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

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Doubt about a probability excercise

I'm a statistics teacher at a college. One day a student came with a doubt about an exercise about probability. The text goes like this: A person has two boxes $A$ and $B$. In the first one has ...
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1answer
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Intuition behind aparticular formulation of Bayes's Theorem

Bayes's theorem states $P(A\mid B) = \dfrac{P(B\mid A)\cdot P(A)}{P(B)}$. The intuition behind this is pretty simple: if $B$ is true, then the probability that $A$ is true is the number of cases where ...
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2answers
50 views

Which one of the following versions of Bayes' theorem is correct?

I've seen two versions of Bayes' theorem: I've seen this very long version from a frequentist probability class: $$ P(B|A)=\frac{P(A|B)P(B)}{P(A|B)P(B) + P(A|B^c)P(B^c)} $$ where $B^c$ is the event ...
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0answers
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Process Transition Algorithm

I have a process with 100 possible states and independent entities going through the process. All the Entities have been observed through a span of 5 years at the end of each month. When the ...
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0answers
12 views

Dynamic Bayesian Networks without restrictions

Normally, when you create a Dynamic Bayesian Network, the restriction is that any random variable in time t depends only on variables in time t-1. There are some other algorithms like AR-HMM that ...
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0answers
29 views

Confused if this is conditional or dependent probability.

John observers the following while driving to work. • 4 were driving a red car. • 3 were driving a blue car. • 3 were driving a black car. He also notices ...
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29 views

Calculating Conditional Probability (Maybe using Bayes Theorem)

Well, I've a bunch of variables, viz $Age$, $Gender$, etc.. Age can take: values from $1$ to $6$ (which are actually coded) Gender can take: $0$ and $1$ I know the $$P(A=1),...,P(A=6), ... P(G=0), ...
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1answer
26 views

bayes rule question regarding exam performance related

In an exam, there is a problem that $60\%$ of students know the correct answer. However, there is $15\%$ chance that a student picked the wrong answer even if he/she knows it and there is also a ...
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1answer
27 views

Bayes' theorem timeline of events

When using Bayes' theorem to find the probability of future events along a tree I have had it explained to me that parent events to the child events have already happened? Is this strictly true or can ...
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1answer
52 views

Conditional and joint PDFs and PMFs

We say that $X ∼ \text{Gamma}(a,b)$ (for $a,b > 0$) if the PDF $f_X$ is given by $f_X(x) = Γ(a)/(x^{a−1}e^{−bx}) $ for $x > 0 $ and 0 otherwise. Suppose $ X ∼ \text{Gamma}(a, b)$ and $Y |X ∼ ...
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1answer
39 views

Prove that if events $A,B$ independent of C then $P(A\cap B\cap C)= P(A\cap B)P(C)$

I am trying to prove why the intersection of two events $A, B$ that are independent of C is also independent of C so that the following equality holds: $$P(A\cap B\cap C)= P(A\cap B)P(C)$$ ...
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2answers
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How to conceptualize “dividing out” a number (e.g. in permutations, Bayes' Theorem)?

I'm trying to achieve a better conception of what it means to "divide out" a variable/number, because I'm currently have a lot of trouble justifying to myself why it actually works the way it does in ...
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3answers
48 views

Each of two evidences increases prior probability but both decrease it. May this only happen if two evidences are dependent?

I noticed this while working on another problem. My intuition is that the statement is true, but I am not sure. Let A is an event. Evidence 1 and 2 are $E_1$ & $E_2$ correspondingly. $$P(A|E_1) ...
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1answer
35 views

Bayes' theorem with multiple variables

On the page: https://en.wikipedia.org/wiki/Bayesian_inference#Formal_description_of_Bayesian_inference there is the result: $$p(\theta \mid \mathbf{X},\alpha) = \frac{p(\mathbf{X} \mid \theta) ...
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2answers
27 views

Having a value, calculating the probability of the number of times a die has been rolled

If I enter a friend's house and he has rolled a dice (d6) on a table which has a value $1$, he asks me the following, "Do you think I rolled this dice once and got $1$ or do you think I rolled it ...
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1answer
8 views

Creating pdfs froms Sample Data and Bayes Theorem for Continuous Probability

I am not much of a math guy, but know some basics of pdfs, pmfs, Bayes theorems, probability distribution and stuffs. I am actually trying to build a Bayesian Network that models the personality of ...
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0answers
37 views

