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

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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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Bayesian Statistics … Γ(α,β) Posterior Probability and Estimators

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 ...
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Markov chain: if $X\rightarrow Y\rightarrow Z$, then why is $Z\rightarrow Y\rightarrow X$ true?

in a Markov chain, given three random variables $X,Y,Z$, we have $X\rightarrow Y\rightarrow Z$, which means $p(x,y,z) = p(x)p(y|x)p(z|y)$. The right arrow symbol $\rightarrow$ is used to denote a ...
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Which Probabilistic model to apply for this problem

I am a little confused about which probabilistic model to apply to this. Say I have an intern who is entering data on a daily basis. I have a clerk who at the end of the week pulls out a random ...
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how to calculate unknown probabilities in the bayesian network

I am working on a bayesian network problem. I read from one of the website the following network. My problem : "as soon as the cold water becomes low, you have at least a 94% chance of a high ...
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Bayes Rule, Probability, determine whether dependant or independant

I need to determine whether or not x and y are independent from each other, using the data supplied below. I think the solution should use bayes rule/product rule/some independence rule. I just ...
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Improper Lebesgue prior normalization in Bayesian filtering

Suppose we have a conditionally Gaussian Linear State Space Model (CGLSSM) where $Y_t=(X_t,S_t)_{t \in \mathbb{N}}$ is the Markov chain of hidden states, where for each $t \in \mathbb{N}$, $S_t \in ...
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Conditional probability of C given two independent events

Suppose that a machine depends on the working state of two components A and B. If both $A$ and $B$ do not function then the probability (say $C$) of the machine to work is $0.3$ If both $A$ and $B$ ...
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Generalization of Bayes' Theorem

Does anyone know of a generalization of Bayes' theorem to multiple conditions? From this answer I can see the definition of conditional probability with multiple conditions, but I couldn't find any ...
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Derive contingency table

For my research I am dealing with deriving contingency tables from some different performance results reported. I am trying to express prevalence in sensitivity, specificity and PPV, where, according ...
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Probability of an event happening

Studying for a mid-term with a practice test, and there's no solution, so I want to make sure I have this right. A fire alarm has the property that it will ring with 99.5% probability, if there is ...
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Verification of a Bayes Theorem related problem

This was an assignment problem given to me by the professor. I have did it (not sure if its correct). My answer is around $2/10^{12}$. I fear this is wrong. Can someone try this and verify if ...
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Posterior probability of a $p$ borderline result

Suppose that we throw 400 coins and found a result 0f 220 tails. Using a simple test of the null hypothesis that $H_0: \pi = 1/2$, we get that the probability of such a result is very close to .05. ...
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Proving $P(A|(B \cap C)) = P(B | (A \cap C)) P(A | C) / P(B | C)$ using Bayes' theorem.

The following equation can be proven rather uglily, provided that $P(B \cap C)$, $P(A \cap C)$ and $P(C)$ are non-zero, by expanding the conditional probabilities. $$P(A | (B \cap C)) = \frac{P(B | ...
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Bayesian update from uniform prior to uniform posterior ?!?

I was working through a signaling game problem recently and the proof suggested the following: Actor A has a type: $\ \mathscr{t} \sim Uniform[-1,1]$ Actor A gives signal $\pi^*$ that perfectly ...
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Conditional Probabilities with Given Data

I posted this question the other day but I believe it got deleted? In any case, I also wasn't very specific with my notation and I apologize for that! Here is the question again. Given event A, B, ...
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Probability of both kids being a girl given one is a girl

So I found this teaser: A mom has two kids. Given one of them is a girl, what's the probability that both are girls? The answer is 1/3. Which I have reconciled as the following. There are four ...
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On a decomposition of a conditional distribution

I am trying to make some sense out of equation (7) in the recent paper of Peter van Leeuwen: "Representation errors and retrievals in linear and nonlinear data assimilation" ...
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Is it true that $P(A \mid B) + P(A^c\mid B) = 1$?

That is, generally is it true that $P(A \mid B) + P(A^\complement\mid B) = 1$? I'm doing a problem involving Bayes' rule and my professor (seemingly) makes use of this fact. Any clarity on this ...
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Probability of picking balls out of bins

Question: You have two bins with four different balls in each bin. Bin A: 2 White Balls and 2 Black Balls Bin B: 3 Black Balls and 1 White ball You cannot tell which bin contains what balls. Given ...
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Bayes theorem with infinitesimal evidence

The Bayes theorem is often stated as $$\mathrm{posterior} = \frac{\mathrm{likelihood}\times\mathrm{prior}}{\mathrm{evidence}}$$ So what happens to the posterior probability when the evidence for an ...
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About Bayesian formula and rating system

I'm building a scoring system with score from 0 to 5) and I would like to sort products according to the number of reviews and their scores. After some research on the Internet I have found two ...
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Homework - Probability of raining

When I was doing my homework in probability, I encounter this question. "The weather is raining 70% of the time. There are two people that provide weather forecast, A and B, which their prediction ...
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Binomial Distribution with Conditional Probability

I am currently going through some sample problems in a book I have and I am unsure on how to approach a solution. I am given the following. The probability that a person will be helped by a ...
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When to use Total Probability Rule and Bayes' Theorem.

