Questions tagged [conditional-probability]

For questions on conditional probability.

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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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Memoryless property with any random wait time

We see here a proof of the random-time memoryless property $P(X>T+s|X>T)=P(X>s)$ where $E\sim Exp(\lambda)$ and $T\ge 0$ is a continuous random variable independent of $E$. The proof, however,...
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Algorithm for Weighted Combination

How can we randomly select $k$ people out of $n$ such that for person $i$, the probability they get selected is $0<p_i<1$, and $\sum p_i=k$? I'm struggling quite a bit on this; you could go ...
username's user avatar
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3 answers
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Conditional expectation of random summation - How to show $E[\sum_{i=1}^{N}\xi_i|\sigma(N)]=pN$?

I am using the formal definition of conditional expectations: $E[X|\mathscr{F}]$ is any RV $Y$ such that $Y\in\mathscr{F}$ and $\int_AXdP=\int_AYdP$ for all $A\in\mathscr{F}$. Suppose that $\xi_1,\...
PorkingBun's user avatar
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Regular conditional distribution of $Y$ given $X=x$ in Klenke's book

In his book "Probability theory", Klenke uses the following definition of transition kernel: and if in $ii)$ the measure is a probability measure for all $\omega_1$ then $K$ is called a ...
StrugglingScholar's user avatar
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I want to know the expectation of a product of independent beta distributed random variables.

I have an equation of the form $$Z = \frac{\prod_{i=1}^p X_i}{\prod_{i=1}^p Y_i}$$ where $$X_i \sim \mathcal{B}(\alpha_{x_i},\beta_{x_i})$$ and $$Y_i \sim \mathcal{B}(\alpha_{y_i},\beta_{y_i})$$ $X_i$ ...
Jake Austria's user avatar
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Probability Density Function of $Y \vert X$ when both $X$ and $Y$ are conditional on $\theta$

In the Bayesian setting, suppose that we know the PDF of $X$ given $\theta$ is $p_X(x \vert \theta)$ and the PDF of $Y$ given $\theta$ is $p_Y(y \vert \theta)$. In the standard fashion, we may assume ...
YessuhYessuhYessuh's user avatar
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1 answer
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Conditional distance distribution between 2 points given the distance to another point

Question: Suppose we have three uniform $[0,1]$ random variables $X,Y,Z$. What is $P(|X-Y|\leq r_1 | |X-Z| = r_2)$? Things I know already / have tried: I have that the CDF of $|X-Y|$ is $P(|X-Y|<r) ...
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How to represent difficulty and success?

This is a revision of another question found here: Consider a criminal pondering the commission of a crime, C. It could be launching a cyber-attack or embezzling funds. How likely is it that the ...
Tupelo Thistlehead'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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In Bayesian probability if P(a|bc)=0 will P(b|a) always be less or equal to P(b)? [closed]

If we have 3 variables a, b and c and we assume that a conditional on the conjunction of bc is impossible so that P(a|bc)=0 then will the posterior probability of b always be lesser or equal to P(b) ...
Lingvistisk Dialektik's user avatar
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Optimal Conditional Distribution for Minimising Information-Theoretic Expression

Consider two countable sets $\mathcal{X}$ and $\mathcal{Y}$. I aim to find the conditional distribution $P_{Y|X}$ that minimizes the following expression for any $x \in \mathcal{X}$ $$\sum_y P_{Y|X}(y|...
pmoi's user avatar
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5 answers
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(SOLVED) Monty Hall: the number of unknowns decreases but probabilities stay the same?

I recently got an explanation of the Monty Hall problem and I thought I understood it but after giving it more thought, it still looks wrong. Instead of using goats and doors, the example used a 52-...
moumous87's user avatar
3 votes
2 answers
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Information-theoretic Inequality

If we have two discrete RVs, X, and Y. How can we show: $$\sum_{x,y} p(x|y)p(y|x) \geq 1.$$ The question goes further with finding a sufficient and necessary condition for equality. My attempt: For ...
Science Addict's user avatar
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If one event influences the other, are they independent?

