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

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How to find the conditional PDF of an order statistic

I suppose that $X_1,\cdots,X_N$ are $N$ i.i.d. variates, each with PDF $f(x)$, $Y_1,\cdots,Y_N$ are also $N$ i.i.d. variates, each with PDF $f(y)$. Further, $X_i,Y_j,\forall i,j$ are independent. ...
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How do I combine probability estimates of two equivalent/mutually inclusive events?

Let's say I'm pregnant with fraternal twins. One of them hangs out in the left side of my womb, and the other hangs out on the right (I have no idea how biology works). We've applied a flaky test to ...
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Help proving $Pr(\mathcal{X})= \phi_1(X,Z)\phi_2(Y,Z)$ if $ P \models (X \perp Y | Z)$ and $\mathcal{X}=X \cup Y \cup Z$

I was trying to prove the following: if $X,Y,Z$ were three disjoint subsets of variables such that $\mathcal{X}=X \cup Y \cup Z$, Prove that $ P \models (X \perp Y \mid Z)$ if and only if we can ...
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How to solve disjoint probability problem?

Suppose the two events “high” and “low” make a disjoint partition of a sample space and “favourable” is any event. If P(high) = 0.3, P(low) = 0.7, P(favourable| high) = 0.9 and P(unfavorable| low) = ...
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(What is the formula to find) What is the probability that the sum of the numbers on the tickets chosen is at least 7?

Senario: Box A contains four equal-sized tickets, numbered 1, 2, 3 ,4 Box B contains three tickets of the same size, numbered 4, 5, 6 An experiment consists of selecting one ticket from the box A ...
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Help with conditional expectation

I need help finding a conditional expectation: Let $X$ be a $(0,1)$ uniform random variable i.e. $\mathbb{P}(X \in A)=\lambda((0,1)\cap A)$ where $\lambda$ is the Lebuesgue measure. We define the ...
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Exponential of Squared Brownian Motion

Long time lurker, first time posting! Have a problem, that looks familiar but I can't put my finger on it. Need to calculate $\mathbb{E} [\exp(aW_T^2)|F_t]$ where $W_t$ is an $F_t$ adapted standard ...
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Conditional random variables

How to generate three random variable $a, b, c$ of some distributions that satisfy: $$a \in [0, 0.5]$$ $$b \in [0, 0.4]$$ $$c \in [0, 0.3]$$ $$a+b+c = 1$$
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Calculating conditional lottery probabilities - and example from DeGroot.

I'm trying to understand this example in Probability and Statistics in DeGroot: http://imgur.com/yjr1vLQ "You learned that the event B = {one of the numbers drawn is 15} has occurred. You want to ...
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36 views

Expected value of the floor function of a sum of two variables

In a recently published paper I have encountered the following equality. Let $U$ be a random variable uniformly distributed in $[0,1]$ and let $Z$ be a Gaussian variable with mean zero and standard ...
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Easier way to solve conditional probability question?

Two digits are chosen at random from a table of random numbers containing the digits 0,1,2,...,9. Find the probability that the second number is 2, given that the sum is of the two numbers is greater ...
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Conditional expectation of the sum of two random variables

I've got some difficulties in calculating the conditional expectation of the sum of two RV. I am not sure if I correctly formalized the scenario I am looking at. So I am trying to describe it first: ...
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Properties of conditional expectation

In the probability book of Bauer it is claimed that for nonegative X and Y, or integrable X and Y we have $$ (1) \quad X=Y \, a.s. \Rightarrow \mathbb{E}(X \mid \mathcal{A}) = \mathbb{E}(Y \mid ...
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Question on an example that $\Bbb E{(X\mid Y=y)}$ always exists, but $\Bbb E(X)$ doesn't

On page 18 of Conditional Measures and Applications(2005) by M.M.Rao, there is an example that $\Bbb E(X^n\mid Y=y)$ always exists, but $\Bbb E(X^n)$ doesn't. It is mentioned that $\Bbb E(X^n\mid ...
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Probability of heads given we observe HTH?