Using Bayes Theorem To Simplify Probabilities

I want to compute P(AB|X) in terms of the (given) probabilities P(B), P(X), P(A|BX), P(AB, !X), P(A|!B,X), P(A|!B, !X) where every letter is some random variable. How would I go about doing so using ...
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1answer
19 views

Determine probability of better player given indirect measurements

Say you have a two-player game and three, fixed-but-different skill players: A, B and C. Player A only plays against C and player B also only plays against C. You want to to determine if player A is ...
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2answers
31 views

Understanding different definitions of bayes theorem

I am taking course on probability and reading about bayes theorem. In Sheldon Ross' book, it given as $$P(E) = P(E|F)P(F) + P(E|F^C)P(F^C)$$ with note: Equation above states that the probability of ...
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2answers
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Baye's Theorem Conditional Probability with multiple conditions

Lets assume I have a supermarket and I track the purchase history of my customers with 2 attributes of each customer - Gender (M/F) & Smiling (Y/N). Assume this is historical data of purchases: ...
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2answers
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Why do We Refer to the Denominator of Bayes' Theorem the “Marginal Probability”?

Consider the following characterization of Bayes' Theorem: Bayes' Theorem: Given some observed data $x$, the posterior probability that the paramater $\Theta$ has the value $\theta$ is $p(\theta \mid ...
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ANSML - Proof of Naive Bayes Derivation

I was working through one proof of the Naive Bayes and got stuck at the last step. The setup is as follows: Given a dataset $\left\{ (x^{(i)},y^{(i)}), \cdots\right\}$ for $i=1,\cdots,m$, $y$ can ...
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Classification Rule for Bayes Optimal Classifier (continuous x var)

I'm having trouble determining the Bayes classifier where x is a continuous variable. Basically the problem I'm trying to solve is this: There are type 1 and Type 2 components. There are 66 Type 1 ...
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1answer
26 views

Bayes estimator under squared error loss

Consider one random variable X from the Bernoulli distribution with parameter θ. Let p, the prior density, be equal to 6θ(1 − θ), for θ ∈ (0, 1). Under squared error loss, L(t, θ) = (t − θ)$^2$, the ...
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1answer
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Stumped on a Bayes Theorem Question

A certain medical syndrome is usually associated with two overlapping sets of symptoms, A and B. Suppose it is known that: P($A|B$) = 0.8 P($B|A$) = 0.9 P($B'|A'$) = 0.85 Find P($A'$|$B'$) From ...
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1answer
26 views

Statistical/Combinatorial: How to analyze?

I'm currently preparing for my exam and in the process trying to solve some statistical problems. The question goes as follows: Q1: A book consisting of 269 pages contains 40 missprints. Only, you ...
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1answer
48 views

unintuitive meaning to Bayes theorem

The equality $\frac{\mathbb{P}(Y\mid X)}{\mathbb{P}(Y)}=\frac{\mathbb{P}(X\mid Y)}{\mathbb{P}(X)}$ means that in a supermarket analysis, knowing that a customer bought milk ($X$) multiplies the ...
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1answer
41 views

Combining probabilities from different sources

Lets say I have three independent sources and each of them make predictions for the weather tomorrow. The first one says that the probability of rain tomorrow is 0, then the second one says that the ...
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0answers
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Bayesian posterior with a constraining equation (slice-projection?)

Prior and signals: Let $y_1$ and $y_2$ be iid signals on $Y$. The intial prior is $Y \sim N(\bar{Y}, \sigma^2_Y)$, where $N(\cdot, \cdot)$ is the normal distribution The signals are independent and ...
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2answers
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How am I misusing the Bayes' rule?

I just started studying probability. I am trying to solve this exercise: When coin A is flipped it comes up heads with probability 1/4, whereas when coin B is flipped it comes up heads with ...
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2answers
55 views

Bayes theorem: Can some one explain in plain words how argmax is used here?

According to Bayes theorem $$p(y |x) = \frac{p(x |y) p(y)}{p(x)}$$ So, to find the maximal value for $p(y |x)$, we want to compute: \begin{align} &\arg\max_{y\in Y} p(y|x) \\ ...
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1answer
30 views

How to use Bayes's rule with mixed distributions?

On page 81 of The Likelihood Principle by Berger and Wolpert (1988) I find the following claim (which references example 20 on page 75). We consider a certain statistical problem from a Bayesian ...
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1answer
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Probability Urn problem - application of Bayes thm

I am learning statistics and I am trying to solve this problem: Players A and B draw balls in turn, without replacement from and urn containing three red and four green balls. A draws first. The ...
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1answer
47 views

Bayes theorem - is it applicable in any case?

