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Questions tagged [conditional-probability]

In probability, conditional probability, is the probability that an event occurs given something else has already occurred.

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Conditional distribution of process $W$ given $\{W_1 = y\}$ is Gaussian.

Suppose that $X=(X_t)_{t \in [0,1]}$ is a continuous Gaussian process, for which $\mathbb{E}(X_t) = 0$ for all $ t \in [0,1]$ and $Cov(X_s,X_t) = s(1-t)$ for all $0 \leq s \leq t \leq 1 $. Let $Y \sim ...
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Conditional expectation and variance for beta RV

A random variable X has the Beta($\alpha$,$\alpha$+ 1) distribution, with $\alpha$ > 0. The parameter $\alpha$ itself is random, and can take the values 1 or 2, with probabilities 1/2 each. Compute E(...
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Conditional expectation of geometric RV

Let X be a geometric random variable whose probability for success is itself random, and has the standard uniform distribution. Compute the pmf of X.
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Marbles Probability question using Bayes Rule

Marbles Probability question using Bayes Rule I have this question from my revision booklet that I cant understand. A bag initially contained 4 white marbles and 9 black marbles. A marble was drawn ...
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Probability of chossing two points from a segment of length L so that one is 2L/3 greater than another.

Two points are selected randomly on a line of length $L$ so as to be on the opposite sides of the midpoint of the line. In other words, two points X and Y are independent random variables such that X ...
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Existence of a joint distribution given the conditional and marginal distribution

Can anyone point me a book where it has a proof of Theorem 1.7 (ii) of Jun Shao's book - Mathematical Statistics? I need this to show that given a distribution on one space and a collection of ...
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Conditional Probability - Independence of result from the number of balls we have in the container?

I saw this question in a textbook and the answer it has, seems pretty weird to me: Suppose we have 12 balls in a container, of which 8 are white and the 4 others are black. We choose 4 of the balls ...
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Find the minimum and maximum possible values of the conditional probability

Given two events $A$ and $B$, such that $P(A) = 0.3$ and $P(A ∩ B) = 0.1$. Find the minimum and maximum possible values of the conditional probability $P(A | B)$.
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What's the probability that the ball (taken from the first box and dropped in the second box) was white?

I have one pretty tricky question: There are two boxes. Each one has 7 balls: 3 white and 4 black. One ball was taken from the first box and dropped in the second one. After that, ball was ...
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One coin chosen between a biased coin and a fair coin, and is tossed n times. Find probability of having gotten the biased coin.

In a different question, I had asked for clarification on the following problem where I wanted to just understand the problem. Now, I have attempted it and wish to know if my solution is right. ...
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$(X|Y=y)\sim N(y,y^2)$, $Y\sim U[3,9]$. Find $Var(X).$

$(X|Y=y)\sim N(y,y^2)$, $Y\sim U[3,9]$, where $N(y,y^2)$ is a normal distribution with mean $y$ and variance $y^2$, $U[3,9]$ is a uniform distribution on $[3,9].$ On this condition, find $Var(X).$ ...
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Condition Expectation of normal variables

Let X,Y be jointly normal. Then I know that $E(X|Y)=E(X)+\frac{Cov(X,Y)}{V(Y)}(Y-E(Y))$. Do I need joint normality for this result? Does it also hold, if just X is normal and Y is normal?
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One of two coins chosen, tossed n times

I am looking at the following problem about a coin toss experiment. I cannot understand the statements in the problem. Problem statement is given below. A drawer contains two coins. One is an ...
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Probability high school math competition problem

Team A and Team B are playing basketball. Team A starts with the ball, and the ball alternates between the two teams. When a team has the ball, they have a 50% chance of scoring 1 point. Regardless of ...
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binary variables : bound on probability of $A=B$ given respective correlation with $C$

Let $(A,B,C)$ be three binary random variables, i.e., $(A,B,C)\in \{0,1\}^3$ Suppose pairwise correlations between $(A,C)$ and $(B,C)$ are respectively given by the following odds ratios $$ \frac{P[A=...
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If $E(Y\mid X)=a+bX$, show that $b =\frac{\mathrm{Cov}(X,Y)}{\mathrm{Var}(X)}$ without assuming $(X,Y)$ has bivariate normal distribution?

If $E(Y\mid X)=a+bX$, show that $b =\frac{\mathrm{Cov}(X,Y)}{\mathrm{Var}(X)}$ where $a$ and $b$ are constants. This question was asked before: Proving $a$ (in $Y = aX + b + e$) satisfies $a = Cov(X,...
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Conditonal probability question concerning vampires. Genetic variation.

