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

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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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5answers
84 views

Deck of Cards Stats Probability Question [on hold]

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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1answer
31 views

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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2answers
52 views

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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1answer
27 views

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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20 views

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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1answer
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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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probablitly using bayes theroem

Three numbered urns contain colored balls as described in the table below. One of the urns is picked at random and a ball is drawn from the urn; the ball is red. What is the probability the ball can ...
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1answer
27 views

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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1answer
33 views

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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2answers
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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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2answers
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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. ...
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1answer
21 views

Can this question be solved parametrically? Bayes rule related

I was wondering if its possible to solve this question in terms of $x,y,z$ (parameters, instead of actual numbers)/ We have a bag of red and blue balls. Suppose red count is $x$ and blue count is ...
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2answers
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Bayes' theorem and total probability problem.

Suppose 5 men out of 100 and 25 women out of 1000 are good orator. An orator is chosen at random. Find the probability that a male person is selected. Assume that there are equal number of men and ...
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1answer
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$P\left(X_1 < X_2 < X_3\right) = P\left(X_1 \le X_2 \le X_3|X_1\ne X_2\ne X_3\right)$?

I have 3 independent random variables, $X_i$, distributed on a continuous uniform distribution between 0 and 1. Does the following hold given the assumptions above? $$ \tag{1} P\left(X_1 < X_2 ...
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4answers
42 views

Law of total probability explanation about sample space

$P(Y=0)+P(Y=1)=1$ in above diagram. Arrows represent probability $P(Y=0\, \text{or}\, 1|X=0\, \text{or}\, 1)$ To use the law of total probability, I know that to find $P(Y=0)$, we need to find the ...
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Girl day problem [duplicate]

There are two children in a family, if one of the girl is born on friday, find the probability that both the children are girls?
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Poisson Conditional Probability with varying $\lambda$

The number of accidents that a given person has in a given year is a Poisson random variable with mean $\lambda$, independently each year. However, suppose that $\lambda$ changes from person to ...
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Which is the best indicator of probability here? Bayes

I am part of a group of teachers in DFW area. We are very competitive when it comes to our profession. So we like to have a little fun throughout the year by having “test battles”. We simply ...
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1answer
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Bayes' rule where the realization of random vector is a subset of the realization of a different random variable?

I have realizations of two different random vectors, where one is a subset (is that proper terminology here?) of the other $$s^t = (x_1,x_2,x_3,\dots x_\tau, x_{\tau +1},\dots x_t)$$ and $$ s^\tau = ...
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1answer
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Conditional joint probabiltiy P(A|BCDEF) from single conditional P(A|B), P(A|C), etc?

I have a matrix of conditional single probabilities, e.g.: P(A|D), P(A|E)...P(C|H); also, P(A) + P(B) + P(C) = 1. How would I express a joint conditional probability, say P(A|DEFGH), if DEFGH are all ...
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Find a conditional probability of a Bayes' Net knowing only the prior probability of the root.

Given three nodes A,B,C that form a Bayes Network as the following: (A)-->(B)-->(C) If we know the prior probability of A is 0.3, i.e. P(A)=0.3, is this ...
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1answer
28 views

Understanding Bayes' theorem through an example

Suppose I have three nodes A,B,C such that A and B are independent and pointed to C as the following: A --> C <-- B Also Suppose that each node takes a peobability between (0,1) so that the ...
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4answers
190 views

Can someone help me on this probability question please?

I'm quite comfortable with probability, but sometimes the wording of the questions REALLY throw me off. Given the following problem: On a production line, $12\%$ of items are imperfect, and $25\%$ ...
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1answer
26 views

Calculating a (Forward Measure) Martingale

Above is my question. I am, unfortunately, stuck on part (a)! Below are my workings. I've just spotted a typo -- at one point, an "$\exp$" is missing, but it's fairly obviously supposed to be there. ...
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2answers
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Bayes Theorem with multiple random variables

I came across this expression in the Intro to Probability book I am studying: $P(A,B|C)=\frac{P(C)P(B|C)P(A|B,C)}{P(C)}$ Could anyone please explain how is this obtained. From a simple application ...
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Bayes classification

What are the synonyms for 1) Bayes classifier 2) Bayes decision rule 3) Bayes decision function for uniform distribution I found many terms in literature and got confused because they look similar ...
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Finding conditional distribution $P(X|Y)$ parameters from concatenated Gaussians $X$ and $Y$ joint distribution

I'm trying to find the parameters of a conditional Gaussian distribution given the joint distribution between two variables and use it in a MAP estimator. Given two multivariate random variables $X$ ...
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Can this question be solved using Bayes's theorem?

Q. Of all the graduate students in a university, 70% are women and 30% are men. Suppose that 20% and 25% of the female and male population respectively smoke cigarettes. What is the probability that a ...
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1answer
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P(A|C)=P(A|B)*P(B|C)?

