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

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Simple Probability of Playing Cards

An ordinary deck of playing cards has four suits: hearts, spades, diamonds, and clubs. Suppose you have a reduced deck of eight playing cards, consisting of four aces and four kings. I draw two cards ...
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interpretting prior and posterior

I am bit puzzled on how we can interpret the posterior. Assume a coin which is 0.1 probable to be unfair. So our prior probability on the coin being unfair is 0.1, and being fair is 0.9. Also by ...
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Conditional Probability of Sinking Ship Question

Question: Two ships. Ship A's missiles have an 80% probability of hitting its target, ship B's missiles have a 50% probability of hitting the target. It only takes one hit from a missile to sink a ...
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Proving Conditional Probability Statement

Question: Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Prove or disprove that $P(A|B \cup C) = P(A|B) + P(A|C) ~~~\forall~~ A,B,C~ \in \mathscr F$ where $B \cap C = \emptyset$. ...
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Interpretation of Bayes Formula Question

This is a simple application of Bayes Theorem, I'm just confused about the labeling. If i let T = test outcome is positive, D = patient has disease, then, I am trying to find $$ P ( D | \bar{T} ) ...
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Bayesian update multivariate normal based on one-dimensional signal: simple rule

Is there a simple rule to update the linear combination of normal distributions based on a one-dimensional signal? The unconditional joint density of $(\eta,\theta)$ is multivariate normal ...
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In Bayes' theorem, what is little $p$?

In Wikipedia's conjugate prior article, Bayes theorem is given as: $$p(\theta|x) = \frac{p(x|\theta) \, p(\theta)} {\int p(x|\theta') \, p(\theta') \, d\theta'}.$$ What is $p$ here? Is it the ...
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Subpopulations of an island (Bayes theorem?)

Help appreciated here. An island with 2 regions, I and II, has 4 types of individuals: AX, AY, BX and BY, for which we know their exact total nos. Here A-B-X-Y are simply traits, e.g., A=Male, ...
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Bayesian Updating with 1 Signal but 2 Unknowns

Suppose I have an unknown variable $X_i = \alpha_i + \beta_i$ where $\alpha$ is one of 2 different values {${\alpha_1, \alpha_2}$} such that $\alpha = \alpha_1$ with probability $p_1$ and $\beta$ is ...
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Bayesian probability and coin toss

Assume that John and Mary, not knowing anything about fairness of the coin, have common prior of obtaining H (heads) in coin toss equal to $\frac{1}{2}$. Before tossing a coin, each of them is allowed ...
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Why does Bayes' theorem work?

Why does Bayes' theorem work? I'm not looking for a cryptic math demonstration. Rather, what I'm interested in is the intuition behind the theorem that allows to obtain the a posteriori probability ...
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Bayes: markov chain, serial connection, marginalization

Goal is to check if p(a) is unconditionally independent to p(c) in the markov chain - serial connection. $$ p(a,b,c) = p(a) p(b|a) p(c|b) $$ $$ p(a,c) = \sum_b p(a) p(b|a) p(c|b) = p(a) p(c|a) \neq ...
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Conditional probability relationship among $\Pr(A|B), \Pr(A|C), \Pr(A|B, C)$

I need some help in deriving some relationships between $\Pr(A|B,C)$ and $\Pr(A|B), \Pr(A|C)$. Specifically, I want to have a proportional relationship between the former and the two in the latter. I ...
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Prove that $P(G \mid E_i) > P(G) \land P(G \mid E_1, E_2) < P(G), i \in {1, 2}$

Let G be the event that a certain individual is guilty of a certain robbery. In gathering evidence, it is learned that an event E1 occurred, and a little later it is also learned that another event ...
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Bayes factor for fair and biased coin

There is the following task: Suppose we toss a coin $ N = 10$ times and observe $m = 9$ heads. Let the null hypothesis be that the coin is fair,and the alternative be that the coin can have any bias, ...
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Bayes Theorem chaining

Is this statement true $P(A|B)=P(A|C) \times P(C|B)$ If yes, how to prove it? If no, what should be minimal conditions on A, B and C for this to hold true? Thanks for help.
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Calculate CPT of bayes net

I have a Bayes net of a pretty simple construction. I need to find the expressions that the CPT's represent and also the number of entries. A--B--C .....| ....D A is the parent node of B. B is ...
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Setting up Posterior Probability and Bayes Theorem

I would just like to ask some questions with regards to Bayesian probabilities and an exercise I am trying to solve. Here is the problem: A footprint was found at the crime scene, which is known to ...
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Basic probability: Is event space in probability poorly defined?

