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

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Determine the probability distribution of a ratio of two random variables?

Setting You are given two independent random variables $X_0,X_1$ with common exponential density $f(x) = \alpha e^{-\alpha x}$. Let $R = \frac{X_o}{X_1}$. Determine $\Pr[R > t]$ for $t > 0$. I ...
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Conditional expectation of an uniformly distributed random variable

Suppose $U_1, \ldots, U_n$ are i.i.d. random variables with $U_1$ distributed uniformly on the interval $(-1, 1)$. Compute $\mathbb{E}(U_1 + \ldots + U_n |\max(U_1, \ldots, U_n) = t)$ for $t \in (-1, ...
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Conditional distribution of geometric variables

Setting Suppose X1 and X2 are independent with the common geometric distribution w(k; p). Determine the conditional distribution of X1 given that X1 + X2 = n. Solution My argument is $$\Pr[X_1| ...
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Conditional Probabilty [on hold]

Could someone please give a short answer, its been a while. ...
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Definition of conditional probabiliy as function dependent on $\sigma$-Algebra

I know that for events $A,B$ with $P(B) > 0$ the conditional probability is defined as $$ P(A | B) = \frac{P(A \cap B)}{P(B)}. $$ Of course by regarding $A$ as constant, and varying $B$ we get a ...
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Expected value of a Poisson variable conditioned on sum [duplicate]

Setting $$X_1 \overset{d}{\sim} \operatorname{Poisson}(\alpha_1)$$ $$X_2 \overset{d}{\sim} \operatorname{Poisson}(\alpha_2)$$ $$S = X_1 + X_2$$ Find $E[X_1 | S =n]$ My argument is that since $X_1 + ...
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Probability coupon collection question - nth coupon is a new type?

I'm just solving some probability problems in preparation for my exam, and I stumbled upon this one which I cannot tackle: Suppose that you continually collect coupons and that there are $m$ ...
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Why does Average Log Likelihood

The average log likelihood $$L(W,X) = \frac{1}{N}\sum_{1}^{N} log(p(x_n;W))$$ as defined by the authors in http://www.gatsby.ucl.ac.uk/aistats/fullpapers/217.pdf (first equation, first page, right ...
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probability and bernoulii random variable? [closed]

A database file has 6,000,000 (six million) records, which occupy disk storage at a density of 12 records per block. A weekly update modifies 6.5 percent of the file and we assume that the changes ...
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Using Bayes' Theorem,compute the probabilities [closed]

A manufacturing process produces computer chips of which $6$ percent are defective. This percent is actually found using a thorough (and expensive test) on a small random sample of chips. The plant ...
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How to prove Bayes formula with additional conditional(s)?

I am trying to prove a version of Bayes formula which is used in Beyond the Kalman Filter: Particle Filters for Tracking Applications, by Branko Ristic and Sanjeev Arulampalam, page 45-47. ...
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Coin Tossing Conditional Probability

On a practice test with no available solutions I was asked the following two-part question: 1) If a coin is tossed until three consecutive heads are shown, what is the probability that one tail is ...
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Probability question involving conditional probability

A medical patient is diagnosed with a condition that is fatal 60% of the time. One possible treatment involves a surgical procedure. Research has shown that 40% of survivors had surgery and 10% of ...
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Expected value using conditioning with exponential distribution

In a mile race between $A$ and $B$, the time it takes $A$ to complete the mile is an exponential random variable with rate $\lambda_a$ and is independent of the time it takes $B$ to complete the ...
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A question about Poisson process such that…

I got the following problem: Suppose that instances of some event occur in accordance with a Poisson process having a rate of 24 instances an hour Suppose we take a time-interval of length 1 hour ...
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Simplifying Conditional Expectation with two Random Variables

In my introductory probability class I ran across these two expressions in a solution to a homework problem. X and Y are two random variables, and f(Y) is any function. ...
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Probability of exactly one event occurring [duplicate]

Event A has probability of 0.7 and Event B has probability of 0.6. If A and B are independent, what is probability that exactly one occurs?
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Which of the following is always true for A and B

