Questions tagged [bayes-theorem]

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

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what is conditional probabiltiy

what is exactly is the conditional probability I saw this this kind of definitions for conditional probability Definition: Conditional probability is the likelihood of an event occurring based on the ...
Qwe Boss's user avatar
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I can't seem to understand marginal probabilities.

In the book Pattern Recognition and Machine Learning, CM Bishop has elaborated on calculating marginal, conditional and joint probabilities by creating a table with rows and columns being outcomes of ...
DeadAsDuck's user avatar
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Is this conditional probability always equal to 1?

Consider $X = \text{number of defective items in bought items}$. Is the probability that $X \geq a$ given $X = a$, always 1: $P(X \geq a| X = a)=1$. I was wondering if the above holds because $A$ (...
DubsVeer23's user avatar
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Why isn't the denominator in bayes rule always one

So, I thought I kind of understood Bayes and total probability, but I see now I am not clear on the calculation and which information to take in. I read an article that was using the Monty Hall ...
Curious student's user avatar
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How to find $P(A|B)$ when we only know $P(A)$ and $P(B)$?

In case you’d want to know: I’m a 6th grade student and I am self-learning probability (that’s one of the things). I know Bayes’ theorem: $$ P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B)} $$ Here’s an ...
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Finding the probability of having a genetic trait given test is positive

I've been doing this question but I'm a little stuck on the second part. The first part is as follows: The probability of a randomly selected person in a population having a particular genetic trait ...
Developer's user avatar
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1 answer
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Is this reasoning about Bayesian Inference on the Lewis Carroll's pillow problem correct?

The Lewis Carroll's Pillow Problem's solution was previously addressed here and made me wonder if it makes any sense to reason that in this context the Bayesian framework allows us to estimate "...
Luciano Dourado's user avatar
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3 answers
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Monty Hall Problem, but the contestant opens the door.

In The Monty Hall Problem, suppose there are three doors A, B, and C. Suppose the contestant chooses the first door A. The variation comes here, suppose that the contestant now has the option to open ...
Notwen's user avatar
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Bayes with LOTP and conditioning

I am trying to solve problem 5 (https://www.probabilitycourse.com/chapter1/1_4_5_solved3.php). Part a) and b) I got right, part c) an extra factor appears in there for me. Here's my work below. Any ...
IGottaLearnMath's user avatar
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Conditional Probability with Three Sets

I'm stuck on finding values related to C. Assume $$ \begin{aligned} P(A) &= 0.3\\ P(B|A) &= 0.75\\ P(B|A′) &= 0.20 \end{aligned} $$ and $$ \begin{aligned} P(C|A \cap B) &= 0....
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Proof Check: Probability of choosing a specific urn when pulling a red ball.

Question: If there are 100 balls in urns A, B and C each (300 total), and the red balls in each urn are 45, 50 and 55, respectively, and the rest being yellow. If we pull a red ball from an urn, what ...
Xerium's user avatar
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Given N observations - Bayesian Posterior for Unknown Variance of a Normal Distribution with a Known Mean?

So, starting from no information besides N trials from a Gaussian with $\mu = 0$, I'd like to know the best Bayesian posterior for the unknown variance, $\sigma^2$. My approach so far as been to ...
SSD's user avatar
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Application of Bayes's theorem to (unfair) coin flipping

I'm trying to solve the following problem: You are given two unfair coins. You flip both of them and ones comes up heads $\frac{2}{3}$ of the time while the other comes up heads $\frac{1}{3}$ of the ...
James Arten's user avatar
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How Do I Calculate The Evidence From Bayes Theorem

In Bayesian Inference, we can calculate the evidence in one of two ways: $$P(E) = \sum_k^K P(H=k)P(E|H=k)$$ or by: $$P(E) = \int P(H)P(E|H) dz$$ We want the probability that our coin toss is fair, ...
Sergio Orozco's user avatar
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Probability of new sample belonging to one set or another

I have a stochastic event with two possible outcomes ($A$ and $B$). I also have two measurement techniques, a direct and an indirect one. The direct one can tell me exactly the outcome, but it implies ...
Franco's user avatar
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Calculate the posterior

I have the following old exam task and I need some help to understand how to solve it. I mean I know how to solve something like that in theory, but their has to be a trick, especially for the ...
aragornthegrey's user avatar
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Can two combined events have a probability greater than the events themselves?

I'm presented with the following conditional probability problem: There are 2 bags with different numbers of lemon and grape candies: Bag #1: 5 lemon candies, 5 grape candies (10 candies total) Bag #...
Clark's user avatar
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Random variables and ordres of joint distributions

I have a question on exchangeable random variables. I have always assumed that given random variables $X$ and $Y$ we have equality of the two joint distributions $p(X,Y)=p(Y,X)$ (don't we need this ...
nomadicmathematician's user avatar
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Would knowing that exactly 14 out of the first 20 coin tosses are heads change the probabilty that exactly 7 out of the first 10 tosses are heads?

