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

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Finding conditional probability [on hold]

Two floppies are selected at random without replacement from a box containing $7$ good and $3$ defective floppies. Let $A$ be the event that the first floppy drawn is defective, and let $B$ be the ...
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Probability of begin cut

Any athlete who fails the Enormous State University's women's soccer fitness test is automatically dropped from the team. Last year, Mona Header failed the test, but claimed that this was due to the ...
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Meaning of the expectation of a conditional variance

I'm reading page 2 of conditional probability. The author states that if $\text{Var(X|Y)}$ is treated as a random variable then the expectation is $\text{E[Var(X|Y)] = E[E[X^2|Y]] - E[E(X|Y)]^2}$ My ...
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Solve my probability doubt? [on hold]

A parent gives birth to two children. One of the child is surely a male, what is the probability of having both male child? Common answer 1/2 Actual answer 1/3
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Conditional Probability Question - on route availability

Hey Guys I am seemingly stumped with this question I have gotten involving conditional probability and routes Suppose route $A$ to $B$ is available 0.5 of the time An alternative route to B from A ...
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What is the conditional mass function of $X$ given that $Y = i$?

First, let me explain the entire problem. Choose a number $X$ at random from the set of numbers $\{1, 2, 3, 4, 5\}$. Now choose a number at random from the subset no larger than $X$, that is, from ...
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Bayes Theorem, The Law of Total Probability, and Trees that Don't Grow.

" A doctor is concerned about the relationship between blood pressure and irregular heartbeats. Among her patients, she classifies blood pressures as high, normal, or low and heartbeats as regular or ...
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Conditional mutual information and Markov chain.

If we have the Markov chain $X \to Y \to Z$, or equivalently $$I(X;Z| Y)=0, \tag{1}$$ where $I(\cdot)$ denotes the mutual information. Does the Markov chain $X \to (Y,W) \to Z$ also hold? Or ...
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Is this correct: $E(Y\mid X,U)=E(Y\mid X,U=0)P(U=0)+E(Y\mid X,U=0)P(U=1)$ if $U$ is a discrete taking values $0, 1$

Is this correct: $E(Y\mid X,U)=E(Y\mid X,U=0)P(U=0)+E(Y\mid X,U=0)P(U=1)$ if $U$ is a discrete taking values $0, 1$. Further more, can I do this, with the same $U$: Let $X=U+V$ $E(Y\mid X)=E(Y\mid X, ...
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Nested Conditional Expectation

Let X, Y, and Z be some random variables. Is the following true $E[E[Z|Y,X]|X]= E[E[Z|Y]|X]$? Cheers
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Are these 2 events independent?

We draw 4 cards from a standard 52-card deck. $A$ is the event that we draw 4 different color cards. $B$ is the event that we draw at most 3 aces. I have calculated $P(A)$ and $P(B)$, and I know ...
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D6 Event Tree Probability Question. [closed]

One has tried looking this one up and Googling it, One is also dispraxic so while math can be tricky if one can get the concept and explanation behind something generally work at it until one ...
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Conditional probability finding another given probability

If i have P(A) = 0.05 and P(B | A) = 0.95 the first question asks 1) What is the P(AnB) I did: P(AnB) = P(B | A) x P(A) and 0.95 x 0.05 = 0.0475 then it asks: 2) P(B' | A) =? I tried doing: 1- P(B ...
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Conditional Probability teenage drivers

Teenage drivers pay more for automobile insurance than older drivers. Many companies offer discounts for teenage drivers good grades. Assume that 20% of all teenage drivers are involved in accidents ...
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Product of two random variables together with conditional density

Let $X_1$ and $X_2$ be two real valued random variables such that we have the conditional density of $X_1$ given $X_2$, i.e. $$\mathbb P(X_1\in M\mid X_2) = \int_M \phi(x_1\mid X_2)dx_1$$Also, let $h$ ...
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Is the probability that I am late if I go via a route conditional probability?

