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

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Expected number of draws of a labeled ball knowing it's already been drawn once

Suppose you are drawing labeled balls from an urn with replacement, and you record what you've drawn. You have $n$ balls and draw $m$ times. My question is, for the balls that you've drawn, what is ...
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Calculation of Conditional Expectation

I have problems with the following exercise: Let $\Omega=[-\frac{1}{3},\frac{1}{3}]$, $\mathcal{F}=\mathcal{B}(\Omega)$ the Borel-$\sigma$-algebra on $\Omega$ and P the Lebesgue-measure. ...
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Proof of Conditional Probability

Given a set of data points D and mean u which is continous - How to prove the fact below p(x=1|D) = Integral( p(x=1|u)*p(u|D) du ) given that x can take x=1 ...
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Alternative Monty Hall Problem

So the typical set up for Monty Hall problem, I there are 3 doors where 2 have goats and 1 has a car. I, the contestant, get to randomly guess a door looking to get the one with the car, after this ...
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Meaning of $P(Y|X=x)$

Suppose that $X$ and $Y$ are two random variables on $(\Omega, \mathcal H, P)$ with values in $(\mathbb R,\mathcal B_{\mathbb R})$. I want to understand what is "formally" the expression $P(Y|X=x)$ ...
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Conditional probability and bayes theorem problem involving a medical test

I have a test that checks if a patient is sick (E = {patient is sick}) and gives either a positive (A={result is positive}) or a negative result. Given that $P(A|E) = 0.95 = P(A^c | E^c)$ and $P(E) = ...
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Calculating Probabilities for a cumulative distribution function within a given inequality

Given that K = 1/36, I require some help understanding (b) • Pr(1/2 ≤ X ≤ 1) Is re-written as such: Pr(X ≤ 1) - Pr(X < 1/2) I do not understand why! Is it because Pr(X ≤ 1) is solved as ...
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Calculating Probabilities using a cumulative distribution function

For (b) Pr(X greater than or equal to 2) = ? The textbook says as such but I am confused: Pr(X greater than or equal to 2) = 1 - pr(X less than 2) I do not understand why they re-write the ...
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Conditioning continuous on discrete random variables

In a class hand-out of "review" material (which, in theory, should simply be concepts that we learned last semester in class but in practice contains a lot of stuff that we never covered due to what ...
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Markov property for discrete Markov chains. A question about “adjacent random variables”

Consider a discrete Markov chain (with values in $\mathbb R$) $\{X_n:\, n\in\mathbb N\}$: namely the state space $S$ is a countable subset of $\mathbb R$ and the random variables are $X_0, X_1, ...
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A problem on verify conditional expectation

Suppose X and Y are independent.Let $\varphi $ be a function with $E(|\varphi(X,Y)|)< \infty$ and let $g(x)=E(\varphi(x,Y))$.The conclusion is $E(\varphi(X,Y)|X)=g(X)$ So the first step is to ...
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I'm not sure if I'm supposed to use a Poisson distribution or Conditional Probability (or both) to answer this question

I have a question that I'm trying to solve. I have the answer but I don't know how they arrived at the answer so I can't compare my work and see where I went wrong. The number of injury claims per ...
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Expected Profit for Binomial Variable

Part (a) I am familiar with: (a) P(batch is rejected) = P(X greater than or equal to 3) and n = 15 and p(defective) = 0.1 This gives me the correct answer of 0.1841 I am stuck at part 2! I have ...
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The relation of $P(X=x+1)$ and $P(X=x)$ in binomial distribution

If I substitute the values to the binomial probability theory, it appears as such $${n \choose x+1} p^{x+1} (1-p)^{n-x-1}$$ I don't know how to move on... What am I doing wrong, or are you ...
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Probability in Medical Testing [on hold]

This is a real life question: A medical diagnostic test, Test 1, will test if a certain infection is present. The test will be positive or negative. Whichever the result, the probability that the ...
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Statistic Probability bays theorem [on hold]

Can someone help me with this question A tree diagram will make me understand but i confused as how to draw it
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How to determine long-run probability using conditional probability?

