Questions tagged [bayes-theorem]

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

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Probability of having lung cancer for smokers

Let L be the event that a randomly chosen person has lung cancer. Let S be the event of a person being a smoker. Suppose that 4% of the population has lung cancer, 30% of the population are smokers, ...
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Non informative prior for location parameter

In "Bayesian Data Analysis" Third Edition by Gelman, there is a small paragraph that I can't fully understand: "If the density of $y$ is such that $p(y−θ|θ)$ is a function that is free ...
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Probability of Lie Detector bayes theorem

Lie detectors have been used during wartime to uncover security risks. As is well known, lie detectors are not infallible. Let us suppose that the probability is 0.10 that the lie detector will fail ...
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Finding Conditional Probabilities, and Independence from Joint Probability Distribution Table

I have the following joint probability distribution table, E E ~E ~E G ~G G ~G F a b c d ~F w x y z a+b+c+d+w+x+y+z = 1 Given that F is True, I need to find, P(E|F,G) P(E|F,~G) P(G|F,E) P(G|F,~E)...
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Problem with conditional probability of three events

I have got some trouble trying to solve this problem that I came up with. I do not even know whether it is solvable or not. So thank you so much if you can help me! Problem: I have got a number of ...
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Home assignment involving Bayes Theorem

I was discussing with a friend the following question and its solution by our teacher: In a dish there are 2 types of cell cultures: $W,V$. $V$ constitutes 20% of the dish. The life expectancy of $V$ ...
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3 answers
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how can I find $P(X|\text{not } A)$ in Bayes theorem?

I have this exercise. it involves Bayes' Theorem: in an exam, $29\%$ of students chose to write essay $A,$ if a student chooses to write essay $A,$ then student had read author $X,$ and $P(X \text{ ...
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1 answer
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Bayes' rule and combinatorics

Given the following information: We have $2$ baskets with red and blue marbles in each basket. In basket 1 there are $3$ red and $4$ blue. In basket 2 there are $6$ red and $8$ blue. and the ...
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Change of measure in expectation of a discrete random variable w.r.t a continuous random variable.

Let us consider that I have two random variables. A discrete one, called $\alpha:\Omega\to\{1,2,3,\ldots,n\}$ and another one called $X:\Omega\to\mathbb{R}$, which is a continuous random variable. I ...
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How to apply Bayes' theorem when the prior is not known

I'm a little confused about how to use Bayes' theorem when I don't have any way to establish an initial prior. Say I have a sensor that can detect whether there is flouride in a water sample. Let's ...
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Probability that coin $2$ is flipped third day

Suppose that coin $1$ has probability $0.6$ of coming up heads, and coin $2$ has probability $0.3$ of coming up heads. If the coin flipped today comes up heads, then we select coin $1$ to flip ...
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Picking balls out of a bag intuition - "Rearranging the combinatorial tree"

This question is about the intuition behind the classic "picking the balls out of a bag" problem. It feels like it flattens out time, which is what I want to understand. First i will layout ...
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Bayes theorem and probability of a man [duplicate]

A man is known to speak truth 3 out of 4 times he throws a die and reports that it is a six. What is the probability that it's actually a six. So I applied Bayes theorem and found the answer as 3/8. ...
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Incorrect answer in a questions linked to 'Sequential Experiments in Probability' due to diff. approach

Two white pawns are placed at random on the third row of a chessboard. Assume that the chessboard is oriented in such a way that the allowable moves for the pawn are in the fourth row (in other words ...
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The gambler chooses dice at random, and rolls it $six$ times. What is the probability that fair die was chosen?

Assume that a gambler has two dice, one of which is fair, and the other is biased toward landing on $six$, so that $0.25$ of the time it lands on $six$, and $0.15$ of the time it lands on each of $1$, ...
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The gambler chooses a dice at random and rolls it once and the dice comes up with a 6. What is the probability that fair die was used?