For any individual x born in Transylvania with a vampire father, there is a 50% chance that x is a vampire, independently for each birth. These are the only conditions under which a new vampire can be ...
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Conditional probability and expected value calculation

A football team, LIBO, wins a match with probability 0.75 irrespective of its opponents. What is the probability that the team wins 4 matches out of 5 matches? In a knockout tournament, LIBO ...
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Conditional probability on joint pdf

The joint pdf of $X$ and $Y$ is given by: $$f(x,y) = \frac{6}{7}\left(x^2+\frac{xy}{2}\right),\quad 0 < x < 1,\quad 0 < y < 2$$ Find $\;\displaystyle P \left( \left. X > \...
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Probability of winning a Craps game

A Craps game consists of throwing 2 dices. If the sum is either 7 or 11, you win. Else, if the sum is either 2,3 or 12, then you loose. If the sum is either 4, 5, 6, 8, 9 or 10, then let's call the ...
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Conditional probability on Sheldon Ross Exercise 48

Sixty percent of the families in a certain community own their own car, thirty percent own their own home, and twenty percent own both their own car and their own home. If a family is randomly ...
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Condtional probability problem for machine learning

I found this problem in a book (relates to ML, $F_1$ and $F_2$ are supposed to be features and $C$ is supposed to be label) Given: $P(F_1=T | C=T) = 0.8$ $P(F_1=T | C=F) = 0.9$ $P(F_2=T | C=T) = 0.4$...
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Intersection of Probabilities and Bayes Theorem

It is known that 70% of women and 60% of men have voted in a poll, in a village where 500 women and 400 men live. If only 80% of the inhabitants tells the truth, what is the probability that a person ...
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Write the expression for $\mathbb{E}(Y|X)$

I have a confusion regarding how to go about solving the following question: Suppose you are invited to play a game where your earnings are given by multiplying the outcome of rolling a fair die ($Z$)...
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The efficacy of mumps vaccine is about 80% that is 80% of those recieving the mumps vaccine will not contract the disease when exposed.

The efficacy of mumps vaccine is about 80% that is 80% of those recieving the mumps vaccine will not contract the disease when exposed. Assume that each person's response to the mumps is independent ...
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“Joined” conditional probabilities

Given $$ P(H|F_1) = 1.1 \cdot P(H|\overline{F_1}),$$ $$ P(H|F_2) = 1.1 \cdot P(H|\overline{F_2}).$$ Determine $k$ in $$ P(H|F_1 \cap F_2) = k \cdot P(H|\overline{F_1} \cap \overline{F_2}).$$ If ...
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Can anyone help me calculate the probabilities of some UEFA champions league draws?

So I've always been highly suspicious of some of the draws in the champions league in the past & I'd love to calculate the probabilities but I can't do it myself. Can anyone help me? Specifically ...
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Conditional Probability Negation

I have the probabilities of event A, event B, and event ( A ∩ B ). The conditional probability equation allows me to find the probability of event A given event B, and the probability of event B given ...
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Conditional Probability and Marbles in different bags

There is a white marble in one bag and a red marble in another bag. You have a white marble in your hand and put it in one of the bags. You draw a white marble from one of the bags. What is the ...
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The face cards are removed from a pack of 52 cards then 4 cards are drawn one by one from the remaining 40 cards

The face cards are removed from a pack of $52$ cards then $4$ cards are drawn one by one from the remaining $40$ cards what is the probability that four cards belong to different suits and different ...
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Basic Question on Conditional Expectation (conditioning with different sub-$\sigma$-algebras)

I have a very basic question concerning conditional expectation. Let $(\Omega, \Sigma, p)$ be a (finite) probability space and let $\Sigma_k$ be a sub-$\sigma$-algebra (with $k \in K := \{1,2\}$), ...
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P(X+Y<a | X<a) for X, Y normal and independent distributions

let $a\in \mathbb{R}$ and let X and Y be independent normally distributed random variables, with mean $0$ and respective variances $\sigma^2_X$ and $\sigma^2_Y$. Can we express $$P(X+Y<a\,|\,X<a)...
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Retrieving a matrix from its Schur complement

I came across a problem that pertains to Schur complements and Gaussian conditional law. Consider $x \sim \mathcal{N}(0, \Sigma_{xx})$ an $n$-variate Gaussian random variable and an independent (of $x$...
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Probability of two events with mutually exclusivity among them adding to 1

Okay, I thought I had this, but it seems I have no idea. We have two events, A and B. And each event has 3 "states". We will call these states 1,2,3. And we have the following rules: For each event, ...
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Existence of something like regular conditional distribution

This exercise is from Durrett's Probability: Theory and Examples, Exercise 5.1.16. Suppose $X$ and $Y$ take values in a nice space $(S,\mathcal{S})$ and $\mathcal{G}=\sigma(Y).$ Then prove that there ...
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difference in conditionnal probabilities in markov chain

I am studying Markov chain and i want to know the expression of $P(X_{n+1} =j)$ but i don't understand the difference between : $\sum_{i} P(X_{n+1} =j | X_{n} = i). P( X_{n} = i)$ and $\sum_{i} P(X_{...
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Finding entries of a joint distribution table.