Consider the following information about travelers on vacation: $40$% check work email, $30$% use a cell phone, $25$% bring a laptop with them, $23$% both check work email and use a cell phone, and ...
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What is the complement of conditional probabilities?

I am working with a problem that uses Bayes Theorem and conditional probabilities. I have the conditional probability that a plane has an emergency locator $(E)$ given that it was discovered $(D)$ ...
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Proving a conditional probability

*Let A; B; C be events of a common probability space. (a) Prove the following equation holds: $$ \mathsf P(A\cup B\mid C) = \mathsf P(A\mid C)+\mathsf P(B\mid C)-\mathsf P(A\cap B\mid C) $$ (b) ...
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Comparing models to smoothed data

I am attempting to fit a model to a noisy data set. I am performing this modeling in two stages - first, smoothing it out by fitting an analytic mixture model to it, and second, fitting my final model ...
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Probability Proof - Bayes law multiplicative rule

I have been trying to prove that P(A∪B|C) = P(A|C)+ P(B|C) - P(A∩B|C) I have gotten to P(C|A∪B)*[P(A) + P(B) - P(A∩B)] = P(C|A)*P(A)/P(C) + P(C|B)*P(B)/P(C) + P(C|A∩B)*P(A∩B)/P(C) but I am ...
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Pattern Recognition terms a little fuzzy

I've been trying to learn more about probability and the websites I have visited are not describing the relationship to each other that well (when to use what, for what purpose in conjunction with ...
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Probability of one biased and two fair coins

You have three coins. Two of them are ‘fair’ while one of them is biased in that, for the biased coin, P{Head} = 2/3 and P{Tail} = 1/3. All three coins look alike, so that you don’t know a priori ...
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Probability that I am better that my opponent, given $n$ wins out of $m$

So I'm assuming that I have some static probability ($p$) of winning a given game against my opponent. I am also assuming that the wins are independent of one another. My intuition is that this can ...
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Minimum of a random variable sequence

$S_{n}$ model the price of a financial asset. The recurrence relation is given by: $$ S_{n+1} = (1 + r\Delta t_{n} + \Delta W_{n})S_{n}, n = 0, \dots, N $$ where $\Delta W$ has a normal ...
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Confusing term in total probability rule

I'm working on the following probability problem: I'm using Bayes' Rule and am having one problem. Using the total probability rule, the denominator of my Bayes' Rule expression looks like $$ ...
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Trouble getting probabilites for Bayes Theorem

I'm trying to think of a way to calculate the probability of P(A & B), where: A = {a company makes me an offer} e.g. 1/20 B = {I accept the offer} e.g. 1/5 Assuming that the denominator of the ...
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Probability question with two groups of students

Thirty percent of the students in a calculus course and 20 percent of students in a statistics course receive A's. Furthermore, 60 percent of the students with an A in calculus receive an A in the ...
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Conditional probability and bayes theorem problem involving a medical test

I have a test that checks if a patient is sick (E = {patient is sick}) and gives either a positive (A={result is positive}) or a negative result. Given that $P(A|E) = 0.95 = P(A^c | E^c)$ and $P(E) = ...
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Conditional probability that a randomly chosen detail was made by Y, using Bayes's theorem

I found the below question on the internet while working through a conditional probability questionnaire. An automobile plant contracted to buy shock absorbers from two suppliers X and Y. X ...
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How to compute this conditional probability in Bayesian Networks?

I met a problem related to conditional probability from the article "Bayesian Networks without Tears"(download) on page 3. According to the Figure 2, the author says $$P(fo=yes|lo=true, ...
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Bayesian Inference Problem

We have a Bayesian Network that A to D is Boolean variable. we want to calculate the probability which C and D be True and A be false. my answer sheet calculate the last result and is 0.0424. any ...
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Bayes Theorem with multiple observations

Let $H \in \{1,..,K\}$ be a discrete random variable and $e_1, e_2$ be observed values of 2 other random variable $E_1$ and $E_2$. We wish to calculate the vector ...
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Solving for the difference between the posterior and prior

a = (bc) / [bc + d(1-c)] Solve for (a-c) as a function of (b-d). You may recognize the expression above as Bayes' Theorem, where: a = P(A|B) b = P(B|A) c = P(A) d = P(B|not A) 1 - c = P(not A) I'd ...
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Where can I find a set of probability problems?

Is there a database of solved probability problems available? I am currently studying probability (and statistics) and, while I think I have a decent grasp of permutations, combinations, conditional ...
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rationalwiki on “Extraordinary claims require extraordinary evidence”

I don't have a strong background in probability/statistics and I'm trying to understand the example at ...
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Bayes' formula application to the probability of a horse winning, depending on the jockey

I have a problem that I have been racking my brain to figure this out and I just don't have the background to know if I am correct or not so I am hoping that you can help me out, OK here it goes so ...