Assume that a gambler has two dice, one of which is fair, and the other is biased toward landing on $six$, so that $0.25$ of the time it lands on $six$, and $0.15$ of the time it lands on each of $1$, ...
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Prove $P(B|A) = P(B)$, if $A$ and $B$ are independent [closed]

How can I show that $P(B|A) = P(B)$, given that $A$ and $B$ are independent?
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Calculating Probabilities and Using Binomial Distribution

A recent survey of residents in Texas concluded that 55% of Austin city residents and 46% of Houston city residents broke a bone at some point during their childhood. a. Let’s say Austin has 5200 ...
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Simple Problem using Bayes Rule

This is Exercise 1 in Chapter 2 of the Probabilistic Robotics book by S. Thrun etal. Problem. A robot uses a range sensor that can measure ranges from $1$m to $3$m. For simplicity, assume that actual ...
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The probability of topics conditioned on words in Latent Dirichlet Allocation

Let $w_i$ denote word $i$, $z_k$ denote topic $k$, and $d_j$ denote text $j$. In the LDA method, it is assumed that text generates topics and topics generate words. Thus the $\text{P}(z_k|d_j)$ and $\...
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Conditional probability (Bayes theorem)

One of the 256 subsets of {1,2,3,4,5,6,7,8} is chosen uniformly at random. Let X be the number of elements of this subset. Let Y be 0 if the subset is empty and be the least element of the subset ...
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1 answer
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Backwards Bayesian argument for alien visitation?

Let $A$ = the hypothesis that aliens are visiting Earth Let $E$ = evidence that aliens are visiting Earth The posterior probability that aliens are visiting Earth, given some evidence, $P(A|E)$, can ...
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Conditional probability sample space

I was solving a probability question. Here are the screenshots for the problem. I want to know , In the second image when we are calculating P(D | E1) , why it is being divided by 100. The sample ...
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Marginalizing Probablistic Graphical Models/Bayes' nets

I wanted to verify my approach to marginalizing out variables in a Bayes' net or Probabilistic Graphical model. Consider the following model: $$\require{enclose}\begin{array}{c} \enclose{circle}{~\...
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Conditional Probability and Bayes Nets

I just started learning Bayes Net and conditional probabilities, and I was having trouble understanding how to determine if two variables are conditionally independent. Consider the simple Bayes Net ...
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Incorrect conditional probability when expanding expression (bayes theorem)

Given the following probabilities: $P(A) = 0.5$, $P(¬A) = 0.5$ $P(E|A) = 0.1$, $P(F|A) = 0.8$ $P(E|¬A) = 0.3$, $P(F|¬A) = 0.2$ I am trying to solve for $P(A|Z)$ where $Z = (E ∩ F)$. Please note that ...
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Unable to prove(may be under some condition) the identity: for random variables $X,Y,W$, we have $p(x|y) = \int p(x|w)p(w|y)dw$.

Let $X,Y,W$ are random variables. Let $p(x)$ denotes the pdf $X$. We try proving the identity: $p(x|y) = \int p(x|w)p(w|y)dw$. We start with $$p(x|y) = \int p(x|y,w)p(w)\,dw,$$ $$p(x|y) = \int \int p(...
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Conditional Probability in poker, Bayes Theorem

In a four person pokergame you are dealt a "king of heart" and "8 of spades". Three cards lay on the table none being a king or 8, meaning there is 41 cards left. Q: What is the ...
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Bayes theorem false positive and false negative example

I'm learning about the conditional probabilities and need some help in solving a example: Suppose a test method gives positive results for the infected person 65% of time and negative results for ...
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1 answer
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Classic Picking Balls from 2 Urns Question

The question involves 2 urns, URN1 and URN2. There are 18 balls of 4 variations in URN1: 3 Red Balls, 5 Orange Balls, 5 Yellow Balls and 5 Green Balls. There is 1 ball of unknown color in URN2, i.e. ...
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Calculating the posterior distribution - missing dependency

I am reading an old paper from https://www.jmlr.org/papers/volume1/tipping01a/tipping01a.pdf. In that paper, specifically, Equation 10 says $$ p(w|t,\alpha,\sigma^2) = \frac{p(t|w,\sigma^2)p(w|\alpha)}...
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Why is my probability value greater than 1?

Question Given the observation $\{H,T,H,H,T\}$, use the Bayes rule to find the probability that the coin is unbased? My approach Let the probability of seeing heads be $f$. For unbiased coin, $f=.5$. ...
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1 answer
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Bayesian updating with Binomial data and a Uniform([0,$\theta$]) prior

Assume $p$ has a $\text{Uniform}([0,\theta])$ prior, where $\theta\in(0,1)$ is a known constant. Let $X\sim \text{Binomial}(n,p)$ for some $n\in\mathbb{N}$. What is posterior $\mu(p|X)$? (In the $\...
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3 votes
3 answers
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$30\%$ students have glasses. $20\%$ of students with glasses play. $60\%$ of students without glasses play. Probability, student without glass plays.

Problem: In a school, $30\%$ of students have glasses. $20\%$ of students with glasses play sports. $60\%$ of students without glasses play sports. If we randomly choose a student, find probability ...
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0 answers
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How to prove this limit result?