How to determine long-run probability on a calculator and manually? For example: Ben plays a tennis match every day. If he wins on one particular day, the probability that he wins the next day is ...
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Conditional probability that a randomly chosen detail was made by Y, using Bayes's theorem

I found the below question on the internet while working through a conditional probability questionnaire. An automobile plant contracted to buy shock absorbers from two suppliers X and Y. X ...
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The probability of having a disease, conditional on the presence of a symptom [closed]

I have been having some problems with this: Suppose the following facts to be true: $\bullet$ The probability of a random kindergartener having chicken pox at any given time is 2%. $\bullet$ ...
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Random Variable Problem with unrestricted Parameters Worded Problem

I have no idea how to go about solving (a) -> (c) For (a) Is $k=0.2$, because $\frac{k}{1-0.8}=1$ Hence, $P(Z=z) = 0.2(0.8)^x$ But How do we determine the mean or variance with unrestricted z ...
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Expectation of Random Variable - Probability Worded Problem

The part I am confused with is (c) I found part (a) which is: p(0) = 7/24, p(1) = 21/24, p(2) = 7/40 and p(3) = 1/120 How do we find the values for a and b, for part (c) ?
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Expanding the expected value

How to expand: $E(Y+1)^2$ my working out: $E(Y^2)+E(1^2) = E(Y^2)+1$ (I'm not sure why this is though..) Can someone link to or list the rules for expanding the expected value ......
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Finding values of a constant in a probability distribution

A probability distribution for the random variable $X$ is defined by: $$\mathbb{P}[X=x] = K\cdot(0.9)^x,\quad x = 0,1,2,\ldots$$ It is asked to find $\mathbb{P}[X\geq 2]$. When there is a domain for ...
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If the two-engine plane cannot take off unless both engines are operating properly, which plane is safer on takeoff?

I am practicing a bunch of probability problems I find through random sources and I am stuck with this one. Suppose the probability that the engine in a single-engine fighter will fail on take-off is ...
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Probability of getting a certain group of students when choosing three at random out of 25

A teacher randomly chooses a group of three students from her class of 25 students. Find: a) Probability that friends Suri, Lily and Violeta are chosen for the group? b) If he ...
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Conditional probability with a normal distribution

Given that Y and L are normally distributed, the expectation of L given Y is $\mu (Y)$ and the variance of L given Y is $\sigma ^2 (Y)$, why is the conditional probability $P(L > x| Y) = \Phi ...
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Conditional probability for random variables with different distributions

Random variables $X$ and $Y$ are independent, where $X$ is exponentially distributed with parameter $1$ and $Y$ has uniform distribution on $[-1,1]$ interval. Find $\mathbb{P}(Y>0|X+Y>1)$. My ...
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Probability of guessing correctly after x tries with multiple right answers

I'm afraid I'm stuck on something that seems way too simple. I'm trying to calculate the chances of a brute force attack succeeding within a given time period. Simplified problem statement: Say I ...
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How to compute this conditional probability in Bayesian Networks?

I met a problem related to conditional probability from the article "Bayesian Networks without Tears"(download) on page 3. According to the Figure 2, the author says $$P(fo=yes|lo=true, ...
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Simple conditional probability inequality

I'm reading on some branching process theory in Harris' Theory of Branching Processes and encountered an inequality which looks simple but is eluding me. The full version is a bit complicated to ...
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Max of independent and identical random variables is Markov

I'm supposed to show that given a sequence $\{Y_n\}$ of i.i.d the stochastic process $$X_n=\max(Y_0, Y_1...,Y_n)$$ is a Markov of chain. I think I could do it by induction but I would rather see how ...
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Estimating conditional probability

This may be a noobie question, any help is greatly appreciated. Is there a way to estimate $P(X\mid A,B)$ given ONLY an estimate for $P(X\mid A)$ and $P(X\mid B)$? if not, what other information is ...
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$\mathbb E[\mathbb E(X|Y, Z)|Y]$ or $\mathbb E\{\mathbb E[(X|Y)|Z]\}$?

To begin with, the standard iterated law of probability is as follows. $$ \mathbb E X = \mathbb E [\mathbb E(X|Y)]. (1) $$ I am perfectly happy with $(1)$ and there is also some quite good ...
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notation for minimum and maximum?

