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

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Newspaper confusion and statistics.

As part of my new university course I am taking I need to do some statistics so I decided to do some practice. As I said I am new to statistics and I am currently very stuck on the following question: ...
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2answers
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Bayes theorem and conditional probability [on hold]

I have a problem like this: Seventy-eight percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, ...
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1answer
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Conditional probability using set notation [on hold]

Got this wrong on a quiz and i don't have the answers. Need to figure this out for a test coming up. \begin{align} P(A) &= 0.75 \\ P(B\mid A) &= 0.9 \\ P(B\mid A^c) &= 0.8 \\ P(C\mid A\...
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Knight (Chess) Problem on telephone keyboard

There is phone keyboard with Knight on 0 (as shown below). 123 456 789 0 Knight moves as per the rules of chess (2 straight and one turn). T is no. of moves ...
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Given $n$ heads out of $n$ tosses. What is the posterior probability that coin is fair? [closed]

I am given an $\sigma$-fair coin with the probability of head $(\theta)$ being in the interval $[\frac{1}{2} - \sigma, \frac{1}{2} + \sigma]$. Also I am given: For a Bayesian analysis of the ...
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1answer
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Use Bayes rule to test whether patient has disease after several positive tests

I have solved one of those standard bayes rule application exercises a la: Given a prevalence value of a disease, the sensitivity and the specificity of a test, calculate the probability that the ...
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1answer
24 views

Conditional Probability that sum of dice is even

If two dice are rolled and let $ X and Y $ be the two random variables. What is the conditional probability that $X+Y$ is even when $X$ is odd. And when $X$ is odd? What is the total Probability of $X+...
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Loaded Dice Conditional Probability

If I have two dice, one regular and one loaded. The loaded die has the probability 1/2 of landing a six and rest of the numbers are equally probable. If you select a die randomly and throw it and it ...
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3answers
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Probability using Bayes formula

Two masked robbers try to rob a crowded bank during the lunch hour but the teller presses a button that sets off an alarm and locks the front door. The robbers, realizing they are trapped, throw away ...
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2answers
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Dash and dot probabilities

This is problem 1.41(b) in Casella and Berger's Statistical Inference. Consider telegraph signals "dot" and "dash" sent such that $$\mathbb{P}(\text{dot sent}) = \dfrac{3}{7}$$ and $$\mathbb{...
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Compute conditional probability

If a conditional probability table is given for $P(S_t|M,E)$. How to compute the value for $P(S_t = x | M,E)$ ? where $E$ is binary (0 or 1) and $M$ is ternary ?
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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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2answers
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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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2answers
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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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1answer
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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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1answer
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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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2answers
37 views

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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2answers
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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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1answer
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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
55 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
32 views

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
29 views

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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1answer
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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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1answer
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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
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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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1answer
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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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1answer
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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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1answer
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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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2answers
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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 ...