Questions tagged [conditional-probability]

In probability, conditional probability is the probability that an event occurs given another (conditioning) event occurred.

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19 views

If you roll a fair 6-sided die and then flip a fair coin that number of times, what is the probability that you will get at least two heads?

My idea is to use disjoint events and calculating the probability of getting at least two heads for each number rolled. For example, if I roll a 3, I would calculate the probability with the ...
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Finding a set of independent Bernoulli random variables of the same joint distribution of non-i.i.d. Bernoulli random variables.

Let $\mathbb{P}^n$ be the joint distribution of non-i.i.d. Bernoulli random variables $X_1, \dots,X_n$ such that $X_i\in\{0,1\},\ \forall i$. Is it possible to find independent Bernoulli random ...
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Flip a fair coin 5 times. $A$ is the event that the first flip is heads, and $B$ is the event that you get at least three heads. What is $P(A|B)$?

I'm trying to use the conditional probability formula, $P(A|B)= \frac{P(A \cap B)}{P(B)}$. I've determined that $P(A)=\frac{1}{2}$ and $P(B)=\frac{1}{2^5}(\binom{5}{3} + \binom{5}{4} + \binom{5}{5}) = ...
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Relationship between circular real ensemble and Gaussian orthogonal ensemble?

Suppose I draw from the Gaussian orthogonal ensemble an $N\times N$ matrix $\mathbf{H}$ with eigenvalues $\lambda_1, \ldots, \lambda_N$. If I condition these $\mathbf{H}$'s so that $\lambda_1 \in [L_1 ...
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Finding conditional distribution in matching ordering situation

Suppose we draw two values $x_1,x_2$ according to a CDF $F$. Independently, we draw another two values $y_1,y_2$ according to another CDF $G$. Both $F$ and $G$ has support $[0,1]$. Among those four ...
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The stock of a warehouse consists of boxes of high, medium and low quality light bulbs in respective proportions 1:2:2.

The stock of a warehouse consists of boxes of high, medium and low quality light bulbs in respective proportions $1:2:2$. The probabilities of bulbs of three types being unsatisfactory are $0.3, 0.1$ ...
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If $X_n$ is an $L_1$-bounded martingale, show that $\sum_n(X_n-X_{n-1})^2< \infty$ a.s.

I had asked this question previously in the following post but there were no replies. Recently, I found a two page article(free download) with a possible alternate line of proof to the one suggested ...
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A couple has 2 children. What is the probability that the couple has one boy and one girl given that they have at least one girl? [duplicate]

This is what I have so far: All possible combinations: BB GG BG GB $P$(Boy $\mid$ Girl)$= 1/2$ given that the sample space is reduced to GG and GB However, I am not sure if this is the right way ...
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Birthday $pmf$ and joint $pmf$ for $n$ people

Let a discrete r.v be denoted $X_1,..,X_n$ denote the birthdays of $n$ people in a room. Assume that $X_1,..,X_n$ are mutually independent and that $X_i$ is a distribution such that $X_i \sim U(\left \...
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I understand the intuition of the Monty hall problem, I do not understand the math

Take the famous Monty Hall problem. There are three doors, one has a prize. If you pick the door with a goat, he reveals the other door with the goat. If you picks the door with the car, he randomizes ...
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conditional expectation with updated information

Let $\epsilon_{t+1} = \rho\epsilon_t + \eta_{t+1}$ $E_t[r_{t+k}|\eta_t] = \phi^k \eta_t$ Can we say that $$E_t[r_{t+1}] = \sum_{k=0}^\infty \phi^k \eta_{t-k}$$ ?? Are there any conditions for this to ...
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Using Martingales to find expected values.

Suppose we have i.i.d random variables $X_1,X_2,...$ s.t $P(X_i=1)=\frac{1}{2}=P(X_i=0)$. Let $$\Omega=\inf\{n\geq 5|(X_{n-4},X_{n-3},X_{n-2},X_{n-1},X_{n})=(1,0,1,0,1)\}.$$ I would like to use ...
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Probability Problem on Gates in Subway station

A subway station in a metro city has 10 gates, five for entering into the subway station, and five for exiting the subway station. The number of gates observed in each direction is observed at a ...
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Memoryless of Geometric Distribution [closed]

Hi I am really stuck with this question. I have no idea where to even start Consider $(\mathbb N, \mathscr P(\mathbb N),\mathbf P)$, where $\mathbf P=\operatorname{Geo}(p)$ is the geometric ...
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Why is $\frac{Z_n}{\mu^n}$ a martingale? (Galton-Watson Process.)