I'm trying to figure out the correct notation for this situation for use in Machine Learning. I have various ratings (for texts): ...
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Probability Distributions and Random Discrete Variables

How do you read this? For (a) do we let $X= 1/6, 1/2, 1/5$ and $2/15$ and sub into the equation, $$ Y=X^2-2X. $$ How do we go about solving this?
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Conditional Probability using a Matrix

I understand how to find P1: that is simply: P(D1|D0)=0.8 P(W1|D0)=0.2 P(D1|W1)=0.4 P(W1|W0)=0.6 I do not however, understand how to find P2 using the matrix. Normally I would solve it as ...
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How to characterize $F(x \mid Y = y)$ satisfies $\exists ! G(\cdot)( G(x) = \int F( x \mid Y = y) dG(y))$?

$F(x \mid Y = y)$ is the conditional distibution function $P(X \leq x \mid Y =y)$. $G(x)$,$G(y)$ represent $P(X \leq x)$ and $P(Y \leq y)$. Is there a charaterization of $F(x \mid Y = y)$ such that ...
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Proving the Probability of an Event Through Bayes Theorem.

The question goes as such: An event A can occur if only one of the mutually exclusive events B1, B2, or B3 occur. Show that P(A) = P(B1)P(A|B1)+P(B2)(A|B2)+P(B3)*(A|B3) my working out: P[A|(B1 U B2 ...
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Confusion about continuous Bayes

I'm going through Bertsekas' and Tsitsiklis' "Introduction to Probability" and I'm stuck understanding a problem. This is not homework, and I have the solution. But I don't understand it, and I ...
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What's the probability of third coin being of 50 cent?

I was asked this question in an interview for Data Scientist position: I have 1 coin of say 10p,20p and 50p each in my pocket. I then draw out of my pocket the coin of 10p. So now I'm left with 1 coin ...
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Strange conditional probability problem

Not sure if this problem even makes sense, but anyway: Lets say you have a button which switches on a light. The light lights green with probability $p$ and red with probability $1-p$. If you push ...
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Conditional expectation for Poisson process

Let $X(t)$ be a Poisson process with rate $\lambda = 6$ describing arrivals per hour of customers at a bank. Let the probability of a customer being male be $2/3$. Suppose 10 males has arrived during ...
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Die Probability Question + Basics of Conditional Probability

A die is rolled twice. What is the probability of observing: a) a four and a three P (obtaining a four and a three) or P(obtaining a three and a four) therefore P(obtaining a four)* P(obtaining a ...
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Difficult Compounding Probabilities Problem (Though Independent, the Previous Probabilities MUST be considered).

I could really use some help solving this problem. Here is the fact pattern: There are four (4) balls in a bag from which to choose (A, B, C, & D). The picking is 100% random, so the probability ...
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Does interpretation affect the conditional probability distribution function?

Suppose X and Y are independent standard normal random variables. We are finding the conditional probability distribution function (pdf) of Y given Y=X. However, there can be different ...
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Law of total expectation: $\mathbb{E}( X | A )$ and $p_{A|B}(a|b)$

I've got a question regarding the law of total expectation and conditional probabilities. Assuming I know the EV: $\mathbb{E}(X | A)$ and the PDF $p_{A|B}(a|b)$ (A,B are not independet but binomially ...
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Conditional Expectation: How to get $ \mathbb{E}( X | B ) $ from $ \mathbb{E}( X | A ) $ with $C=A+B$

I'd like to calculate some conditional expected values and I'm facing some problems. Here is what I've got: Known is: $ \mathbb{E}( X | A=a ) $ And I'd like to calculate: $\mathbb{E}( X | B=b )$ ...
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Problem regarding Conditional probability

Let $\mathbf{X}$ be an $n-$ dimensional random variable. This variable can be written as $\mathbf{X} = \left[\mathbf{X}_1^T\hspace{5pt}\mathbf{X}_2^T\right]^T$. where, $\mathbf{X}_1$ is $m-$ ...
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Combining classifiers

How can I compute $P$($X$ | $f_0$, $f_1$, $f_2$), given the following pdf's: P($X$|$f_0$), P($X$|$f_1$).P($X$|$f_2$). I am interested in the case of discrete probabilities, but I guess it will be ...
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Conditional probability of someone being a terrorist, based on the output of a prevention system

I've come up with a solution to this book exercise. My answer is different to the one given by the book by exactly one order of magnitude, unfortunately I can't figure out what I'm doing wrong. The ...