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Questions tagged [bayes-theorem]

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

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Probability apple is delicious given a red apple is produced in a green apple orchard?

An example scenario of what I am trying to model: A farmer has a large apple farm, producing green apples (C=baseline apple color ?). Sometimes, a red one is produced (B=apple is red). I am wondering ...
learning_physics's user avatar
-1 votes
1 answer
35 views

Likelihood of Bayes' theorem [closed]

When estimating the parameter (hypothesis), I thought it was correct to compare the values of "P(hypothesis_i | observed data)" by changing i for each hypothesis However, when applying Bayes'...
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Experimental Design: Selecting value of $n$ given desired width of credible interval

Suppose I have $n$ IID Bernoulli trials with $k$ successes. Assume that as a prior we are assuming that $P(\theta)$ is uniform on $[0,1]$. We can pretty easily use Bayes theorem to represent the ...
wjmccann's user avatar
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Gaussian as mixture with uniform weights

Suppose $W$ is $\operatorname{Uniform}[0,1]$ distributed. Suppose $Y|W$ has known mean $\mu(W)$ and variance $\sigma^2(W)$, where $\mu$ and $\sigma$ satisfies some regularity conditions: (1) $\mu$ and ...
yrq's user avatar
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Need Clarity in Finding Expectation of X, and Y

If there are four buses each bus has 40,33,25,50 students respectively. If X,Y are two random variables, X represents the no of students in the bus of the selected student , Y represents the no of ...
vishnu_vardhan's user avatar
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Deriving posterior distribution with explicit constants in closed form using Jacobian method compared to Bayes' rule

Question on Bayesian Inference Deriving posterior distribution with explicit constants in closed form using Jacobian method compared to Bayes' rule I am working on a Bayesian ...
Alireza Ghazavi's user avatar
1 vote
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21 views

Solving the marginal likelihood integral and approximating closed-form solutions for that

Given the likelihood function: \[ p_{y_s|r_s,x}(y_s|r_s,x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{(y_s - \alpha \frac{\rho_s}{r_s^4})^2}{2}} \] and the prior distribution for \( r_s \): \[ p_{r_s}(r_s) = \...
Alireza Ghazavi's user avatar
4 votes
1 answer
78 views

Best strategy to determine which of two coins is biased using only two flips

We have two coins - one fair, and one biased with a probability of 0.6 for flipping a head. We get to make exactly two flips before making a guess as to which coin is biased. The question is which ...
Abhinav Sood's user avatar
1 vote
2 answers
69 views

I don't understand why this solution used Bayes Theorem instead of the regular formula for conditional probability. [closed]

When I view the solution to this problem, I understand the calculations but I don't understand why those calculations were done. Below is the question that I'm reading A theme park conducts a study ...
j.jerrod.taylor's user avatar
-1 votes
1 answer
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Simple example of Bayesian probability, picking from a bucket

Bowl I contains six red chips and four blue chips. Five of these 10 chips are selected at random and without replacement and put in bowl II, which was originally empty. One chip is then drawn at ...
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A mapping is selected at random from all defined on set A . If the selected mapping is injective, the probability that only one element maps on itself

Now my attempt : Total mappings $=n\times n\times n.....=n^n$ Since every element maps to only one other element and none is left untouched in injective: Number of injective mappings $=n(n-1)(n-2)........
Aurelius's user avatar
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How to Derive the Binomial Coefficient Upper Bound and Final Inequality in "Scheduling Multithreaded Computations by Work Stealing"?