I am observing a sequence of heads and tails and trying to deduce the bias of my coin. Let's say I observe HTH. Can I estimate the bias of my coin $p$ ? ...
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Probability that exactly 2 of 3 objects are in 1 of 3 baskets with sizes 5, 8, 2

I want to calculate the probability that some mutation occurs on a certain DNA section by a given number mutations. I rephrased it into this problem: Three (identical) persons enter a train (section ...
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What is the “average time remaining” when guessing a random value?

I lack the terminology to ask this question "properly", so to illustrate what I'm bouncing around in my mind, let's take a story example: John wrote a script which guesses passwords. He has a list ...
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Gambling interpretation of conditional probability

In Billingsley, when defining conditional probability the following property has been given a gambling interpretation : $$ \int_G P[A||\mathscr{G}]dP = P(A \cap G), G \in \mathscr{G} $$ where at ...
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Multivariate normal conditioned on sum of squares

Suppose that $X_i$ are i.i.d. N(0,1) random variables, and set $S = \sum_{i=1}^n X_i^2$. Then $S \sim \chi^2_{(n)}$, the $\chi^2$ distribution with $n$ degrees of freedom. Compute the induced ...
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How is conditional probability being used here?

Because of conditional probability: $P(A\mid B)=P(A,B)/P(B)$, $$P(C(t)\in dt\mid x(T^+_{i-1}),x(T^-_{i}))=\dfrac{P(C(t)\in dt,x(T^-_{i})\in dx\mid x(T^+_{i-1}))}{P(x(T^-_{i})\in dx\mid ...
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Probability of socks with different colors [closed]

A drawer contains $7$ different pairs of socks. You have to pick $4$ singles. What is the probability they are all different colors.
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Real-world problem: Behavior of a conditional probability

Apologies for the rather vague phrasing of the title; I'm trying to formulate this problem as exactly as I can and perhaps not succeeding very well. The problem is inspired by a real-world situation: ...
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Joint probability of two conditional probabilities [closed]

I have calculated two conditional probabilities p(A|B) and p(B|A). I would like to know p(A|B ∩ B|A). My initial hunch was that these two probabilities could be treated as independent events, meaning ...
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Conditional Probability - Solving for Unknown

Given: P(A) = 0.3 and P(A | B) = 0.1 Desired: Value of P(B). What is the proper way of going about to solve this with only the two pieces of information given? Thanks
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What is the probability when i attempt twice?

// I have never got probabilities' lessons , I only know some basics Let's say the probability of me hitting the target ( let's consider hitting a bottle with a soccer ball ) in ONE ATTEMPT is ...
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What does the decomposition, weak union and contraction rule mean for conditional probability and what are their proofs?

I was reading Koller's book on Probabilistic Graphical Models and was wondering what the decomposition, weak union and contraction properties of conditional probability mean. Decomposition: $$(X ...
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Application of Bayes' Thm.

I know that for this problem you would use Bayes' Theorem, but I am having issues figuring out which pieces would be of value. So far I have: P(cancer) = .008 P(accurate test given cancer) = .95 ...
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Further Conditioning upon already Conditional Expectation

Let's say that $Y$ as a sample space of $\{1,2\}$ and $Z$ has a sample space of $\{3,4\}$. I know that $E[X]=E[X|Y=1]P(Y=1)+E[X|Y=1]P(Y=2).$ Now suppose I now want to further condition upon $Z$. I ...
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Calculating Conditional Probability of stock prices

I have a data set of stock prices, with a set of events and their probabilities as follows: A = Change in price between current day and previous day $$ P(A>0) = 0.548 $$ B = Correlation between ...
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Probability of two cot deaths in one family

In the UK, the probability of a cot death is about 1:8000. In the case of Sally Clark, two of her children died, apparently of cot death. But she was prosecuted for murder. An expert at her trial ...
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Conditional probability $P(A|B \cap C).$

I am trying to calculate $P(A|B\cap C).$ From my data set I have calculated: $P(A|B) = 0.58$ $P(A|C) = 0.44$ However, there is not enough data in the data set to calculate $P(A|B\cap C).$ Is ...
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Combinatorics and conditional probability

Assuming seven standard dice are rolled, what is the probability their sum equals 17? Show a general approach to solving this problem analytically, using conditional probability, combinatorics, etc
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what is wrong in my answer and what will be the correct solution for this probability question ?