Given that: $ P(A) = 0.5$ $P(B) = 0.7$ $P(A \cap B) = 0.3$ I have to choose one option that is true... However they all seem to be false which means I am possibly making a mistake.. The only option ...
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Conditional Probability in Poker

I'm thinking of a ten person Texas hold'em game. Each person is dealt 2 cards at the start of the game. The question is: GIVEN that you have been dealt 2 hearts (Event B), what is the probability ...
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Mixed Conditioning - Two Normal Distributions

Let $Z \sim \mathcal{N}(0,1)$ and $Y|Z \sim \mathcal{N}(Z, 1)$. Show that $f_{Z|Y}(z|y)$ is a normal density, and find the parameters of this density. What I have so far: \begin{align*} ...
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Conditional expectations of joint normal distribution

$u_1$ and $u_2$ are jointly normal, with zero means, unit variances, covariance $\sigma _{12}$. I know $E(u_1|u_2)=\sigma _{12}u_2$, but why $E(u_1|u_2<c)= \sigma _{12}E(u_2|u_2<c)$ ?
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Comma vs pipe/vertical line in notation for conditional probability

What is the difference between the following expressions: $$P(X_1 < X_2 \mid \min(X_1, X_2) = t) \qquad \text{and}\qquad P(X_1 < X_2, \min(X_1, X_2) = t)$$ For context, I am trying to ...
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Conditional probability intuition

Regarding this question and its highest-voted answer: Conditional probability intuition. So I get the idea of using Venn-Diagrams and limiting our "universe" to a new subset, but what happens if we ...
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Picking balls blindfolded without replacement

Assume we have $n$ red and $n$ green balls in a box. What is the probability that blindfolded, you will pick a red ball on the third pick, if you learn that at least one red ball was picked on the ...
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Finding Joint Probability of a Binomial Tree Model given the stock price , then Conditional Probability

Consider a T-period binomial tree model with stock price $S_{t,n} = S_0u^nd^{t-n}$ at each node $(t,n)$ of the binomial tree for every $n = 0,1,...,t$ and every $t = 0,1,...,T$. a) Let $v,t \in ...
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Conditional expectation for exponential random variables

A man puts his house for sale, and decides to accept the first offer that exceeds the reserve price of $£r$. Let $X_1,X_2,...$ represent the sequence of offers received, and suppose that the $X_i$ are ...
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Conditional Expectation on Bernoulli Variables

From Basic Probability Theory by Ash: Let $R$ be the number of successes in $n$ Bernoulli trials, with probability $p$ of success on a given trial. Find the conditional expectation of $R$ given that ...
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Check my reasoning on three conditional probability problems?

Please remember that I am supposed to use conditional probability!! First problem: S is a set of permutations of (1, 2, 3, 4, 5, and 6). The first term of the permutations cannot be 1. If we ...
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Given that two people were at a location at the same time, find the probability they were there at a particular time

Roger and Stacy each go to the county fair on the same day. They each separately show up at a random time between $12:00$ PM and $9:00$ PM. Roger stays for $2$ hours and Stacy stays for $3$ hours. ...
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conditional probability question - Sanjay the kindergartener with chicken pox

Suppose the following facts to be true: -- The probability of a random kindergartener having chicken pox at any given time is 2%. -- Among kindergarteners who have chicken pox, 75% have red spots. ...
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Proof of Independence of two functions of random variables

If we have two random variables $X$ and $Y$ ($0<x<y$) and we know the conditional density $f(x\mid y) = \frac{3x^2}{y^3}$, how can we show that $Z = \frac{X}{Y}$ and $Y$ are independent? ...
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Probability with qualifications and gender

Qualification Female Male Degree 5 1 None 5 4 School 8 12 Vocation 8 7 I've been going through some ...
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Given the joint pdf $f_{X,Y}(x,y) = 2e^{-(x+y)}$, $0 \leq x \leq y$, $ y\geq 0$. . Find $P(Y < 1| X = 1)$.