I'm studying the Bayes' Theorem and I have a doubt. In this wikipedia page there's an example of application for the following events: ...
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2answers
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In Bayesian Statistic how do you usually find out what is the distribution of the unknown?

To estimate the posterior we have $$p(\theta|x) = \frac{p(\theta)*p(x|\theta)}{\sum p(\theta ')*p(x|\theta ')}$$ $x$ is usually the experimentally sampled data, and $\theta$ is the model, but both ...
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1answer
47 views

Validity of conditional CDF proof via PDF integral

Given the question: $$\text{Show that}\ F_X(x\mid A) = \dfrac{\Pr(A\mid X\leq x)}{\Pr(A)}\cdot F_X(x)$$ I have seen the solution via probabilities 'directly'. My question is whether the following ...
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Proof using Bayes rule?

In Statistical analysis of randomized experiments with non-ignorable missing binary outcomes:an application to a voting experiment by Kosuke Imai a proof is given referring to Bayes rule. Let: ...
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1answer
54 views

Bayes Rule in 2 Fair and 1 Biased coin

I was watching this video on Khan Academy about condition probability where they demonstrated a problem using a tree. I tried to solve that problem using Bayes rule, but my answer doesn't match the ...
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2answers
43 views

Bayes' rule with 3 variables

I have been using Sebastian Thrun's course on AI and I have encountered a slightly difficult problem with probability theory. He poses the following statement: $$ P(R \mid H,S) = \frac{P(H \mid ...
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1answer
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Bayes' theorem to find $P(A\cap B), \,P(B\mid A),\,P(A\cup B)$

Given, $P(A)=0.4, P(B)=0.5,P(A\mid B)=0.3 $. Need to find $$P(A\cap B), \,P(B\mid A),\,P(A\cup B).$$ So far I did $$P(A\cap B) = P(A\mid B) P(B) = ...
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1answer
49 views

Conditional probability, Baye's rule, prisoner Ural / Siberia + coat exercise

I am currently statistics and probability course. One of the questions in the textbook is following: A prisoner will be sent to either Urals or Siberia, but he does not know where. He knows, ...
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3answers
48 views

Probability of coin flips conditioned on an assistant lying about the results

During a project researching coin flips, an assistant is asked to flip a fair coin twice. But the assistant is feeling lazy, and instead of following the directions, he does the following: He starts ...
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1answer
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Probability question for economics that I'm struggling with. Please help.

(There are 4 districts in the land of Oz. At home, the inhabitants of each region wear ties of a special colour, Munchkins (M) wear blue, Scarecrows (S) wear purple, Tin Men (T) wear red and Wizards ...
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1answer
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Please help me to answer this conditional probability question

Question: So, for part (a), my answer is: $$ {10 \choose 8} (0.6)^8 (0.4)^2 + {10 \choose 9}(0.6)^9 (0.4)^1 + {10 \choose 10 } (0.6)^10 (0.4)^0 = 0.1673$$ I am not sure how to answer part ...
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3answers
150 views

Probability of balls drawn with replacement

We have two bags, Bag A has 40 red balls and 15 blue balls, Bag B has 40 blue balls and 10 red balls. One of these bags is selected at random and from it five balls are drawn at random, replacing each ...
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Normalizing factor for product of Gaussian densities - interpretation with Bayes theorem

The normalizing factor for the product of two multivariate Gaussian densities, $f(x)$ and $g(x)$ with mean vectors $a$ and $b$ respectively, and covariance matrices $A$ and $B$ respectively, is itself ...
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Simple Probability of Playing Cards

An ordinary deck of playing cards has four suits: hearts, spades, diamonds, and clubs. Suppose you have a reduced deck of eight playing cards, consisting of four aces and four kings. I draw two cards ...
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2answers
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interpretting prior and posterior

I am bit puzzled on how we can interpret the posterior. Assume a coin which is 0.1 probable to be unfair. So our prior probability on the coin being unfair is 0.1, and being fair is 0.9. Also by ...
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1answer
37 views

Conditional Probability of Sinking Ship Question

Question: Two ships. Ship A's missiles have an 80% probability of hitting its target, ship B's missiles have a 50% probability of hitting the target. It only takes one hit from a missile to sink a ...
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1answer
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Proving Conditional Probability Statement

Question: Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Prove or disprove that $P(A|B \cup C) = P(A|B) + P(A|C) ~~~\forall~~ A,B,C~ \in \mathscr F$ where $B \cap C = \emptyset$. ...