I am in a probability class and we are just getting into random variables. A random variable to me is a function that maps event to some real number. But the concept of event is difficult to swallow. ...
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Bayes probability with unfair coin - what went wrong

A box has 1000 pennies. One penny in the box has 2 heads. A coin is selected at random and flipped 5 times. If the coin comes up heads each time, what is the probability that the selected coin had two ...
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Probabilities, dependence and independence

Given the following about the three events $A$,$B$, and $C$: $P(A \cap B) = P(A)P(B)$, $P(A \cap C) = P(A \mid C)P(C)$, and $P(B \cap C) = P(B \mid C)P(C)$ In other words, events $A$ and $B$ are ...
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Probability of drawing a yellow ball after 1 or 2 have been drawn.

The question is as follows: An opaque urn contains ten balls, five red, four yellow, and one green. Three balls are drawn from the urn in sequence but at random, without replacement. ...
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Calculating posterior probabilities when 2 independent events occur?

This problem is 6.1 from Tom Mitchell's Machine Learning. The givens: $P(\text{cancer}) =.008$ $P(\neg \text{cancer}) = .992$ $P(\text{test} =+ \mid \text{cancer}) =.98$ $P(\text{test}=-\mid ...
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conditional probability question Given bayesian network could not understand solution given

Given Bayesian network can't understand the two last steps in why the p(C=c|E=e,~H) can get out of the e sum? and why sum p(E=e|A,S,~H) and sum p(C=c|E=e,~H) and been neglected? Thank for the ...
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A Bayesian problem of coin tossing

A box contains $3$ coins . Among these three , each of two coins have the probability of giving head $\dfrac 23$ and the remaining one have the probability of turning head $\dfrac 12$ . One coin is ...
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Bayes theorem - posterior probability

Let's suppose, that we have 6 aces and two players. Two aces are hearts, two diamonds and two spades. The exercise says, that the first player picked two spades. The exercise asks me to determine ...
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How can I reword this problem illustrating a scenario that needs Bayes Theorem to solve?

Taken from Stat Trek, an example explaining Bayes Theorm http://stattrek.com/probability/bayes-theorem.aspx Marie is getting married tomorrow, at an outdoor ceremony in the desert. In recent ...
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Probability Formula for Posterior With 3 Variables

First post on math.stackexchange; pardon me if this is naive/a repeat. I'm following this document here by Prof. David M. Blei: ...
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Poker odds: Chances of a straight flush, given H4,H5

I'm trying to learn Bayes's formula, and am coming up with some poker problems to learn this. My problem is as following: given a $H4,H5$ ($4$ of hearts, $5$ of hearts) hand, what are the odds that ...
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Conditional probability, and Bayes rule (and Bayes rule with background knowledge)

So, I've got a few questions: $1)$ is $P(a,b) = P(b,a)$ ? $2)$ How do I get some intuition for Bayes rule? I know don't really understand what is happening. $P(h|d) = (P(d | h) P(h) / P(d))$. I get ...
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Probability that the fish that set off metal detector is the one true fish

I have exams in Machine Learning coming up and I need help answering this question. There are a million identical fish in a lake, one of which has swallowed the One True Ring. You must get it ...
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A result regarding Hierarchical Bayes

I have the following, $$x_i \mid \theta_i \sim \text{Bin}(m, \theta_i), ~i=1,\dots,n,$$ $$\theta_i \sim \text{Beta}(\alpha,1),$$ $$ f(\alpha) \propto 1.$$ I wish to compute ...
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Bayesian inference and new information

The Bayesian inference [1] tells how we can update the prior probability based on evidence. My question is that, in real world, we also update our prior probability of an hypothesis based on new ...
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Probability exercise

I have an exercise, where I now know the correct answer, but I just don't know how to get that result. This is the exercise: I have tried using Bayes' Theorem, but I still cannot get the same ...
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Is this form of bayes rule valid?