Assume that the coin is fair and is tossed an infinitely number of times, independently. In the original version of this question, the former event is "9 out of the first 20 coin tosses are heads&...
ensbana's user avatar
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Conditional probability with two dice

I am trying to solve a problem. Two dice (1 and 2) are rolled. Let's define the following events $$ A_{1} = \text{the result of dice 1 is even}\\ A_{2} = \text{the result of dice 2 is even} \\ A_{3} = ...
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Turning over blue and pink cups until the gender of the baby is known.

I'm trying to do the following problem, and want to know why my approach does not work. A “gender reveal” party is held to announce the gender of an expected newborn. 15 cups are filled in advance ...
ensbana's user avatar
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Finding Posterior Distribution for Simple Linear Regression

A simple linear model is given as $$ Z_i = \gamma_1 + \gamma_2 y_i + \epsilon_i$$ $i=1, \ldots,n.$ Let $\mu = (\gamma_1, \gamma_2)'$. Assuming that $\epsilon_i \sim N(0,1)$ and using a noninformative ...
holala's user avatar
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Finding the posterior and Bayes estimator with a beta prior

Let $Y_i, \ i =1,2,\ldots n$ be a random sample from the probability function $$f(y\mid p) = \frac{2y}{p^2}, \quad 0 < y \le p$$ where $p\sim Beta(2n+1, 1)$ is the prior, find the posterior ...
holala's user avatar
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Work someone check my work using Bayes Classifier?

Background: Suppose we have two classes for p=2 variate observations, where the probability for class 1 follows MVN($\mu_1$, $\Sigma$) and the population for class 2 follows MVN($\mu_2$, $\Sigma$) ...
Calum's user avatar
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Expected number of heads after 4 coin tosses plus an additional condition.

Let's say we take the simple question: "What is the expected number of heads after 4 coin tosses?" In this case, the answer is straightforward. For $1≤i≤4$ let $X_{i}$ be a random variable; ...
Mkion57's user avatar
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Trouble understanding a Bayesian update over multiple hypotheses

I have found the following formulation for Bayes' Theorem for multiple hypotheses: $P(h \mid e) = \Large {P(e \mid h) P(h) \over \sum_i P(e \mid h_i) P(h_i)}$ Suppose I have three hypotheses and three ...
James Watson's user avatar
7 votes
1 answer
270 views

Bayes' theorem and card colors

This is an expansion/generalization of a previous question I've asked here. Some of the simplifications I made in the original question turned out to be too simplifying, so I'm trying again. The most ...
mikev's user avatar
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Binary Classification Problem finding thetha

Consider a binary classification problem to be solved using Bayes decision rule. It is given that $P(x | y_0) = \sigma(0.9x_1 + 0.3x_2 + \theta)$ and $P(x | y_1) = \sigma(0.3x_1 - 0.1x_2 + 0.7)$, and ...
IsraKing10001's user avatar
1 vote
1 answer
117 views

Find the Bayes estimator under mixture normal distributions.

Consider the following model: $$\mathbf{X}=(X_1,...,X_p)|\theta \sim (1-\theta)N_p (\mathbf{0},\mathbf{\Sigma})+\theta N_p (\mathbf{\delta}, \mathbf{\Sigma})$$ $$\theta \sim \textrm{Bernoulli}(1-\pi)$$...
Sofia Delacruz's user avatar
1 vote
1 answer
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Why can't I apply the odds version of Bayes' to this exit poll problem?

This is a question about an alternative solution to the problem stated in Probability and exit polls The problem is: Consider an election with two candidates, Candidate A and Candidate B. Every voter ...
matto's user avatar
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2 votes
3 answers
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Sheldon Ross' A First Course in Probability - Section 3.3., Example 3f: Further questions on conditional probability.

Here's the statement of the problem. At a certain stage of a criminal investigation, the inspector in charge is 60 percent convinced of the guilt of a certain suspect. Suppose, however, that a new ...
fresh_start's user avatar
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Help me understand the application on Bayes' Theorem in this case

I am studying a paper (https://www.jstor.org/stable/1804124) which discusses a two period revenue optimization problem as follows We are trying the sell a single unit of product over two periods, i.e. ...
Ricky Doo's user avatar
3 votes
1 answer
116 views

Conditional Probability of Finding a Defective Item amongst $k\times m$ Items

There are $k$ packages, each with $m$ items. One of the $k \cdot m$ items is a defect. To find the defect, $n$ items are randomly selected from each package. I wish to determine the probabilities that ...
V. Elizabeth's user avatar
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2 answers
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Where did I go wrong with my reasoning of the blue cab/green cab problem?