I have been given the question: I travel to work via route J or K. The probability that I chose route J is $\frac{1}{4}$. The probability that I am late for work if I chose route J is $\frac{2}{3}$. ...
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Probability without replacement questions

A Bag contains 4 red balls and 6 green balls. 4 balls are drawn at random from the bag without replacement a) Calculate the probability that i) all the balls are green; ii) at least one ball of ...
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Is there any definition for conditional probability over conditional probabilities? e.g. $P((A|B) | (B|A))$

Does this $P((A|B) | (B|A))$ meaningful? $P((A|B) | (B|A)) \cdot P(B|A)=P(A|B, B|A)$. I just figured in some scenario the probability $P(A|B, B|A)$ may signify some interior correlation between the ...
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Brownian Motion Conditional Probability Question

If $X_t$ is a standard Brownian motion, how does one calculate a conditional probability. Specifically, $P(X_2\gt 0 |X_1\gt0)$. I am thinking that the two are independent so I can just calculate ...
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Calculating probabilites

We know the following probabilities: $P(A)=0.25$ $P(A|B)=0.25$ $P(B|A)=0.5$ The question is: $P(\overline{A}|\overline{B})=?$ I have calculated: $P(B)=\frac{P(B|A)P(A)}{P(A|B)}$ and from this ...
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The probability of the first roll having the highest number in n consecutive rolls of one 6-sided die

What is the probability that if we roll one six-sided die for n times - whatever the number we see in the first roll, no other following roll has a number higher than that (that is all the numbers we ...
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Need help with this conditional probability question

There are 4 alternatives on a multiple choice test. Let suppose that a student learned 70% of the material. If she doesn't know the answer, she picks one randomly. If she picked the right answer, ...
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Conditional Expectation on similar sigma algebras

I'm trying to prove the following, (or to find a counterexample): Let on a probability space. $Y$ be a Bernoulli variable, $X\in L^1$ be another random variable, let $\mathcal{G}$ be some ...
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Inequality: $\mathbb E[(X-\mathbb E[X\mid \mathcal H])^2] \le \mathbb E[(X-Z)^2]$

Let $X$ be a random variable on the probability space $(\Omega, \mathcal A, P)$. Suppose $X$ is square-integrable and $\mathcal H$ is a sub $\sigma$-algebra of $\mathcal A$. Then $\mathbb E[X\mid ...
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Conditional Expectation Problem With Noise

I was given the following problem: let $X,N\sim \mathcal{N}(0,1)$ and let $A$ equal $1$ w.p $p$ and $0$ w.p $1-p$. Also, let $X,N,A$ be independent. Define $Y=AX+N$. Find $\mathbb{E}(X\mid Y)$. My ...
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Regression function - conditional mean

I am trying to understand the statistical fundamentals behind linear regression, and i have never been able to intuitively understand the following; really would appreciate if someone could give an ...
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Bayes' theorem and conditional probability?

This is a bit of a soft-question, which I just happened to overhear, so please, bear with me. "How can one derive Bayes’ theorem from the definition of conditional probability?" After hearing said ...
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throwing a dice repeatedly so that each side appear once. [duplicate]

Pratt is given a fair die. He repeatedly throw the die until he get at least each number (1 to 6). Define the random variable $X$ to be the total number of trials that pratt throws the die. or ...
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How can I calculate the probability of multiple events assuming they may or may not be independent?

I have this problem in my work. Assume I have multiple events, say A, B, C. I want to calculate probability P(ABC) But among A,B,C some of them may or may not be dependent to each other. With ...
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Conditional-Probability Question

Consider two random variables X and Y with mean 0 and variance 1, and with joint normal distribution. If covariance(X,Y) = 1/sqrt(2). what is the conditional probability of P(X>0, Y<0| Y<0) ? ...
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Similar formulas to that of “conditional distribution of a normal given another normal”?

We know that if $X$, and $Y$ are two (vector) normal distribution then we have a formula to obtain $E(X|Y)$, and $cov(X|Y)$. Are there any similar formulas to give $E(X|Y)$ and $cov(X|Y)$ that given ...
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Prove that $P(x,y|z)=P(x|z)P(y|x,z)$

This is my attempt, but I'm stuck: $P(x,y|z) = P(y|x,z)P(x)=P(y|x,z)\sum_{z}P(x,z)=P(y|x,z)\sum_{z}P(x|z)P(z)$ I think the steps I took thus far have been correct, although I'm not sure whether they ...
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obtain conditional probability

Two common coins are tossed. a) Obtain the condiitional probability that both coins resulted in tails knowing that the first coin resulted in tail. b) What is the conditional probability that we ...
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Probability with conditional substitution

