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

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Bayes Theorem - one event dependent on two

I do have problems to put up the Bayes Theorem for one event depending on two: $\mathbb{P}(q|z,x)$. Two events depending on one was still ok, \begin{equation*} \mathbb{P}(x,q|z) = \frac{\mathbb{P}(z|x,...
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Choose 3 cards from a deck, if last two are spades what is the chance of first card being a spade?

I'm stuck on this problem, and need some explanation if possible : From 52 cards we take 1, after that we take 2 more, both of which are spades. What are the odds that first card is also a spade? ...
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Bayesian posterior probability [closed]

Let's say that you are in a casino and you have played on 3 different slot machines following this flow: Slot machine A, play 10 times, win 2 times for a total of 2$ Slot machine B, play 100 times, ...
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Bayes Theorem problem, from Finan #9.4: $P(A\mid B ∩ C)$

The Problem: You are given $\Pr(A) = 2/5, \Pr(A ∪ B) = 3/5, \Pr(B\mid A) = 1/4, \Pr(C\mid B) = 1/3,$ and $\Pr(C\mid A ∩ B) = 1/2$. Find $\Pr(A\mid B ∩ C)$. My work: I know that $\Pr(A\mid B) \Pr(B)...
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How to calculate the denominator of Bayes' Rule

I'm currently working through the example of applying Bayes rule to a scenario in this chapter on page 491. This is the information I've been given: $$\operatorname{Pr}(B) = 0.8$$ $$\operatorname{Pr}...
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Question related to Bayes theorem

I spent some times on this website trying to figure out how Bayes Theorem works. Now, from that example, consider The Yankees has a 72% of winning next game based on previous results. When someone (...
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1answer
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probability - Total probability and Bayes theorem

Suppose you're at a college campus. $3/4$ of the people on the campus are students or professors from that college, and the rest $1/4$ aren't. When asked a question, students and professors from that ...
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Assumptions with Bayes's Theorem

After reading extensively on the subject I would like to clarify this apparent problem with "Bayes Rule". Namely the notation often used P(A and B) = P (B and A) has a big assumption that I will try ...
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1answer
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Bayesian Nets and weird probability

I have to solve the following problem: Suppose we have a bayesian net in which we have the following variables: R, PA and PR Let: P(R) = 0.1, P(PA) = 0.5, P(PR|R, PA) = 0.6, P(PR|¬R, PA) = 0.4, P(...
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Calculating total probability given some conditions

One machine element is being produced in $3$ series, each series consisting of $20$ elements. In the first series, there are $15$ elements that work correctly, in the second series there're $18$ and ...
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Probability of not $A$ and knowing that $B$ has not happened.

I'm clear that $$P(A\mid B) = \frac{P(A∩B)}{P(B)}$$ But how can you calculate the following: $P(\bar A \mid \bar B )$? Thanks
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Solving for a Conjugate Prior in search of MAP estimator

I am trying to prove that if a given random variable $X \sim Exp(\lambda)$ and $\lambda \sim Gamma(\alpha,\beta)$ hen $\lambda | X \sim Gamma(\alpha^{*},\beta^{*})$ for some parameters $\alpha^{*}$ ...
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1answer
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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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1answer
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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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2answers
51 views

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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1answer
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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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1answer
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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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1answer
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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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1answer
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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 C)...
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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. $$0.3*...
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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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1answer
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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 $6$...
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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
20 views

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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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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1answer
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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 S_1}\bar{...
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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: $P(G) = 0.7$ $P(M) = 0.3$ $P(ET|G) = 0.3$ $P(ST|G) = 0.5$ $P(NT|G) = 0.4$ $P(ET|M) = 0.0$ $P(ST|...
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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}), p(X=0|w=\...
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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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1answer
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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) =...