In the paper Scheduling Multithreaded Computations by Work Stealing under the section "Atomic accesses and the recycling game", it mentions the binomial coefficient approximation: $$ \binom{...
grzhan's user avatar
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Derivation of Inequality 3 from Inequality 4 Using Bayes' Theorem in "Scheduling Multithreaded Computations by Work Stealing"

In the paper "Scheduling Multithreaded Computations by Work Stealing" under the section "Atomic accesses and the recycling game", it is mentioned that inequality 4: $$ \Pr \left\{ ...
grzhan's user avatar
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2 answers
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Bayes theorem tricky example

In a certain population, 1% of people have a particular rare disease. A diagnostic test for this disease is known to be 95% accurate when a person has the disease and 90% accurate when a person does ...
samsamradas's user avatar
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Prove or disprove: $P(A\cap B|C)=P(A\cap B)$ given $P(A|C)=P(A)$ and $P(B|C)=P(B)$. [duplicate]

Prove or disprove: $P(A\cap B|C)=P(A\cap B)$ given $P(A|C)=P(A)$ and $P(B|C)=P(B)$. Since \begin{align} P(A \cap B|C)=P(B|A \cap C) P(A \cap C)/P(C)=P(B|A \cap C) P(A), \end{align} the statement is ...
kenji's user avatar
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Conditional probability density function of the parameter $\lambda$, given a data set.

Consider the one-sided conditional exponential distribution: \begin{equation} f_X(x|\lambda)=\frac{\lambda}{Z(\lambda)}\exp(-\lambda x), 1\leq x\leq 20, \end{equation} where $\lambda>0$ and $Z(\...
SecretKeeper's user avatar
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Apply Bayes' Rule to a joint, prior distribution?

The problem is how to update a joint prior distribution consisting of two independent variables with observed evidence? For the prior distribution, data exists for average event frequency (i.e., ...
shammi's user avatar
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How to get an intuitive viewpoint towards Bayes theorem

At the time of shuffling 52 playing cards, 5 cards are dropped accidentally. If it is known that 2 of 5 cards are red, find probability that all dropped cards are red cards. I cannot seem to ...
Mitansh's user avatar
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1 answer
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Determine the distribution of $X$

Let $N$ a random variable such that $N \sim \operatorname{Pois}(\lambda)$. Furthermore, let $X$ a random variable such that $$\mathbb{P}( X = k\ | \ N=n)=\binom{n}{k}p^kq^{n-k},\ \ 0\leq k \leq n, \ \ ...
Nicolas Rodriguez's user avatar
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Relationship between the limit of the CDF and the PDF evaluated at the limit value.

Apologies if this is a dumb question, but I haven't been able to find an answer. Suppose an event induces a posterior belief: $$ \frac{p*\int_{0}^{x}f(t)dt}{p*\int_{0}^{x}f(t)dt+(1-p)*\int_{0}^{x}g(t)...
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Question 29 from Chapter 3 of A first course to probability from Sheldon Ross ed. 10

Xiku Road has $n_1$, $n_2$, $n_3$, and $n_4$ houses with $1$, $2$, $3$, and $4$ occupants, respectively. Two random selection without replacement strategies are being contemplated to obtain a sample ...
Abhishek Singh's user avatar
2 votes
1 answer
34 views

Conditional Probability in Selection of Nuts from Different Weight Distributions

I'm struggling with solving a problem involving conditional probability. Here's the problem: There are three species of nuts: those from the canton of St. Gallen (Switzerland), whose weight in grams ...
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Bayes' formula for distriibutions

In my statistics course, we have discussed Bayes' theorem for distributions: $$ p(\theta | y)=\frac{p(y|\theta)p(\theta)}{\int p(y|\theta)p(\theta) \mathrm{d}\theta} $$ Where $p(\theta)$ is the prior ...
sodium-hydroxide's user avatar
1 vote
1 answer
60 views

Calculate the posterior distribution

How can I solve the letter (a)? Discrete sample spaces: suppose there are N cable cars in San Francisco, numbered sequentially from $1$ to $N$. You see a cable car at random; it is numbered $203$. You ...
Siqueira's user avatar
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1 answer
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A conditional probability problem about a fiber cabel and bits, Bayes' theorem

I've been racking my brain about this problem for way longer than I should: "Through a fiber cabel, information is sent in the form of bits, that can take the values 0 or 1. Sometimes, there is ...
Edward Chen's user avatar
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false negative as a conditional probability

