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

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Deriving the the conditional PDF from Bayes' Rule

I am having trouble getting the conditional PDF from Bayes' Rule for the following problem: Fred wants to sell his car, after moving back to Blissville (where he is happy with the bus system). He ...
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Bayes' Theorem and Law of total propability for CDF

The calculation of conditional probability is the same for conditional PDF and CDF(according to a number of questionable sources: first, second) (I will use rough notation, with just $x$ and $y$): ...
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Chance of getting a car toy in a chocolate

A mom brings her child every day a chocolate with a toy inside, the toy is random. The boy is happy when he gets a car as a toy. His mom decided to look in which supermarket the probability of getting ...
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Silly question regarding probability?

A probability exercise which I believe that it is written in slightly dodgy manner as I have trouble finding a solution for it. The way I go about the solution is to have the problem split into three ...
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How can we prove this equation using marginalization and conditioning? [closed]

I want to prove $$P(A|C) = \sum_{B} P(AB|C) $$ How can we prove this using marginalization and conditioning?
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Bayes' rule / Probability Tree question

One percent of a population suffer from a certain disease. A diagnostic test gives a positive indication 97% of the time when an individual has the disease, and a negative response 95% of the time ...
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Conditional probability problem with Bayes' Rule

The question is as follows: A crime is committed by one of two suspects, A and B. Initially, there is equal evidence against both of them. In further investigation at the crime scene, it is ...
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Question involving Bayes' Rule and the Law of Total Probability

The question is as follows: A bag contains one marble which is either green or blue, with equal probabilities. A green marble is put in the bag (so there are $2$ marbles now), and then a ...
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Applying Bayes Theorem to find the probability of a finite intersection of events

I have a notation problem. Let $A$ and $B$ be events such that $P(B)\ne 0$. The by Bayes' theorem, $$ P(A \cap B) = P(A|B)P(B)~~~~~~~~~(1) $$ Similarly, $$ P(A\cap B \cap C) = P(A|B \cap C)P(B\cap ...
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Statistics SOS - Using Bayes Theorem [closed]

Please see attached image. I'm generally confused when I see a Statistics problem like this, as I can't seem follow a logical route to the answer. Can you please advice me on the step I should take as ...
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1answer
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Simple Bayes? Probability of a state at time t in hidden markov model

Suppose we have a HMM with $2$ states -- $A$ and $B$, with $P(A) = 0.4$ and $P(B) = 0.6$. $A$ has a probability of $0.9$ of outputting "hot," and $B$ has a probability of $0.1$ of outputting "hot." ...
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Application of Bayes theorem and Partition Law, total probability

Hi guys, preparing for my finals and trying to get this question out for practice. The exam is in a couple of hours so apologies for being brief. I think I have computed parts 1 and 2 fine. ...
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bayesian posterior of truncated normal distribution with uniform prior

Let $N_T(\mu,\sigma)$ be a truncated normal distribution with support on $[0,1]$. Draw $x \sim N_T(\mu,\sigma)$ (What I want to model is, I have a unknown quantity $\mu \in [0,1]$, but I only ...
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Conditional Probability Problems (visualizing problem)

I am having some difficulties with conditional probability problems in terms of "visualizing the problem". What I mean by this is that instead of being able to directly apply the bayes rule, I find ...
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Two regular dice are rolled. Given that one of them is a $6$, what is the probability that the other is also a $6$?

I'm having trouble with the following question for and I have an exam in two days: Two regular dice are rolled. Given that one of them is a $6$, what is the probability that the other is also a ...
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1answer
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conditional probability: Bayes or not?

I'm learning about conditional probabilities and the Bayes theorem, but I don't know how to really think about it. The problem I have is the following: There are 50 men and 40 women at a workplace. ...
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1answer
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Probability with a joined condition

I want to know the probability $P(A|X,Y)$, given that I know $P(A|X)$, $P(A|Y)$, $P(A)$, $P(X)$, $P(Y)$ and given, that $X$ and $Y$ are independent. I'm also going to assume that $X$ and $Y$ are ...
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Simplifying conditional probabilities

Is there a clever or neat way to simplify the following, $$ \frac{ P(F,G|E)}{P(G|E)} < \frac{ P(F,G) }{ P(G) } $$ We have $ P(G,F | E) = P(G|E) P(F|E) $.
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Need to find P(M|¬F)

Jon is in a pit with 100 deadly scorpions, 60 of which are male and 40 of which are female. The male scorpion's bites are fatal 70% of the time and the females scorpions' bites are fatal 90% of the ...
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189 views

Bayes Theorem probability

Past Exam Paper Question - Prof. Smith is crossing the Pacific Ocean on a plane, on her way to a conference. The Captain has just announced that an unusual engine fault has been signalled by the ...
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Proving $\displaystyle P(A|B \ \mathrm{and} \ C) = \frac{P(A|C)P(B|A \ \mathrm{and} \ C)}{P(B|C)}$

Problem Prove that $\displaystyle P(A \mid B \cap C) = \frac{P(A\mid C) \cdot P(B\mid A \cap C)}{P(B\mid C)}$. Thoughts I'm having some trouble interpreting $\displaystyle P(A\mid B \cap C)$, and ...
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unsure about variables in Bayes' Theorem question

I would just like to double check that I have completed this question correctly. I am new to Baye's theorem and find the variables a bit confusing, particularly what a general rule is for determining ...
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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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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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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. ...