Here’s the statement of the problem. "A cab was involved in a hit-and-run accident at night. Two cab companies, the Green and the Blue, operate in the city. You are given the following data: 85% ...
fresh_start's user avatar
1 vote
0 answers
63 views

Probability of rain in Tokyo

Three friends in Tokyo told you that its rainy. Each person has a probability of 1/4 of lying. What is the probability that it is raining in Tokyo? Assume $Pr(rain)$ in Tokyo = 1/5. Given options: A) ...
Tuhin Dutta's user avatar
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How to solve example with multiple events using Bayes product rule

I have some confusion regarding one of the examples from probability and statistics course. The translation of the examples goes something as: The chef knows how to prepare 4 different meals. He ...
Mire's user avatar
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Probability of composition of two random algorithms working on sets

Suppose we have a random algorithm $A$ that builds a set $\mathcal{S}^n$ of $n$ elements from a domain $\mathcal{H}$, and, in particular, the probability of a particular $\hat{h} \in \mathcal{H}$ to ...
Antonio Ferrara's user avatar
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0 answers
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3 prisoner problem slightly reworded. Who is right?

Three prisoners, A, B and C, have been told by their jailer that one of them, chosen at random, will be executed, and the other two will be freed. Prisoner A says to the jailer,“I know that one of the ...
i.diazr's user avatar
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1 answer
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Changing events when given additional information in Bayes' Theorem

I encountered a problem as follows: You have a white cube with side length 3. You paint all the faces red, then cut it into 27 small cubes with side length 1. Then, you randomly pick one small cube, ...
LBD635's user avatar
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-2 votes
2 answers
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Is there something wrong with Bayes' Theorem

Bayes' theorem (simplified): $P(A|B) = \frac{P(A) \times P(B|A)}{P(B)}$ Per the books/teachers I've had the honor to meet, the probability of an event $P(E)$ is such that $0 \leq P(E) \leq 1$ i.e. $P(...
Agent Smith's user avatar
1 vote
2 answers
117 views

An urn has 5 blue, 3 green and 7 yellow marbles. two marbles are drawn without replacement. probability that the 3rd one will be a green marble?

Probability of 3rd ball being green while drawn without replacement, in a case where there are 5 blue 3 green and 7 yellow marbles in an urn. I'm having a hard time coming to a conclusion about the ...
Suburban13's user avatar
3 votes
2 answers
395 views

Probability of drawing from box B

I have the exercise: Consider the following scenario. There are two boxes, Box A and Box B. Initially, Box A contains 3 red balls and 3 white balls; Box B contains 6 red balls. We swap the balls in ...
mads grønbeck's user avatar
0 votes
5 answers
100 views

How do I solve this problem using bayes theorem?

The problem is as follows A secret agent, disguised as a waiter for the evening, momentarily observes a note with a highly confidential password over the shoulder of an invited military commander. ...
Pete's user avatar
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0 answers
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Determining the probability of 6th round championship win

You are betting on a tournament, teams X and Y play until first to four wins. You want to place a bet on the 6th round. What is the probability of team X winning the tournament in the 6th round? My ...
Jared 's user avatar
2 votes
1 answer
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Number of problems a student knows on an exam given the conditional probability of the student knowing the answer given the answer is correct.

A student takes an examination consisting of $20$ true-false questions. The student knows the answer to $N$ of the questions, which are answered correctly, and guesses the answers to the rest. The ...
Numerical Disintegration's user avatar
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2 answers
53 views

Why does this solution use the simplified version of Bayes' Formula?

For the following problem In a given health insurance group $20$% of policyholders have high blood pressure and $30$% have high cholesterol. Of those with high blood pressure $25$% have high ...
Numerical Disintegration's user avatar
1 vote
1 answer
118 views

Gaussian distribution using Bayes' rule

I have this exercise: An explosion was detected by two sensors. Each sensor is only able to output a noisy estimate of the location of the explosion due to measurement noise. Assuming the two sensor ...
mads grønbeck's user avatar
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0 answers
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Question about computing denominator in Bayes' theorem

I have a question which I found on the web: Two boxes are placed in a cupboard out of which the first box contains 1 black and 3 red balls and the second box contains 4 black and 2 red balls. A ball ...
user5954246's user avatar
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2 votes
2 answers
141 views

8 Drawer Desk Conditional Probability Problem: Use Bayes' Theorem using the given statement

A desk has eight drawers. There is a probability of 1/2 that someone placed a letter in one of the desk's eight drawers and a probability of 1/2 that this person didn't place a letter in any of the ...
Adrian Fletcher's user avatar
1 vote
2 answers
226 views

A more complicated conditional probability of drawing balls from an urn

Suppose there are 12 balls in an urn: 4 red, 4 yellow and 4 black. We draw at random without replacement. What is the probability that the 6th draw will be a yellow ball, given that in the first 5 ...
Jan Stuller's user avatar
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