For questions related to Bayes' theorem, a result about conditional probabilities.

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Use of Bayes theorem in the Lovásk local lemma

Here's a line from the proof on Wiki I don't understand. $$\Pr(A\mid\bigwedge_{B\in S}\bar{B}) =\frac{\Pr(A\bigwedge_{B\in S_1}\bar{B} \mid \bigwedge_{B\in S_2}\bar{B})}{\Pr(\bigwedge_{B\in ...
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Probability problem involving Bayes' Formula

Two brothers share a car. They each have n keys in their pocket. They try one key at random and discard it until they can get the right one to start the car. Brother $A$ has only $1$ compatible key in ...
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Bayesian urn questions

There are two urns, each with four ping-pong balls. In one urn, three of the balls are red, and one is white; in the other, three are white, and one is red. Without knowing which urn you are choosing, ...
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Bayes' Rule Question

I am reading about Bayes' rule, I can solve all the exercise but this one. Suppose you had a checkup, and there is a bad news; you tested positive for "the giggles" and that the test is 99% accurate( ...
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Incremental Bayes Rule

I want to use Bayes theorem on a problem multiple times. I am given the following information: $P(G) = 0.7$ $P(M) = 0.3$ $P(ET|G) = 0.3$ $P(ST|G) = 0.5$ $P(NT|G) = 0.4$ $P(ET|M) = 0.0$ ...
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Help with Bayes Rule and Probability

I've seen many similar problems to this one, but I'm still a bit confused as to how to solve it: Two buckets: bucket $1$ has $3$ black balls, $7$ blue balls bucket $2$ has $7$ black balls, $3$ blue ...
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Possibility of analogy of the Bayes theorem for expectations

I recently found a scenario where I wanted to find the relation of $ E[X|Y] $ and $ E[Y|X] $ for $ X,Y $ two random variables. For probabilities and densities we have the Bayes theorem which is well ...
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Probability and Bayes Theorem

I have been trying to solve this problem using Bayes Theorem all week with no luck. A study showed that in​ 1990, $45​\%$ of all those involved in a fatal car crash wore seat belts. Of those in a ...
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posterior probability of bag given ball (evidence)

Question: Given the distribution of the coloured balls in three different bags: - Bag A: 1 Red 2 Black 2 Blue - Bag B: 2 Red 4 Black 4 Blue - Bag C: 10 Red 2 Black 3 Green we carry out ...
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Marginal probabilities

I am stuck on the following problem of calculating marginal probabilities, which I have highlighted in yellow: Given the information below, how do we calculate$ p(X=0|w=\frac{1}{4}), ...
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Bayes risk and Bayes decision

We are considering a sample of size $n$ from an exponential distribution, with parameter $w >0$. We wish to produce an estimate for $d$, for $w$ , with loss function: $L(w, d)=w(w-d)^2$ The prior ...
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Bayes and James-Stein estimators for the means of independent Gaussian random variables

I have tried to solve thas question but I cant find the way to solve it. this is the question: Bayes and James-Stein estimators for the means of independent Gaussian random variables thanks for any ...
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1answer
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Understanding Bayes' theorem with uniform distribution

Station $X$ begin to transmit a message in $[0,20]$ with uniform distribution, and $Y$ also want to transmit a message in $[6,14]$ with uniform distribution. Assume that transmission takes $2$ ...
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Online Password Probability Question!

Information An online security firm has surveyed customers from a large bank to investigate the quality of their passwords. The survey classifies passwords into three categories. Bad: 20% use this ...
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Bayes–If the prior is increased by a factor of n, what happens to the posterior? If using a threshold, does higher prior mean more “false positives”?

