# Questions tagged [conditional-probability]

For questions on conditional probability.

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### what is conditional probabiltiy

what is exactly is the conditional probability I saw this this kind of definitions for conditional probability Definition: Conditional probability is the likelihood of an event occurring based on the ...
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### Memoryless property with any random wait time

We see here a proof of the random-time memoryless property $P(X>T+s|X>T)=P(X>s)$ where $E\sim Exp(\lambda)$ and $T\ge 0$ is a continuous random variable independent of $E$. The proof, however,...
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### Algorithm for Weighted Combination

How can we randomly select $k$ people out of $n$ such that for person $i$, the probability they get selected is $0<p_i<1$, and $\sum p_i=k$? I'm struggling quite a bit on this; you could go ...
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### How to represent difficulty and success?

This is a revision of another question found here: Consider a criminal pondering the commission of a crime, C. It could be launching a cyber-attack or embezzling funds. How likely is it that the ...
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### Is this conditional probability always equal to 1?

Consider $X = \text{number of defective items in bought items}$. Is the probability that $X \geq a$ given $X = a$, always 1: $P(X \geq a| X = a)=1$. I was wondering if the above holds because $A$ (...
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### In Bayesian probability if P(a|bc)=0 will P(b|a) always be less or equal to P(b)? [closed]

If we have 3 variables a, b and c and we assume that a conditional on the conjunction of bc is impossible so that P(a|bc)=0 then will the posterior probability of b always be lesser or equal to P(b) ...
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### Finding the probability of having a genetic trait given test is positive

I've been doing this question but I'm a little stuck on the second part. The first part is as follows: The probability of a randomly selected person in a population having a particular genetic trait ...
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### Proving $P(A\mid B,C)=1$ given $P(A\mid B)=1$ [duplicate]

I'm trying to prove the following statement formally: if $P(A\mid B)=1$, then $P(A\mid B,C)=1$. I can see why logically speaking the statement is correct but I'm lacking a formal argument, using ...
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### Clarification on independence and conditionality of coin tosses and the resulting PMF

I'm confused about independence of events in the following question from Blitzstein and Hwang, chapter 3 (Random Variables): Let X be the number of Heads in 10 fair coin tosses. Find: (a) the ...
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1 vote
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### Probability of winning a game that ends after one player gets 3 wins in a row

Question: In a game with two people, there is a draw half of the time. The other half of the time, player A wins with 2/3 and player B wins with 1/3. The game ends once a player wins three games in a ...
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1 vote
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### Gun standoff game I devised for myself (basic probability problem)

I devised a probability problem for myself to help me better understand conditional probability (because I am evidently missing something very crucial). The rules: $k$ players are arranged in a ...
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### Probability of an event conditional on union of two events

Let $A,B,C$ be three events such that $A,B$ are independent and $A,C$ are not independent. Now I am wondering if $P(A\mid B\cup C) = P(A\mid C)$? First of all, I think we cannot say $B$ is ...
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I have the following problem An honest coin is tossed. What is the conditional probability that it appears head for the first time on the $N$ th toss, knowing that it has come up at least one head ...
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### Lower and upper bounds on conditional expected value.

Suppose we have a random variable $X$ which can take negative values. I am wondering if it is possible to find exact or high-probable lower and bounds for $\mathbb{E}(X|X\ge a)$ based on $a$, $var(X)$ ...
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### True and False Positives

I am getting confused on what should be a fairly simple concept. A flu test gives true positives 90% of the time, and true negatives 85% of the time. 5% of people have the flu. We want to find the ...
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### Do I need additional assumptions for this equality to hold?

Suppose I have three random variables $X_1,X_2, V$, and I want the following condition to hold: $E[X_1^2|X_2<V<X_1,X_1,X_2]=E[X_1^2|X_1]=h(X_1)$, i.e., I want conditioning varibles $V$ and $X_2$...
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### Conditional Probability of Intersections in the Given

A construction firm purchased 3 tractors from a certain company. At the end of the fifth year, let $E_1$, $E_2$, $E_3$ denote, respectively, the events that tractors no. 1, 2, and 3 are still in good ...
1 vote
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### Conditioning reduces entropy, directly from Khintchine--Shannon axioms

Assume there is a function ${\bf H}\{X\}$ on discrete random variables $X$ that satisfies the following axioms: a) (Invariance.) If $X$ takes values in $A$, $Y$ takes values in $B$, $\phi:A\to B$ is a ...
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### Expressing the probability of a random vector being in a random set as a conditional expectation

Let $X$ be a random vector in $\mathbb{R}^n$ and let $\mathbb{P}_X$ be its distribution. Let $\{A_y : y \in \mathbb{R}^m\}$ be some collection of Borel subsets of $\mathbb{R}^n$ and let $Y$ be a ...
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### Conditional Expectation of a product of r.v.

I am exploring a work in probability theory that states $$E[U1U2 | V1, V2] = E[U1| V1]E[ U2 | V2]$$ for independent pairs of random variables $(U1,V1)$ and $(U2,V2)$. I understand the intuition behind ...
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