)}{f_Y(y)}$$ this is the basic definition I know about conditional distribution Consider n + m trials having a common probability of ...
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Conditional probability combining discrete and continuous random variables

Consider the following problem, from Tijms's Understanding Probability: A receiver gets as input a random signal that is represented by a discrete random variable $X$, where $X$ takes on the value +...
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Conditional probability of min and max of two dice

Consider the following problem, from Tijms's Understanding Probability: Two dice are rolled. Let the random variable $X$ be the smallest of the two outcomes and let $Y$ be the largest of the two ...
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Conditional expectation in RA-DFM model

I have found such an equality in Bank of England paper (https://www.bankofengland.co.uk/-/media/boe/files/working-paper/2018/uncertain-kingdom-nowcasting-gdp-and-its-revisions, equation (15), page 15):...
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Probability of X being a trick coin (heads every time) after heads is flipped k amount of times

A magician has 24 fair coins, and 1 trick coin that flips heads every time. Someone robs the magician of one of his coins, and flips it $k$ times to check if it's the trick coin. A) ...
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Calculating probability when failure rate is known

I've been been developing a web application that features barcode scanning. Before implementing a fix, the failure rate was approximately 50%. The solution implemented involves scanning the barcode 5 ...
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Probability of colour with putting back twice.

I'm having a bit of trouble with following probability-related question: A container contains $5$ red and $10$ black balls. Take a ball out of the container at random and note its color. After ...
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Joint conditional probability of geometric and poisson distributions

I'm new to the concept of joint conditionals, and I want to make sure that a move I made is valid and logic. $$X\sim Geom(0.21) \rightarrow P(X=k)=(0.79)^{k-1}\times0.21$$ $$Y|X\sim Poisson(x+1)\...
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Expected value of conditional probability event

This is in continuation to Expected value of conditional events In a cricket match, event A is a subset of all possible moves taken up by the fielding team. The batsman has a set of moves, and, ...
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Find P(B) given the following values: [closed]

P(A) = 0.56 P(B|A) = 0.85 P(¬B|¬A) = 0.32 What is P(B)? Please answer using at least two decimal places. The correct answer is supposed to be 0.78 but I am completely stumped on finding this. ...
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Expected value of conditional events

This question is in continuation to: Obtaining binomial distribution from normal distribution, and repeated events. Let us consider a random variable, $x$ chosen from a normal distribution. Define, $...
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Conditional probability find P(C)

A and B are mutually exclusive events, and $P (A)=0.2$ and $P (B) = 0.8$ Find &P (AIC) when $P (CIA) = 0.4 $ and $P (ClB) = 0.5. $ From $ P (CIA) , P(C \cap A) =0.08$ From $P (CIB) , P (C \cap ...
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Independence of Events and Conditional Probability

A person tried by a 3-judge panel is declared guilty if at least 2 judges cast votes of guilty. Suppose that when the defendant is in fact guilty, each judge will independently vote guilty with ...
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How do I visualise dependency of two events?

Background/Context: I tried to solve the question For a certain probability experiment, the probability that event $A$ will occur is $\frac 12$ and the probability that event $B$ will occur is $\frac ...
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A person has five coins, two double headed, one double-tailed, two normal. Conditional probability with discard

A person has five coins, two of which are double headed, one of which is double-tailed, and the remaining two are normal. The person shuts their eyes, picks a coin at random and tosses it Suppose now ...
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calculate join probably of 3 variables

I have part of the join probability table here, x, y, z are binary variables. I also have P(x=0)=0.3, full probably table of P(Y|X) and P (Z|Y). I wonder how can I complete the rest of the joint ...
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Probability conditional on entry of exchangeable random vector: does the index matter?

I have a random varbiable $Y\in\{0,1\}$, which is dependent on a random vector $X \in \{0,1\}^n$. Therebey, the entries in $X$ are exchangeable. That is, $P(X_i=1)=P(X_j=1)$. I want to compute/...
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Conditional probability for 4 events

Alice tries for 4 job interviews A,B,C,D with the probabilities to pass the interviews $Pr[A]=0.1$ $Pr[B]=0.2$ $Pr[C]=0.3$ $Pr[D]=0.4$ There are no dependence to get accepted for different ...
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Scott Steiner's WWE promo

In Scott Steiner's WWE promo video, Scott mentions that in a three way match, everyone has a 33⅓ chance of winning, which is fair if you consider all fighters to be equal. But then, he proceeds to say ...
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Basic probability question on a joint conditional distribution

Why does $$\frac{Pr(A, B, C, D)}{Pr(D)}=\frac{Pr(A, B, C, D)}{Pr(C, D)}\frac{Pr(C, D)}{Pr(D)}$$ ? Edit - elaboration I do not understand the step-by-step operations (probability algebra?) that ...
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Factoring a joint conditional probability into a specific form.

I have an equation of the form: $$ Pr(A, B, C|D) = Pr(A, B|C,D)Pr(C|D)$$ but I'm having difficulty justifying why it is correct. How can you get the right hand side from the left?
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Conditional probabilities for sequence of events

The following table represents all possible paths of dichotomous events at 5 time moments. At each time moment either 1 or -1 event occurs with probabilities $p$ and $q$. Time stops when one observes ...