If we're using Bayesian inference in two situations where everything is the same, except that the prior in one is n times the prior in the other, is there anything we can say about how the posteriors ...
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Applying Bayes theorem to a simple problem

When a test for steroids is given to soccer players, 98% of the players taking steroids test positive and 12% of the players not taking steroids test positive. Suppose that 5% of soccer players take ...
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Bayes Rule for Multiple Dependend Random Variables for parameter estimation

During implementation of Expectation Maximization algorithm I got stock on this one P(X|Y,Z, theta), which I tried to solve as follows however I do not know if it is correct $P(X=x | Y=y,Z=z, theta) ...
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Bayes theorem for joint probabilities

I want to use bayes theorem to essentially 'swap' the $A$ and $B$ without moving $C$ as in the sense of, $$ p(A|B,C) \propto p(B|A,C) $$ can this be done and what are the other terms which make this ...
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what is the probability that drug was effective for this person knowing the Poisson distribution of the number of time that person get sick

The number of colds person gets in a year has ${\rm Poisson}(3)$ distribution. A new drug lower it to ${\rm Poisson}(.75)$ and is effective for $8$ out of $10$ people. The entire population was ...
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Bayes Theorem - Having trouble applying it

I'm trying to apply Bayes Theorem in a assignment's problem but I'm having trouble with it. Here is the question: "A insurance company classify its insureds into two risk categories: 80% of them are ...
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Deformations using the Bayes' theorem

I couldn't do the following deformations using the Bayes' theorem: $$p(\phi|x,\eta) = \frac{p(\phi,x|\eta)}{p(x|\eta)} = \frac{p(x|\phi)p(\phi|\eta)}{p(x|\eta)}$$ I can understand that the first ...
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Bayes Theorum with Multiple conditions with independent ancenstors

I have the following Bayes' Net: {D}->{C}<-{A}->{B} I need to find P(A|B,C) with B and C both being true. I have calculated the probability of A given B using the formula $$ ...
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Why this equation holds, using Bayes formula?

My mysterious equation is: $$p(x|\chi)=\int_{\theta\in\Theta}p(x|\theta)p(\theta|\chi)d\theta$$ where $\chi$ is some samples drawn from sample space parameterized by $\theta\in\Theta$. Follows the ...
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Deriving the joint posterior pdf as a decomposition in terms of…

Really struggling with how to approach this question. The lecturer, as per usual, has provided us with the bare minimum in terms of hints on how to approach this. I know how to do it when we want in ...
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Calculating the PPV and NPV using Bayes' Theorem [closed]

A kit manufacturer markets a diagnostic kit for use in mass screening for Thyroid disease. The diagnostic kit has the following specification: Diseased Population: Has response variable of ...
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2answers
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Is it true that $P(A|B)+P(A|\overline B) = 1$?

tl;dr $P(A|B) + P(A|\overline B)=1$. My question is, is this true? More detail The book I'm reading (Statistics for Business and Economics, Paul Newbold et al) has this example (paraphrased a ...
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Bayes' theorem assumptions (using continuous distributions)

I have computed $\frac{g(y\mid x)f(x)}{g(y)}$, where $g(y\mid x)$, $f(x)$, and $g(y)$ are density functions (i.e. they integrate to 1 and the functions are always $\geq0$). What assumptions must I ...
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Probability of selecting balls from Bag B

There are two bags A and B. Bag A contains $3$ white and $4$ red balls whereas bag B contains $4$ white and $3$ red balls. Three balls are drawn at random (without replacement) from one of the bags ...
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Bayesian inference for sum of random variables

Assume that we have a random variable $Z = X + Y$ for $X$ and $Y$ independent. Then if w use two independent data-sets $D_1$ and $D_2$ to try and approximate the distribution of $Z$, i.e. ...
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How to approach this kind of (conditional) probability problem?

Edit: I would appreciate any kind of input. This is an old homework task I'm still not able to solve: A DNA sample was taken from a murder scene. Statistically, only 1 in a million people is a ...
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Bayes' Net Conditional Probability

I have the following Bayes Net. And I need to calculate $P(R\mid W)$ and $P(S\mid W)$. For, $P(S\mid W)$, is it $.1 \cdot .9$ because I multiply the probabilities of those two events that the ...
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Bayesian probability on Bernoulli distribution

Let $D$ be a Bernoulli distribution with $P[X=1] = \theta$ (and so $P[X=0]=1-\theta$). Let $\chi = \{0,1\}$ be an iid sample drawn from $D$. Assume a prior distribution on $\theta$, with $\theta$ ...
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Calculate the probability that a certain percent of the products supplied by a factory are sold

The problem: A shop sells 2600 pieces of a certain product, supplied by factories A, B, and C, in the following quantities: 3000 pieces from factory A, 2600 pieces from factory B, and 4200 pieces ...
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Are specificity and selectivity independent variables in regard to bayes's rules?