Consider some human action, C. It could be to commit a crime like launching a cyber-attack or embezzling funds. How likely is it that there is an actor that intends to attempt C in the next year? Call ...
Tupelo Thistlehead's user avatar
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Follow up question on application of Fubini in an old answer

I am working with the same problem and have a follow up question to an old solution: https://math.stackexchange.com/a/1842770/1100158. Here they use Fubini's, but nowhere does it say that the random ...
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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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2 answers
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Probability of two specific cards to be in your hand in a game of bridge

Assume that a 52-card deck is distributed among 4 players, each with 13. What is the probability that two specific cards, say the Ace of Hearts and Ace of Diamonds to be in my hand? I have two ...
Willow's user avatar
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2 answers
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Maximum after m tries is the same as maximum after m+1 tries, without replacement

This is a question in a quant interview. Given a bag of $n$ marbles each of distinct weights, perform the following process. Pick two marbles at random, keep the heavier one and return the lighter one....
I_cosine_this's user avatar
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Is This Summation Expression Equivalent to the Conditional Probability ($ P_{Z|X} $)?

I have encountered an expression in my studies and am trying to determine if it correctly represents the conditional probability ($ P_{Z|X}(z|x) $). The expression is as follows: $$ \sum_{s \in S} P_S(...
Alireza Ghazavi's user avatar
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1 answer
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Conditionally independent coordinates relative to sum

Let $X$ and $Y$ be random variables taking values in the same finite abelian group $G$, so that we can define their sum $X+Y$. Let $(X_1, Y_1)$ and $(X_2, Y_2)$ be conditionally independent copies of $...
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How do you get joint probability of two events conditional on the same event?

I want to get the joint probability of two independent events, both conditional on another event. What I know: Probability of A is 0.2375. Probability of B given A is .8. Probability of C given A is ....
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Algorithm for Weighted Permutation

Suppose we would like to permute a list of $n$ elements, such that for each element $a_i$, it has a $p_{ij}$ probability of being in the $j$th slot in the permutation, and we assume $\sum_{i=0}^np_{ij}...
username's user avatar
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1 answer
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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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Proving $P(A\mid B,C)=1$ given $P(A\mid B)=1$ [duplicate]

I'm trying to prove the following statement formally: if $P(A\mid B)=1$, then $P(A\mid B,C)=1$. I can see why logically speaking the statement is correct but I'm lacking a formal argument, using ...
Roberto Rozzi's user avatar
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Clarification on independence and conditionality of coin tosses and the resulting PMF

I'm confused about independence of events in the following question from Blitzstein and Hwang, chapter 3 (Random Variables): Let X be the number of Heads in 10 fair coin tosses. Find: (a) the ...
matto's user avatar
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Probability of winning a game that ends after one player gets 3 wins in a row

Question: In a game with two people, there is a draw half of the time. The other half of the time, player A wins with 2/3 and player B wins with 1/3. The game ends once a player wins three games in a ...
user2330624's user avatar
1 vote
1 answer
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Gun standoff game I devised for myself (basic probability problem)

I devised a probability problem for myself to help me better understand conditional probability (because I am evidently missing something very crucial). The rules: $k$ players are arranged in a ...
Ooga Booga Beluga's user avatar
3 votes
1 answer
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Probability of an event conditional on union of two events

Let $A,B,C$ be three events such that $A,B$ are independent and $A,C$ are not independent. Now I am wondering if $P(A\mid B\cup C) = P(A\mid C)$? First of all, I think we cannot say $B$ is ...
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Probability of head given that heads already appeared in the first N+M tosses

I have the following problem An honest coin is tossed. What is the conditional probability that it appears head for the first time on the $N$ th toss, knowing that it has come up at least one head ...
daniel's user avatar
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Lower and upper bounds on conditional expected value.

Suppose we have a random variable $X$ which can take negative values. I am wondering if it is possible to find exact or high-probable lower and bounds for $\mathbb{E}(X|X\ge a)$ based on $a$, $var(X)$ ...
Amin's user avatar
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1 answer
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True and False Positives

I am getting confused on what should be a fairly simple concept. A flu test gives true positives 90% of the time, and true negatives 85% of the time. 5% of people have the flu. We want to find the ...
mintteaplease's user avatar
1 vote
1 answer
77 views

A and B flip a fair dice until 6 occurs. What's the expectation of flipping conditioned on A wins?

A and B flip a fair dice until 6 occurs. What's the expectation of flipping conditioned on A wins. My solution is to calculate it by $$\begin{align*}E(6\text{ occurs}) &= E(E(6 \text{ occurs}|A \...
Xu Shan's user avatar
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0 answers
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In a Bayesian network, does the removal of an edge ever remove existing conditional independences?