The following images taken from Durrett Pg 200 explain what a Galton Watson Process is and its corresponding martingale $\frac{Z_n}{\mu^n}$. However, I don't see why $\frac{Z_n}{\mu^n}$ is a ...
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Variance of Sum of N Discrete Uniformly Distributed Random Variables. [duplicate]

A box contains $N$ identical balls numbered $1$ through $N$. Of these balls, $n$ are drawn at a time. Let $X_1,X_2,...,X_n$ denote the numbers on the $n$ balls drawn. Let $S_n = \sum_{i=1}^{n}X_n$. ...
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Independence of functions of independent random variables.

Suppose $X_1,..,X_n$ are independent random variables and $X_i'$ is an independent copy of $X_i$, then how does one show that $$E[f(X_1,..,X_n)|X_1,..,X_{i-1},X_{i+1},..,X_n]$$ and $$E[f(X_1,..,X_i',....
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Machine Learning: Incomplete, Positive-Only Dataset

Reading through Lemma 1's proof on page 214 of Learning Classifiers from Only Positive and Unlabeled Data research paper. Relevant information: consider the scenario for training a binary classifier ...
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Efron-Stein Inequality proof clarification.

I am currently going through the proof of the Efron-Stein inequality in this set of notes (http://www.econ.upf.edu/~lugosi/mlss_conc.pdf)(P.g. 219-220). However, I have an issue with the final part of ...
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If $(X_n)_{n\in \mathbb{N}}$ is a martingale s.t. $\sup_n E[|X_n|]\leq M < \infty$, then $\sum_{n\geq 2}(X_n-X_{n-1})^2<\infty$ almost surely.

I need to prove the above statement. The hint provided was to consider the stopping time $T_l=\inf\{n\in \mathbb{N}||X_n(w)|\geq l\}$ where $l\in \mathbb{N}$ and then show that $E[\sum_{n=2}^K(X_n-X_{...
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Three bowls with balls problem [closed]

Consider 3 bowls. Bowl A contains $2$ blue balls and $4$ red balls. Bowl B contains 8 blue balls and $4$ red balls. Bowl C contains $1$ blue ball and $3$ red balls. What is the probability to draw a ...
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Conditional expectation of one random variable

Let $X$ be a continuous random variable whose probability density function is $$ f(x)= \left\{ \begin{array}{lcc} \alpha^2xe^{-\alpha x} & if & x > 0 \\ \\ 0&...
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Why does the second equality hold?

A sequence of n independent experiments is performed. Each experiment is a success with probability p and a failure with probability q = 1 - p. Show that conditional on the number of successes, all ...
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Are the events $S>s$ and $T=1$ are independent [duplicate]

$$X,Y\sim \exp(\lambda)$$ $$S=\min(X,Y)$$ $$T=\mathbf1_{X\le Y}=\begin{Bmatrix} 1 & \text{ if } X\le Y \\ 0 & \text{ if } X\gt Y \end{Bmatrix}$$ I have that $P(S>s)=e^{-2\lambda s}$ (so $S\...
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Is there such thing as “homogeneity” for conditional probabilities?

I noticed, several times, that there seems to be some “homogeneity principle” (like in physics) when computing conditional probabilities. It seems that most of the time, formula implying sums of ...
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1answer
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Conditional Binomial Probability

Haven't looked at statistics in a while, and came across the following problem: It is estimated that approximately 20% of marketing calls result in a sale. Assume that 10 out of 12 marketing calls on ...
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Confusion over conditional probabilities

The Question A company is considering a marketing campaign for a product which currently is purchased by $4\%$ of people. If the company decides to run a newspaper campaign then based on previous ...
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Probability of picking $z-n$ uniquely colored balls successively and $n$ repeats without replacement

A follow up on this post. In a bucket, there are $x$ different colors of balls, $y$ of each color ($x \cdot y$ total balls). If you pick $z$ balls at random without replacement, what is the ...
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Derivative of conditional expectation of integral of stochastic process

Let $T>0$. Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathbb{F}=(\mathcal{F}_u)_{u\in[0;T]}$ be a filtration such that $\mathcal{F}_T=\mathcal{F}$. Let $(\alpha_u)_{u\...
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Find the probability that exactly $2$ urns remain empty, given that not every box receives a ball.

$10$ indistinguishable balls are placed in $5$ distinguishable urns. Find the probability that exactly $2$ urns remain empty, given that not every box receives a ball. So, this is a conditional ...
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If a policyholder filed a claim, what is the probability that it was not a low claim?