A and B play a game where each is asked to select a number from 1 to 5. If the two numbers match, both of them win a prize. The probability that they will not win a prize in a single trial is - ...
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27 views

calculate channel capacity and maximum conditional entropy

i want to know when it is equal channel capacity or $I(X,Y)$ maximum or where $I(X,Y)=H(X)-H(X\mid Y)=H(Y)-H(Y\mid X)$ now if we have two random variable with some specific distribution ...
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A simple conditional probability problem

Assume that two fair dice are rolled one at a time. Given that the sum of the two numbers that occured was at least $7$, compute the probability that it was equal to $7$. I tried computing the ...
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Basic question about the probability and expectation of a bijective function.

Dear stackexchange community, I am still unskilled in the language of mathematics, in fact probability theory to be precise. In my spare time I like to do some research of my own, and I am having ...
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R squared (Proportion of variance explained) in terms of conditional variance?

My question concerns a comparison between 2 models in terms of proportion of variance explained. Let $y_{t+1}$ denotes the variable I want to explain or predict and $\mathcal{F}_t$ the information ...
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when it is conditional entropy minimized?

for example let us consider following table know that entropy of variable is maximum when it is equally distributed,all of it's variable has equal probability,but what about joint entropy or ...
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Calculation of conditional variance

I'd like to ask, whether I did this task correctly. We have two r.v. $X,Y$. Firstly, $Y$ has Bernoulli distribution $\mathbb{P} \left( Y=1\right)=a \; \; \mathbb{P} \left( Y=2\right)=1-a$ Moreover ...
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Intuition of Markov structure and how variables relate in the structure

Assume three r.v. $X_1, X_2, X_3$. They are conditionally independent in the following way: $ X_1 \perp X_3 | X_2$ We have that: $$P(X_3 | X_2) = P(X_3| X_2, X_1)$$ In the notes I am reading it ...
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conditional entropy calculation

let us consider following table i am asked to calculate conditional entropy,from table i have understood everything,for instance how to calculate marginal probabilities,also i know formula for ...
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Probability of winning given pair probabilities

Given an analysis of every pair of competitors in a race, how may I determine the probability of any given competitor winning the race? For example, what is the probability of competitor 2 winning ...
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law of total probability and conditiona probability exercise.

Exercise: Let $X$ be an uniform discrete r.v. with four possible values: 1, 2, 3, 4. Let $Y$ be an exponential variable whose parameter is the value taken by $X$. So, if $X = 3$, $Y$ is Exp (3). ...
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Conditional Probability problem - have solution but trying to figure how event derived probability

I have one problem where I have the probabilities assigned, but I don't understand how an event got the probability. Any help much appreciated so I can understand it. The problem: Doctor X ...
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Conditional Independence, Variable Replication

I'm not sure if this statement is correct. Can anyone explain why or why not? If $Y \perp\!\!\!\perp X | Z$, then $Y \perp\!\!\!\perp {X,Z} |Z$. It seems right, since $Z$ should be able to explain ...
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Unusual conditional probability problem

I came across this task in an exam a few days ago: There are 4 men. The first man receives a signal (a "YES" or a "NO"), and tells it to the second man, the second to the third and the third to the ...
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Measures in conditional expectation.

I always make confusion when a measure has to be changed in some other measure. This time I'm stuck on a change of measure in the definition of conditional expectation of a random variable. If $Z$ is ...
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Compute $E(X\mid X+Y)$ for independent random variables $X$ and $Y$ with standard exponential distributions

Being $X$ and $Y$ independent random variables distributed as $\mathrm{Exponential}(1)$ and let $T=X+Y$ Calculate $E(X|T)$. This is my attempt at solving this: Being $Z=\sum_{i=1}^{n}P_{i}$ where ...
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Given every horse's probability of winning a race, what is the probability that a specific horse will finish 2nd and 3rd?

This question is a follow-on from this question . I am trying to determine the probability of each horse finishing 2nd and each horse finishing 3rd. I have developed code to calculate the ...
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What is a fair game?

Suppose $X_n$ is the fortune of a gambler after $n$ th game. Then the game is called fair (Breiman 1968) if $$E[X_{n+1} \mid X_1, \dots, X_n] = X_n \forall n$$ My question is why a fair game is not ...