I was trying to refresh my memory of bayes rule. As far as I understand, P(test is true| X) = sensitivity. P(test is false|X) = specificity. (Let me know if ...
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Integrating over parameter in Bayes

I am going over the paper "Sparse Bayesian Learning and Relevance Vector Match" by Michael Tipping. There is one equality there which I do not fully understand. He states: $$p(t | \alpha, \sigma^2) = ...
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How do I find the following probabilities? (The details are below)

An auto store has two suppliers for a given part. The store orders from Supplier A 70% of the time, and supplier B 30% of the time. Supplier A's parts are defective with probability 0.02, and supplier ...
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Bayesian update and discontinuous cdf

I have an uniform random variable $x\sim U(0,1)$. I receive a signal $z$ about $x$ that is given by $$ z=y(x)+\varepsilon $$ where $\varepsilon\sim U(-\frac{1}{2},\frac{1}{2})$ (independent from $x$) ...
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Combining n simultaneously occuring probabilities of an event occuring into one summative probability

I am a bit lost with regards to the problem described a bit further down, because though many methods to approach it are documented in available literature, the verdict as to which model is the most ...
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Deck of Cards Stats Probability Question [closed]

Randomly select two cards in sequence from a full deck of 52 cards, what i s the probability that the first one is a King given that the second one is a King. If someone can please help me with this ...
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How do you picture: $\Pr(B|A)$ shrunk down by $\Pr(A)$?

Though understanding these diagrams, I do not understand how to visualise the following explanation: $\color{green}{[P1.]}$ Suppose you were to grab the edges of $A$ and stretch it out so it ...
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Expected Value Problem Using Confusing Conditional Probability

I am trying this question: We have a bag with 10 blue jacks and 20 red jacks. We pick 3 jacks from the bag at random and with replacement. We are told that at least one jack is red. Compute the ...
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Why total probability is the sum of conditional probabilities?

Consider the following question taken from this link, question number $25$: We have four boxes. Box $1$ contains $2000$ components of which $5$ percent are defective. Box $2$ contains $500$ ...
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What is the Bernoulli class conditional distribution?

What is the Bernoulli class conditional distribution? I am trying to implement a procedure for computing a naive Bayes classifier for binary features with a Bernoulli class conditional distribution. ...
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Test predictability with Bayes' Theorem

Say we have a disease and a test for it. P(A :=a person has the disease)= 0.01 ( example) P( B:=test is positive | A )=0.95 Is this enough information to calculate the probability that a person has ...
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An explanation of how this solution is derived

I am having difficulty understanding the solution to this problem. Since the solution is in the form of Bayes theorem I expected something along the lines that looked similar to Bayes theorem. ...
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If the $\Pr($hypothesis) is due only to chance, then what is the problem?

Source: p 224, Think: A Compelling Introduction to Philosophy (1 ed, 1999) by Simon Blackburn. I capitalised miniscules, which the author uses for variables. I pursue only intuition; please do not ...
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Why is the accuracy of $\Pr($hypothesis) in Bayes's Theorem less important than apparent?

Source: p 224, Think: A Compelling Introduction to Philosophy (1 ed, 1999) by Simon Blackburn. I capitalised miniscules, which the author uses for variables. I pursue only intuition; please do not ...
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For the same Conditional Probability, why does Bayes's Theorem differ from a direct calculation?

Abbreviate: S = a person is diseased, + = the test is positive. Presume: $\Pr(D) = 0.001, \; \Pr(+|D)=0.99, \; \Pr(+|D^C) = 0.01 \qquad ($$\iff$ $ \Pr(-|D^C) = 0.99).$ 1. Use Bayes's Theorem: ...
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Lower bound of Nearest Neighbour Rule

It is stated often as a matter of fact that the lower bound for Nearest Neighbour rule is the Baye's rate. However when I tried to mathematically prove it,I hit a dead end. For reference : Error for ...
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$P(A=0, B=0)$ is what given the following graph?

Graph and Probabilities Given this graph and respective probabilities, what would be the value for $P(A=0, B=0)$? I computed $P(A=0, B=0)=P(A=0)P(B=0)=0.24$ because A & B are independent of D. ...