An insurance company classifies its incoming claims as low if they under $10,000$ USD, medium if they are $10,000$ USD or above but under $20,000$ USD, and high otherwise. During the year, $79.2$% of ...
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how to Prove P(A/B∩C) = P(A/B).? [closed]

Given P(A∩B∩C) ≠ 0 and P(C/A∩B) = P(C/B); prove P(A/B∩C) = P(A/B) this is all the information that I have :C, I´m stuck, thank you in advance!
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Cluedo probabilities as information progresses

I am trying to calculate the probability that each card is inside the envelope as the information revealed progresses during the game. The deck is composed of 21 cards, which are divided into three ...
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Finding pmf by using law of total probability

A television quiz game operates as follows. In the first part of the game, a contestant is asked a series of difficult questions; the probability of answering any question correctly is $p$, ...
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Prove that $P(A|B) = P(A|B,C)P(C) + P(A|B,\overline C)P(\overline C)$

Let $(Ω,S,P)$ be a probability space. Let $A, B, C ∈ S$ with $P(B)$ and $P(C) > 0$. If $B$ and $C$ are independent show that $$P(A|B) = P(A|B,C)P(C) + P(A|B,\overline C)P(\overline C)$$ $\textbf{My ...
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What is the contrafactual distribution of $\mathsf{P}(F'|do(S'=s'))$?

$F'$ and $S'$ are from a counterfactual scenario, where $S'$ is conditioned to $do(S'=s')$, and we know that in the factual scenario $F$ took the value $f$ when $S$ was conditioned as $do(S=s)$. My ...
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Independence of two intertwined random variables

Let $X_0$, $X_1$ and $X_2$ be three mutually independent random variables. We define two more random variables $D_1$ and $D_2$ as follows: $$D_1 = X_1 + X_0\\[1ex] D_2 = X_2 + X_0$$ We're interested ...
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Probability of car starting first time

The Question One a given day, Claude's car has an 80% chance to starting first time and Andre's car has a 70% chance of the same, Given that one of the cars has started first time, what is the chance ...
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Two fair dice roll. Lower score discarded. In case of tie, either die is discarded. Compute the mean value of the remaining number

"Two fair dice are thrown and the one with a lower score is discarded. In case of a tie, either one of them is discarded. Compute the mean value of the remaining number." I know how to ...
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Probability of 4 drivers in the same group

Yesterday I asked a question that a colleague quickly answered @mjw but some things were not clear and the question was closed. I will reformulate it and count on everyone's support again. The problem ...
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1answer
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Why do we need to calculate this as a conditional probability?

I am trying to solve this problem: A motorist just had an accident. The accident is minor with probability $0.75$ and is otherwise major. Let $b$ be a positive constant. If the accident is minor, ...
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1answer
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E[E[A|B]|C], law of total expectation with two conditional expectations

I have a conditional expectation inside of a conditional expectation. Let A, B, C be real-valued random variables. I am trying to simplify E[E[A|B]|C]. I am not sure much can be done in general; ...
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If I roll 3 die at the same time and know that two of them are 3's, what is the probability the 3rd is also a 3?

This is not a homework problem, I'm just trying to better my understanding of this concept because it's so interesting yet counterintuitive to me and. I'm hoping someone will confirm that this is ...
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Conditional probability lie detector

A lie detector returns a positive result when a person is lying in 90% of all cases. Unfortunately, it also results a positive result when a person is telling the truth in 20% of all cases. ...
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A conditional probability problem where the next day depends on the last 3 days

For many years, Meteorologists have spent long visits (5 days) at the Bigtown. They have observed that, for three consecutive days, if there are EXACTLY two sunny days, the next day is a sunny day*, ...
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Conditional Probability using an urn.

An urn contains $5$ white balls and $6$ black balls. $3$ balls are drawn sequentially without replacement. What are the probabilities... a. $P(\text{2nd is black} | \text{1st is white})$? For this one ...
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Conditional Probability Distribution of Max$(X, Y)$ given $X$

Say $X$ and $Y$ are I.I.D uniform variables in $[0,1]$. Conditional Probability Distribution of Max$(X, Y)$ given $X$. I think it would be simply $P(X)$ when $X>=Y$ and $P(Y)$ the other way around. ...
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If $W_1,W_2$ are independent and $P(W_1)=P(W_2)$, does $P(W_2|W_1)=P(W_1)=P(W_2)$?

I know this problem is on Math SE, but my question has nothing to do with the mathematics itself but rather the intuition/"feel" for the problem. Specifically, my question is on part (b) and ...
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Conditional probability based on signals with common underlying distribution

I have this problem I am not sure how to approach: There is a state 𝜃 ∈ {a, b}, Where a,b are real numbers and equally likely. There are 2 signals s1 and s2. s1 ∈ {a, b}, there is another signal x1 $...
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1answer
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Writing Probability distribution in terms of a trace over a density matrix

I have been given and expression for a probability distribution precisely, $P(x,y,z)= \sum_\lambda P(x|y,\lambda)P(y|\lambda,z)P(z)P(\lambda)$ and I have been asked to show that the above expression ...

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