I am wondering if the removal of any edge in a acyclic Bayesian network ever removes an existing conditional independence? Intuitively, I would think not, but I was wondering if there is a formal ...
ajl123's user avatar
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2 answers
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Optimal Strategy for a Combinatorial Game with Asymmetric Information: Warrior Selection Tournament

Game Setup Bob and Alice are managing a team of warriors. Alice's team consists of warriors with strengths $3, 4, 6, 7$, and Bob has $4$ warriors with strengths $9, 8, 4, 2$ respectively. If a warrior ...
TranscendentalX's user avatar
1 vote
2 answers
55 views

Probability of successfully deciding which of the two six-face dice (Fair and Rigged) was rolled?

Question statement: You are handed a dice which is either fair (all outcomes with probability $\frac{1}{6}$) or produces only the outcomes 2, 4, 6, each with probability $\frac{1}{3}$. If you can roll ...
Jared's user avatar
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Condition on more than one random variables or sigma-algebra

The first question is: Assume we have four random variables $A,B,C,D$, each one is a positive random variable with support $\text{supp}X$ but they are not independent. Given that if $A,B$ are given, ...
jerry's user avatar
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1 answer
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Am I computing conditional expectations correctly?

I am building a working model and would like to know whether my computations so far look correct. I am working with a generally bivariate distribution of two variables $\displaystyle X\sim [ 0,1]$, $\...
Weierstraß Ramirez's user avatar
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1 answer
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Find regular conditional distribution

Question 1.1) Let $X_1 = 1_{A}$ for $A\in \mathbb{F}$ and $X_2$ be a R.V. Where $X_2$ takes value in an arbitrary Borel Space. Let $X_1,X_2$ take values in $(A_1, \mathbb{A}_1)$, $(A_2, \mathbb{A}_2)$ ...
Overkill123's user avatar
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2 answers
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Conditional probability of two continuous random variables

"EDIT: Changed my $k'=k$ to $k'=1$, hope the question does not change alot, due to comments" I have: $X$~$U[0,1]$, as $k$ is just a number, could be $1,2,3$ or even a million. $Y$~$U[0,X]$ $...
Ben Shaines's user avatar
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1 answer
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What is a precise description of the total probability space in the selection problem?

The selection problem tries to mathematically model the following situation (Chapter III in Dynkin-Yushkevich Markov Processes): A bride-to-be is to choose among $n$ suitors who are presented one by ...
Eddie Torres Jr.'s user avatar
3 votes
1 answer
84 views

Splitting $2n + 1$ cards, what is the expected ratio of a size of the smaller deck to the bigger deck?

Assuming equal probability of the deck where the split is. I can find the expected size of the smaller deck of card is $\frac{1+n}{2}$, but I am not sure how the find the expected ratio in a closed ...
Lind G'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
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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....
nickalh's user avatar
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Do I need additional assumptions for this equality to hold?

Suppose I have three random variables $X_1,X_2, V$, and I want the following condition to hold: $E[X_1^2|X_2<V<X_1,X_1,X_2]=E[X_1^2|X_1]=h(X_1)$, i.e., I want conditioning varibles $V$ and $X_2$...
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Conditional Probability of Intersections in the Given

A construction firm purchased 3 tractors from a certain company. At the end of the fifth year, let $E_1$, $E_2$, $E_3$ denote, respectively, the events that tractors no. 1, 2, and 3 are still in good ...
cdohara's user avatar
1 vote
1 answer
25 views

Conditioning reduces entropy, directly from Khintchine--Shannon axioms

Assume there is a function ${\bf H}\{X\}$ on discrete random variables $X$ that satisfies the following axioms: a) (Invariance.) If $X$ takes values in $A$, $Y$ takes values in $B$, $\phi:A\to B$ is a ...
marcelgoh's user avatar
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Expressing the probability of a random vector being in a random set as a conditional expectation

Let $X$ be a random vector in $\mathbb{R}^n$ and let $\mathbb{P}_X$ be its distribution. Let $\{A_y : y \in \mathbb{R}^m\}$ be some collection of Borel subsets of $\mathbb{R}^n$ and let $Y$ be a ...
VKV's user avatar
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1 vote
1 answer
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Conditional Expectation of a product of r.v.

I am exploring a work in probability theory that states $$E[U1U2 | V1, V2] = E[U1| V1]E[ U2 | V2]$$ for independent pairs of random variables $(U1,V1)$ and $(U2,V2)$. I understand the intuition behind ...
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2 answers
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Show that if $\mathbb{P}(A|B) = \mathbb{P}(A |\Omega \setminus B)$ and $ 0\lt \mathbb{P}(A),\mathbb{P}(B) \lt 1$ that A and B are independent events

I for some reason can't solve this and I have issues visualizing solutions that come with stochastical independence. It seems intuitive for me though, that if $\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)...
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