In the bayes rule, does \begin{equation} \Pr((A\mid B)\mid C) \end{equation} have a meaning? is it a valid form of a probability? if $(A\mid B)$ ( I do not mean $\Pr(A\mid B)$ ) is an event, the ...
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Posterior distribution of bernoulli distribution with multiple observations

I'm just learning Bayes's Rule so this question might be really simple Suppose I have a random (real) variable $X$ over $[0, 1]$. I assume a uniform prior. In successive rounds, I sample a value ...
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Where does this conditional probability law come from?

I was trying to follow a computation done in my class notes, and was having difficulty seeing the inspiration for a part of the manipulation in a question regarding probability. I did some Googling, ...
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How do I find (E|F')?

Assume ' is equal to not or complement here. Alright, you are given the following information: p(E)= 1/3 p(F)=1/2 p(E|F)=2/5 You are asked to find ...
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Bayes Theorem for Probability of Manufacturing Process

A manufacturing process produces computer chips of which 6 percent are defective. This percent is actually found using a thorough (and expensive test) on a small random sample of chips. The plant ...
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How to prove Bayes formula with additional conditional(s)?

I am trying to prove a version of Bayes formula which is used in Beyond the Kalman Filter: Particle Filters for Tracking Applications, by Branko Ristic and Sanjeev Arulampalam, page 45-47. ...
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Conditional Probability Question - Please Help

Messages relating to the status of an industrial system are transmitted to a monitoring station via an internal transmission network. During periods of low network traffic, 1.2% of these messages have ...
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Gaussian processes versus Bayes rule misinterpretation

I would like to use Gaussian processes (GP) for Bayesian classification of medical data. I think I already understand the basic stuff but I have some uncertainties that are perhaps partly related to ...
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Probability that a cow is black given that I've observed at least one side is black

I'm on a farm with six cows; three are white, two are black and one is completely black on one side and completely white on the other. I see one cow from the side, who appears to be black (that is, ...
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Extended Bayes' theorem: p(A | B, C, D)

I'm having some difficulty understanding Bayes' theorem with multiple events. I'm trying to put together a Bayesian network. I have four independent probabilities but I have found that A, B and C can ...
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Bayesian Statistics: Estimators and Posterior Probability

If I let $M ∼ Γ(α,β)$ (where $α, β$ are known) Let $X_1,...,X_n$ be discrete random variables such that $X_i$|$θ$ ∼ i.i.d. Poisson with parameter $θ$, where $θ$ is a realization of $M$. I have two ...
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Finding Conditional Expectation and variance E(Y|X=x)

I want to find the conditional Expectation and variance of random function Y for a given value of random function X, i.e. E(Y|X=x). Here X is x(t) and Y is x(t+τ). Also, x(t) is a stationary Gaussian ...
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Calculating $P(A\cap B)$ from $P(A)$ and $P(B)$ without knowing $P(A\mid B)$ or $P(B\mid A)$

Is there a way to calculate $P(A\cap B)$ from $P(A)$ and $P(B)$ without knowing $P(A\mid B)$ or $P(B\mid A)$? I'm asking because it seems to me that a primary function of Bayes' theorem is to ...
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what is the probability (related bayes theorem)

Question is There are 8 fair coins (C0)and 3 unfair coins(C1) (80% chance get head) tosses it n times without looking at it , and reports: n=2, got 1 T and 1 H , don't know the order. what is the ...
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How to write this conditional probability equation?

Question is Alice tosses a fair coin, then rolls a fair die. If the result is <=2, she tosses fair coin ,otherwise she tosses an unfair coin(80% gets head). .what is the probability that she gets ...