A random variable $X$ takes values in $\{0, 1, 2\}$, while another random variable $Y$ takes values in $\{−1, 1\}$. Furthermore, $P(X = 1) = P(X = 2) = \frac{1}{3} = \frac{3}{4} \cdot P(Y = 1)$...
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Finding conditional expectation for 3 tosses of a coin.

We toss a symmetric coin three times. Let $X$ denote the number of tails in the first two tosses, and $Y$ the number of tails in the last two tosses. Find the conditional distribution of $X$ ...
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Coin flipping and Bayes' theorem… but where does binomial theorem come in?

Consider the following question: You are face with two identical coins. One is fair, and the other comes up Heads 90% of the time. You flip coins, which results in THHHTHHHTH (seven heads, ...
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Conditional problem with Bernoulli variables

Let $S_{n}=\sum_{k=1}^{n}X_{k}$ and $T_{n}=\sum_{k=1}^{n}Y_{k}X_{k}$ with all $X_{k}$ and $ Y_{k}$ are mutually independent and of law Bernoulli respectively of parameters p and q. Let $N=inf\{n>0 ...
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conditional probability and Bayes' rule

What's the between $P(x|y)P(y|z)$ and $P(x|y,z)P(y|z)$, it seems to me they both equal to $P(x,y|z)$. Does any condition should be satisfied if they both equal to $P(x,y|z)$.
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Is there a more general form of the conditional probability?

Say you have $P(A_1, A_2, \dots, A_n | B_1, B_2, \dots, B_m)$. Is there a general way to break this down into combinations of $P(X|Y)$'s and $P(Z)$'s? I understand that $P(X|Y) = \frac{P(X,Y)}{P(Y)}$...
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effect of knowing birthday of one person and the gender on probability of the second gender [duplicate]

Consider there are two applicants for a job. One of the applicants is a man, and it is known that he was born on a Wednesday. What is the probability that the second applicant is also a man? (I am ...
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Computing probabilities of standard Brownian motion

Let $W_t$ be a standard Brownian motion. a. Find $P(-3\leq2W_2-3W_5\leq5)$ b. Find the variance of $W_2-3W_3+2W_5$ c. Find $P(-2\leq W_2\leq 3\mid W_1=1)$ d. Find $P(-2 \leq W_3-...
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How to interpret the proof that information cascades will form?

I am reading the 1992 paper of Bikchandani, Hirshleifer and Welch on information cascades. They claim and prove that, given an environment of sequential decision making, an information cascade will ...
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$P(Y \le X)=\int_0^{\infty} P(Y \le X | X=x)f_X(x)dx$

I was looking at a solution of a probability exercise and the author of the solution uses the formula $$P(Y \le X)=\int_0^{\infty} P(Y \le X | X=x)f_X(x)dx$$ where $X$, $Y$ are the random variables $...
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Let $B \subseteq \biguplus^{\infty}_{n=1} A_n$, show that $\mathbb{P}(B)=\sum^{\infty}_{n=1} \mathbb{P}(A_n) \mathbb{P}(B|A_n)$

Question: Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space with events $A,B\in\mathcal{A}$. Now, let $B \subseteq \biguplus^{\infty}_{n=1} A_n$, where $A_n \in \mathcal{A}$ for ...
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Minimum Independence Assumptions needed for statement to be true?

I am a bit confused on conditional independence: Given the following independence assumptions: 𝑋⫫𝑌, 𝑋⫫𝑍, 𝑌⫫𝑍, 𝑋⫫𝑌|𝑍, 𝑋⫫𝑍|𝑌,𝑌⫫𝑍|𝑋 What is the minimal set of independence assumptions ...
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Show that given n coins, if the probability of getting heads an even number of times is 1/2 then there is at least one fair coin

So the set up is as follows: We have n coins being flipped independently, not necessarily all fair. I know that if there is at least one fair coin then the probability of getting an even number of ...