An urn contains 6 balls, 1 purple, 2 blue, and 3 brown. When a ball is selected it is replaced with a green ball unless the ball drawn is green, in which case the green ball is simply returned to the ...
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Conditioning on a random variable

The number of storms in the upcoming rainy season is Poisson distributed but with a parameter value that is uniformly distributed between (0,5). That is Λ is uniformly distributed over (0,5), and ...
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Probabilities of birthdates of married couples

So random, I know. My birthday is December 3 (12/03). My wife's birthday is March 12 (03/12). What the possibilities of a couple having birthdays like ours that use the same digits but in reverse. ...
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Convergence of conditional distributions

Let $(X,Z)$ be a bivariate continuous random variable over, say, $(0,1) \times (0, 1)$. Let $z_1,\cdots,z_T$ be a sample of size $T$ of $Z$, and define $X_1,\cdots,X_T$ as $X_t=X \mid[Z=z_t]$. That ...
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Dependence between a joint probability distribution

Suppose $X$ and $Y$ are correlated random variables in a finite set ${\mathcal A}$, and let $f, g$ be functions that map elements from ${\mathcal A}$ to ${\mathcal B}$ for some finite set ${\mathcal ...
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Tossing a coin k times

We toss a fair coin $k$ times and we set the following possible results: $A$: both heads and tails appear at least 1 time. $B$: tails appears at least one time. For which values of $k$ are $A$ and ...
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Conditional Expectation of rolling two dice.

I have to find $E(X\mid Y)(y)$ where $X$ is the value of the first roll and $Y$ is the sum of the two dice. I know that $$E(X|Y)(y) = \sum_x{xP(X\mid Y)}=\frac{\sum_xxP(X=x, Y=y)}{P(Y=y)},$$ but ...
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Random variable with infinite expectation but finite conditional expectation

I've been very stuck on a question from Probability and Random Processes by Grimmett and Stirzaker for ages - so stuck that I flicked to the back to have a look at the answers. But, I can't seem to ...
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Finding the density function from joint density function

I'm reading the conditional distributions section of Probability and Random Processes by Grimmett and Stirzaker and I've come across a brief exercise I can't seem to figure out. We're given earlier ...
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drop independent random variables in conditional expectation expression

Is the following true for discrete random variables $K,Y,X_1,...$ where $K$ is independent of $\sigma(Y,X_1,...)$? $$E[\sum_{i=1}^K X_i | K,Y] = \sum_{i=1}^K E[X_i|K,Y] = \sum_{i=1}^K E[X_i|Y]$$ ...
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Marbles drawn from a jar problem

Trying to understand better the maximum likelihood method, i've end up finding a very interesting marbles from a jar problem that gives intuition about why the MLE is used. BUT there's a step in the ...
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Bayes' Theorem Question, with a twist

I have a very old past high school exam question I am trying to solve (for interest only). It's a straightforward application of Bayes' Theorem, with the last part of the question containing a slight ...
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There is two boxes with one with 8 balls and one with 4 balls

We have two boxes: $A$ - with two 8 balls, and $B$ - with 4 balls. we choose randomly box and pull out a ball. We do it again and again until box $B$ will be empty. What is the probability that box ...
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Conditional expectation of an exponential random variable

Let $X$ be an exponential random variable with rate $\lambda$ Use the identity below to solve for $E[X|X < c]$ $$E[X] = E[X|X < c]*P(X < c) + E[X|X > c]*P(X > c)$$ So right off the ...
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Can we infer a set has positive measure from the conditional probability formula?

First question: Suppose that I know $P(x\in A|\;y)>0$ where $y$ is a realization of some random variable. Then, in general can I infer using the conditional probability formula $P(x\in A|\; ...
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Probability and Intersections

I'm having trouble understanding the difference between conditional probability and dependent events. Even then, I'm not sure if that's what I'm having issues with. NB This is not a homework problem. ...
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Conditional expectation of a stochastic process in filtered space

It was suggested* to me that if we have a stochastic process with independent increments, and $T > t$, then $$ E(X_{T-t}| \mathcal{F}_t) = X_{T-t} $$ where $\mathcal{F}_t$ is the filtration at time ...