I'm quite sure the following is true, but I'm not sure how to prove it. Fix some $p\in(0,1)$. For any $p'\in[0,1]$, there exists sequences $\{x_n,y_n\}_{n=0}^\infty \subset (0,1)$ s.t. $$\lim_{n\to\...
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2 votes
2 answers
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Bayes's rule and probability theory

I'm trying to solve such a task: Preliminary tests of students showed that 30% of them are very well prepared to the exam, 50% are prepared quite well and 20% are prepared somehow. For the first group ...
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In what situations is this true: $P(A|B) = 1 - P(A|B^c)$? [duplicate]

I am working on some software that uses Bayes' rule to find $P(B|A)$ It asks the user to define $P(A|B)$ but it says that $P(A|B^c)$ is optional, if the user chooses not to define it then it sets $P(A|...
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2 answers
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Probability of a family having 2 boys and 1 girl

I'm working with the following problem: Your new neighbors have three children. You are told that they have three children, but without knowing their gender. If you are told about three independent ...
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5 votes
1 answer
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What if I have different information from the Bayes Theorm diseases question?

I'm studying Bayes' Rule and came across the diseases problem. And wonder if it also works in other circumstances. Let us say $D$ is the event having the diseases, $T$ is the event testing positive ...
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Conjugate prior for estimating one "component" of a merged Poisson process.

Given Information: Let $N_1(t)$ and $N_2(t)$ be two independent Poisson process with arrival rate $\lambda_1$ and $\lambda_2$, respectively. Let $N(t):=N_1(t)+N_2(t)$ denote the merged Poisson process ...
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1 answer
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Posterior with a Beta prior and a liner function of Bernoulli data.

Let $(a,b)\in\mathbb{R}^2$ be fixed, known constants that satisfy$|a|+|b|<0.5$.$^1$ Let $\theta$ have a $\text{Beta}(\alpha,\beta)$ prior. Given an observation of $X\sim\text{Bernoulli}(a+b\theta)$,...
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1 vote
0 answers
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Sequential Bayesian updating with a four-parameter Beta distribution and Bernoulli data

Suppose data $\{z_t\}_{t=1}^T$ are $z_t\substack{{\small i.i.d.} \\{\Large \sim}\\ \color{white}{.}}$ $\text{Bernoulli}(\theta)$ and a four-parameter Beta prior $\theta\sim \text{Beta}_4(\alpha,\beta,...
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1 vote
0 answers
29 views

Bayes rule with 2 conditions on each side of the bar and repeated events

The binary variables $A,B,C \in \{0,1\}$ are not independent. I need to solve for the probability of $A$ and $B$ occurring simultaneously given that $C$ and $A$ occur: $P[A=1, B=1| C=1, A=1]$. I have ...
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0 votes
1 answer
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Numeric examples for Bayes with three variables (+some basic questions)

I have a couple of questions on conditional probability: First of all, how is p(B|A,C) spoken out correctly? The likelihood in joint with C, or B, given the joint of A and C - or does both mean the ...
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2 votes
0 answers
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Tractable setup for sequential Bayesian estimation with a binary data (in a case where a beta-bernoulli setup seems inappropriate)

An unknown parameter $\theta$ is randomly drawn at time $t=0$ according to prior p.d.f. $\mu_0(\cdot)$ that has support $[L,R]\subseteq\mathbb{R}$. At each time $t\in\{1,2,...\}$ an agent makes an ...
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2 votes
1 answer
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How Do You Write Out Bayes Theorem for 4 variable?

I know how Bayes Theorem would look for 3 variables, but I'd like to know how to do this for n > 3 variables? For example, P(A|B,C,D,...N). I'll include 3 variables here for reference if it helps ...
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Finding the probability that a person is positive given a negative test?

Given the problem of a patient taking a test for a disease where having the disease is denoted by X and the a positive test is denoted by Y, the rate of occurrence of the disease in the general ...
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5 votes
2 answers
133 views

Probability of failure of first unit provided that at least one of the two units has failed

The following question was asked in JEE Main 2021: An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first ...
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1 vote
1 answer
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Is it true that $P[A|B]+P[A|B’]=P[A]$ when B,B’ are exhaustive?

Essentially what the title says. For reference, I am working on the following problem and know that the answer above the solid line is wrong and that the answer below the line is right. However, I am ...
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2 votes
2 answers
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Possible bayes theorem question regarding a shared car

The question: Suppose 2 friends share the use of a car evenly, we will call the two friends Roger and James, respectively. We know that Roger will only use the car to drive to the grocery store, on ...
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