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Questions tagged [conditional-probability]

For questions on conditional probability.

Conditional probability is the probability that an event occurs given that another event has already happened. The probability of an event $A$ given another event $B$ is written as $P(A|B)$, and is related to the marginal and joint probabilities via $$ P(A|B)P(B)=P(A\cap B)$$

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If $A$ and $B$ are independent, when are they conditionally independent on $C$?

This problem comes from Grimmett's Probability and Random Processes Problem 1.5.5. Suppose that A and B are conditionally independent events given $C$, and they are also conditionally independent given $C^c$, and that $0 < P(C) < 1$. Prove that…
kordon
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Conditional Probability with a Ball Chosen From a Random Urn

I have 10 Urns; 9 contain 2 White and 2 Black Balls, one urn contains 5 White and one Black ball. A ball chosen from a random urn is white. What is the probability that it came from the urn with the five white balls? I know I have to use Bayes…
Daniel S.
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Probability after the event

A friend of mine says that you can't calculate the probability of an event after it has happened. The reason for that is that it has a probability of 100% for the event has occurred. Secondly he states that any event is highly improbable, say me…
user100940
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Understanding conditional probability in a problem

The problem is as follows: Senior students tend to stay up all night and therefore are not able to wake up on time in morning. Not only this but their dependence on tuitions further leads to absenteeism in school. Of the students in class XII, it…
Akilan SS
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Conditional independence for 3 events

I have the following statement: Prove or disprove if $P(A_1 | A_2) = P(A_1)$ then $P(A_1 | A_2, A_3) = P(A_1 | A_3)$ I had tried things like: $P(A_1 | A_{2,3})=\frac{P(A_{1,3})}{P(A_{2,3})} = \frac{P(A_3 |…
ESCM
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Conditional distribution of one random variable given another, find $f_{X|Y =1}(2)$

From a well shuffled deck of $52$ cards, four cards are selected at random. Let the random variable $X$ denote the number of queens drawn, and let the random variable $Y$ denote the number of kings drawn. Find $f_{X|Y =1}(2)$. How to solve this…
Abbas
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Need Help Understanding The Three Prisoner's Problem

I am working through the book Statistical Inference by Casella and Berger. While I understand that most of probability theory is done heuristically, on a first passing of material I like to be more formal just to get an idea of things. In one of…
Sprinkle
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How to read this symbol "$\varpropto$" in the following equation and what is its meaning?

I stumbled upon the following equation in a research paper and I don't know how to read the symbol "$\varpropto$" in the context of the equation. Is it read "proportional to"? If so, what does it mean in the following equation? $$p(e\mid a,f) =…
Ragavan N
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Conditional probability in Heckman model

Two regressions: $$ y_{1i}=x_i\beta+u_{1i} $$ $$ y_{2i}=z_i\gamma+u_{2i} $$ Two errors are correlated. $x_i$ and $z_i$ I suppose are not necessarily independent (but it is not so clear from my lecture notes). Some background: The two regressions are…
jasmine
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Conditional Probability P(A ∩ B)

So I found myself in a infinite loop while trying to do some probability. If A and B are independent, calculating P(A ∩ B) is as simple as P(A)P(B). However, how do I calculate P(A ∩ B) if they are dependent? I know…
jathu
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Conditional probability for Bernoulli

Let $W = (1 - Z)X + ZY$ where $X \sim N(0, \sigma^2)$, $Y \sim N(\mu,\sigma^2)$ and $Z$ is Bernoulli with $P(Z = 0) = P(Z = 1) = 1/2$. $X,Y$ and $Z$ are independent. I try to find the distribution $p(Z|W)$. It seems $Z$ and $W$ are not independent.…
eet
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How does probability change as a group becomes less "average"?

500 people are asked asked a question with a difficulty rating of 1 in 5. Meaning, the question is graded as being the type of question that, on average, only 1 out of 5 people asked are likely to get right. Therefore, it can be assumed that around…
RevPhillips
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Confusion related to probability distribution

I was reading a book where it said that if the distribution of a multivariate gaussian with three variables $x,y,z$ given by $\pi(x,y,z) = f(x,z)g(y,z)$ then $x$ and $y$ are conditionally independent given z. How it be true? Can anyone help me?
user34790
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Is Independence of two events generalised by conditional independence of those two events?

If $\mathbb{P}(A\mid C)* \mathbb{P}(B\mid C) = \mathbb{P}(A \cap B\mid C)$ i.e A and B are independent given C has happened, then can we generalise and say that A and B are independent irrespective of C? My explanation is that as A and B are…
pavan sai
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Conditional probability for an exponential variable

Let $X$ be an exponential random variable with parameter $\lambda$. I have to find $\mathbb{E}[X\mid X<\alpha]$ where $\alpha >0$. I must find $\mathbb{P}(X\mid X<\alpha)$ first. What I did is as follows: $\mathbb{P}(X\mid…
Bhavish Suarez
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