Given the joint pdf $f_{X,Y}(x,y) = 2e^{-(x+y)}$, $0 \leq x \leq y$, $ y\geq 0$. . Find $P(Y < 1| X = 1)$. Attempt: $P(Y < 1| X = 1) \frac{P(Y<1, X = 1)}{P(X = 1)}$ Can someone please ...
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Two methods for the Nash equilibrium give different answers; which is correct?

Suppose we have a game, played in which Alice and Bob play mixed strategies: (Sorry about the spacing, but I don't know how to put a table or tab spacing in this text box.) Alice plays Dove with ...
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Basic Conditional Probability Question - Please Check My Working

Messages relating to the status of an industrial system are transmitted to a monitoring station via an internal transmission network. During periods of low network traffic, 1.2% of these messages have ...
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Conditional Probability Question - Please Help

Messages relating to the status of an industrial system are transmitted to a monitoring station via an internal transmission network. During periods of low network traffic, 1.2% of these messages have ...
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How should I approach this Conditional Probability Problem?

Can anyone give a hint on how to begin this problem? Suppose $Y = X^2 + W$ where $W$ is Gaussian $N(0, 1)$ noise. Then derive an expression for $P(Y\mid X)$. I know about Bayes' Rule but I'm not ...
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Gaussian processes versus Bayes rule misinterpretation

I would like to use Gaussian processes (GP) for Bayesian classification of medical data. I think I already understand the basic stuff but I have some uncertainties that are perhaps partly related to ...
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Inverse Probability and conditional probability.

An unbalanced die (with 6 faces, numbered from 1 to 6) is thrown. The probability that the face value is odd is 90% of the probability that the face value is even. The probability of getting any even ...
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Problems relating conditional probability [closed]

We choose two numbers from the set $\{1,2,\dots,100\}$. Say the smaller number is $\leq20$. What is the probability of the second number being $\geq80$?
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Conditional Distribution: how to set up Limit of Integration of a joint density

I have a question in conditional probability. I'm asked to find the conditional distribution, however, I'm unsure about the answer given and would appreciate someone helping straighten out the theory ...
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Support of the conditional distribution of a poisson process

I am working on Problem 5.1.8 of this book. It states: Let $\left\{X(t),t \geq 0 \right\}$ be a Poisson process of rate $\lambda$. For $s,t >0$, determine the conditional distribution of ...
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Properties of Identically Distributed RVs.

I've a little doubt in part (iii) of the question posted above First I wrote the PMF of Z \begin{vmatrix} Z = X+Y & -2 & -1& 0 & 1 & 2\\ P(Z=z) & .09 & 0.24 & 0.34 ...
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Conditional Expectation and Almost Sure Convergence

Say we have $Y \in L^2(\Omega,\mathcal{A},P)$ and that $E(Y|X) = X, E(Y^2|X) = X^2$. Then show that $Y=X$ a.s My approach: Define $\mathcal{C} = \{\omega : X(\omega) = Y(\omega)\}$. Then $Y = ...
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Approximating the Probability of a Hypothesis Given Approximated Conditional Probabilities

I am working on deduping entities as well as problems involving expert system like decision making. I find that I often need to combine approximated probabilities. For instance when handling ...
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How do I work out the odds of choosing five matching pairs from a selection of ten items?

Here's the example. I have 10 boxes consisting of 5 matching pairs of items (red gloves, red socks, blue socks, gold earrings, diamond earrings). What are the odds of choosing two boxes with the ...
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what is the probability that the real estate agent can get into specific home ???

A real estate agent has 8 master keys to open several new home. Only 1 master key will open any given house. If 40% of the homes are usually left unlocked what is the probability that the real estate ...
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How many different ways can a student check off one answer to each question?

If a multiple-choice test consists of 6 questions each with 4 possible answers of which only 1 is correct, In how many different ways can a student check off one answer to each question ?
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Find the probability $P(A^c \cap B^c)$ when only given $P(A), P(B)$.

Given the following question: Given 2 events $A$, $B$ where: $P(A) = 0.4$, $P(B^c) = 0.7$, $P((A \cup B)^c) = 0.3$ Does $A$ and $B$ dependent each other? I infered that $P(B) =1 - ...