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Questions tagged [conditional-probability]

For questions on conditional probability.

Conditional probability is the probability that an event occurs given that another event has already happened. The probability of an event $A$ given another event $B$ is written as $P(A|B)$, and is related to the marginal and joint probabilities via $$ P(A|B)P(B)=P(A\cap B)$$

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Formal definition of conditional probability

It would be extremely helpful if anyone gives me the formal definition of conditional probability and expectation in the following setting, given probability space $ (\Omega, \mathscr{A}, \mu ) $ with $\mu(\Omega) = 1 $, and a random variable $ X :…
smiley06
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Bayes' Theorem with multiple random variables

I'm reviewing some notes regarding probability, and the section regarding Conditional Probability gives the following example: $P(X,Y|Z)=\frac{P(Z|X,Y)P(X,Y)}{P(Z)}=\frac{P(Y,Z|X)P(X)}{P(Z)}$ The middle expression is clearly just the application of…
Dongie Agnir
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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. But before I ask exactly what I am confused about let me introduce some of…
Charlie Parker
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Conditional expectation on more than one sigma-algebra

I'm facing the following issue. Let $X$ be an integrable random variable on the probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{G},\mathcal{H} \subseteq \mathcal{F}$ be two sigma-algebras. We assume that $X$ is independent of…
user73191
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The good, the bad and the ugly with conditional probability/expectation

I thought that I understand conditional probability and expectation until I saw this question: The problem for conditional expectation. Basically, it is given that: $$(X,Y)\sim f(x,y)=\begin{cases} 2xy &\text{ if $0
Kiomi
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Probability that the sum of three integer numbers (from 1 to 100) is more than 100

I have an urn with $100$ balls. Each ball has a number in it, from $1$ to $100$. I take three balls from the urn without putting the balls again in the urn. I sum the three numbers obtained. What's the probability that the sum of the three numbers…
Fabio
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What is the probability of a biased coin coming up heads given that a liar is claiming that the coin came up heads?

A biased coin is tossed. Probability of Head - $\frac{1}{8}$ Probability of Tail - $\frac{7}{8}$ A liar watches the coin toss. Probability of his lying is $\frac{3}{4}$ and telling the truth is $\frac{1}{4}$. He says that that the outcome is Head.…
Hoque
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Four coins with reflip problem?

I came across the following problem today. Flip four coins. For every head, you get $\$1$. You may reflip one coin after the four flips. Calculate the expected returns. I know that the expected value without the extra flip is $\$2$. However, I am…
user107224
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The probability of a drunk person/random walk

A drunk person wonders aimlessly along a path by going forward 1 step and backward 1 step with equal probabilities of $\frac12$. a) After 10 steps, what is the probability that he has moved 2 steps forward? b) What is the probability that he will…
Lorain
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Why do many textbooks on Bayes' Theorem include the frequency of the disease in examples on the reliability of medical tests?

A "standard" example of Bayes Theorem goes something like the following: In any given year, 1% of the population will get disease X. A particular test will detect the disease in 90% of individuals who have the disease but has a 5% false positive…
EJoshuaS - Stand with Ukraine
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Two players toss a coin; probability game doesn't end in 100 tosses?

Player A and B alternate when flipping a coin. If the number of heads is K more than the number of tails, A wins, if the number of tails is K more than heads, B wins. What is the probability that the game is not over after 100 coin tosses? I started…
Redwind
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Conditional distribution in Brownian motion

I need to prove the following: Let $X$ be a Brownian motion with drift $\mu$ and volatility $\sigma$. Pick three time points $s < u < t$. Then, the conditional distribution of $X_u$ given $X_s = x$ and $X_t = y$ is normal; in fact $$(X_u\mid X_s…
BM_guru
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Probability, choose a box and then take exactly two white balls

There are $5$ boxes. There are $5$ white and $3$ black balls in two boxes, and $4$ white and $6$ black balls in the other three boxes. One box is randomly chosen. $3$ balls are randomly taken from the chosen box. What is the probability that…
mak
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Existence of conditional density given conditional distribution

If $P\left(X\in B\mid\mathcal{A}=\omega\right)=\int_B f_\omega\space d\mu_\omega$ can the $f_\omega$ be chosen in a consistent way so that $f_\omega\left(x\right)$ is measurable in $\omega$? To put it more precisely, suppose…
Evan Aad
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Expectation of X increases when conditioning on X being greater than another independent random variable Y?

I'm doing research and for a proof I need the smaller result that for X, Y random, independent (but not identical) variables we have $$\mathbb{E}\left[X|X>Y\right] \geq \mathbb{E} \left[X\right]$$ This seems very intuitive, but I've had no luck…
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