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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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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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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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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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A family has a child. They adopt a girl as well. If one of the children becomes #1 in a women's race, what is the probability it's the adopted girl?

A family has a child. They adopt a girl as well. If some years later one of the children become the champion of women's swimming championship, what is the probability that this girl is the adopted…
Arya369
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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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Expected number of iterations till a walk on unit line exceeds a distance

Problem Each timestep $i$, a uniformly distributed next target $X_i$ is sampled in the range $0$ to $1$. We start out at point $X_0$ on the number line. Each timestep, we go from our current point to the next sampled point. What is the expected…
user1145925
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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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Time-dependent transition probabilities

I need to solve the following question, but I got stuck. I would really appreciate it if someone could help me with it! Here is the question: A well-disciplined man, who smokes exactly one half of a cigar each day, buys a box containing $N$ cigars.…
Beerus
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Confused by Kullback-Leibler on conditional probability distributions

I understand the Kullback-Leibler divergence well enough when it comes to a probability distribution over a single variable. However, I'm currently trying to teach myself variational methods and the use of the KL divergence in conditional…
Lodore66
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How does this "wrong" solution, which doesn't explicitly use conditional probability, lead to the correct answer?

AMC, Fall 2021, 10B, Problem 20: In a particular game, each of 4 players rolls a standard 6-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this…
Starlight
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Rigorous definitions of probabilistic statements in Machine Learning

In a supervised machine learning setup, one usually considers an underlying measurable space $(\Omega, \mathcal{F}, \Bbb P)$ and random vectors/variables $X:\Omega \rightarrow \Bbb R^n, Y: \Omega \rightarrow \Bbb R.$ We can then consider the…
John D
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