Suppose we have a test for covid with a false negative rate of 4%. Let $T$ be a positive test result for Covid and $C$ for a person having Covid. Does that mean $P(C| \neg T)=0.04$, or $P(\neg T| C)=0....
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Proving that $T_{n}(X)=\sum_{i=1}^{n}X_i$ is a sufficient statistic for $p$ given a sample of i.i.d random variables

I am asked to show that the statistic $T(X):=\sum_{i=1}^{n}X_i$ is a sufficient statistic for $p$, where $X_i\sim Geom(p)$ are i.i.d random variables. Given a sample $x=(x_1,x_2,\dots,x_n)$ I have to ...
Tutusaus's user avatar
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1 answer
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Combating a specific argument against the Monty Hall problem .

So to get started , I make it clear that I do know how the Monty Hall problem works and although I had my good share of problems understanding it in the past , I did manage to come to terms with it on ...
Mike Billings's user avatar
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1 answer
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Bayes' Theorem with cases

Exactly $\frac{1}{5}$ people that live on earth have a condition .There are two tests for this condition, the Z1 test and the Z2 test. When a person goes to a doctor to test for this condition, with ...
PsychBit's user avatar
1 vote
1 answer
24 views

(Bayesian probability) Show that $P(H|E) = \frac{h (c +a \overline c)}{hc+a\overline c}$

The question, briefly How does the calculation $P(H|E) = \frac{h (c +a \overline c)}{hc+a\overline c}$ work? Some background I'm trying to work through the proof of the following theorem in ...
snofelet's user avatar
2 votes
1 answer
318 views

Let $S$ be the set of points whose coordinates $x$ and $y$ are integers that satisfy $0\le x\le3,$ $0\le y\le4$. Two distinct points .....

Let $S$ be the set of points whose coordinates $x,$ $y$ are integers that satisfy $0\le x\le3,$ $0\le y\le4$. Two distinct points are randomly chosen from $S.$ The probability that the midpoint of ...
Maths lover's user avatar
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Bayes Method How to draw coefficients summing to 1 and each of them follow exponential distribution

There is a question that how to draw the estimators (ratio estimator) if we know the prior for the numerator or the ratio estimator. Assume that we have three coefficients: $a_1,a_2$ and $a_3$ . And ...
Mike's user avatar
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3 votes
2 answers
68 views

Conditional probability using Bayes's theorem

From Blitzstein, Introduction to Probability (2019 2 edn), Chapter 2, Exercise 25, p 87. A crime is committed by one of two suspects, A and B. Initially, there is equal evidence against both of them. ...
Gerry's user avatar
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1 answer
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Can't understand Bayes theorem

A software company conducted a test on their new platform by exposing their users to two versions of the same product. Number of users that were given version A: 4000 Number of user that were given ...
Musab Gulfam's user avatar
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1 answer
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Exercise involving Bayes' theorem

I want to solve the following exercise: John tells the truth two out of every three times he is asked, while Peter tells the truth four out of every five times. Both agree in assuring that from a box ...
Wrloord's user avatar
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0 answers
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Possible error in exercise on Bayes' theorem

I want to solve the following: During exam time, in a certain school, only 25% of the teachers warn their students in writing that they are not allowed to stand up and ask questions during the test. ...
Wrloord's user avatar
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-2 votes
1 answer
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what is conditional probabiltiy [closed]

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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1 answer
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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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0 answers
97 views

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
1 vote
3 answers
110 views

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 ...
user avatar
1 vote
1 answer
95 views

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
3 votes
1 answer
66 views

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
2 votes
3 answers
377 views

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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1 vote
1 answer
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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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1 answer
50 views

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....
nickalh's user avatar
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1 vote
1 answer
33 views

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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0 answers
86 views

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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1 vote
0 answers
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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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0 answers
54 views

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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