Is this formula P(A|C)=P(A|B)*P(B|C) correct according to Bayes' theorem? I don't think it correct but for a transition matrice system we can have something like thistransition matrice So I don't know ...
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When do we expand the numerator of the Bayes' Theorem

I am trying to understand why the proposed solution below to the following question is wrong:- A box contains three cards: a card that is black on both sides, one that is white on both sides and a ...
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1answer
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Find Probabilty of ~Fever

I have been given this P(Strep(some infection)) = 0.15 P(fever|strep) = 0.6 P(fever|~strep) = 0.3 Find P(strep|fever) I could find this put by ...
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1answer
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There are two local factories that produce microwaves. (Conditional probability and total probability)

There are two local factories that produce microwaves. Each microwave produced at a factory $A$ is defective with probability $0.05$ where as each microwave at factory $B$ is defective with ...
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2answers
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Does Bayes theorem have two formulas?

$$P(AlB)=P(A)P(B\mid A)/[P(A)P(B \mid A)+P(A^c)P(B \mid A^c)] $$ $$P(A \mid B) = [P(B \mid A)P(A)]/P(B)$$ 1) Are these two equations so called bayes formula?? 2) how do you distinguish whether to ...
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Questioning on bayes's theorem

Let $P(E) = 0.4$ and $P(F) = 0.7$, with $E$ and $F$ independent. How to calculate $P(F \mid (E^c \cup F^c))$? So:$$\begin{array}{l}\mathrm{P}(F \mid E^c \cup F^c) \\ = \dfrac{\mathrm{P}(F \mid E^c ...
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Estimating the probability that a datapoint belongs to one of two populations

I have a scoring metric $s$ and two datasets, $X$ and $Y$, such that $s(X)$ and $s(Y)$ are distinct, though overlapping, distributions (imagine a bimodal distribution where one of the modes ...
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Bayesian serial link d-separation?

I don't get how I prove d-separation for a serial link: $$ (A)\rightarrow(B)\rightarrow(C) $$ I am trying to prove that if $B$ is known with certainty (hard evidence), then the probability of $C$ ...
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1answer
46 views

problem with conditional probability and independence

Let $P(E) = 0.4$ and $P(F) = 0.7$, with $E$ and $F$ independent. How to calculate $P(((F \cup E^c)\cap(F^c \cup E)) \mid (E^c \cap F))$?
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Bayes risk with loss function that penalizes all errors equally

Loss function $L(\alpha(x),y = 1$ if $a(x) = y$, else 0. If $y\in \{-1,1\}$, then $\sum_y L(\alpha(x),y)p(y|x) = -p(y \neq \alpha(x) |x)$. (taken from ...
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Probability of alarm being valid (Bayes' theorem).

I have the following problem: your box is achieving a false positive rate of 0.01 and a false negative rate of 0.001. What fraction of the alarms that your box generates are valid alarms? I am ...
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How can I use conditional probability to get: $p(y, x\mid \theta, \phi) = p(y\mid \theta)p(x\mid y, \phi)$, for random variables $y,x,\theta, \phi$

If we have random variables $y, x, \theta, \phi$, how can I use Bayes' Rule to get: $$ p(y, x\mid \theta, \phi) = p(y\mid \theta)p(x\mid y, \phi) $$? I tried to factor it out but was stuck and ...
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1answer
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Please help me with a step by step solution to this question.

An X-ray test is used to detect a disease that occurs, initially without any obvious symptoms, in 3% of the population. The test has the following error rates: 7% of people who are disease free have a ...
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1answer
52 views

Conditional Probability or Bayes Theorem

I have this question and it has to do with either Bayes' Theorem or Conditional Probability. Any help in solving it? There is a 65% chance of John passing mathematics. There is a 35% chance that John ...
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2answers
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Finding the probability of the disease

Only 0.01% of people have triskaidekaphobia. The Dreizehn Club has developed a test for the phobia. If you have Triskadekaphobia, the test is 99% likely to identify that you have the disease. ...
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Combining two Gaussian posterior distributions from different data to refine estimated distribution.

If we apply Bayesian inference to try and determine the distribution of a multivariate Gaussian $\textbf{x}$, and we have two predictions $$ \textbf{x}\sim N(\textbf{a}_1,\Sigma _1)~~ and ~~ ...
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Probability of selecting from Urns

I am just preparing for some interviews and came across this questions: There are 6 X chips in Urn A and 6 Y Chips. 3 X chips in Urn B and 9 Y chips. Given that the chip picked is X, what is the ...
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3answers
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Bayes Theorem: what is wrong in using counts instead, intuitively.

I am reading about Bayes Theorem where the sample example used is: We have a red box with 2 apples and 6 oranges, and a blue box with 3 apples and 1 orange. The probability of selecting a red box and ...