Questions tagged [conditional-probability]

In probability, conditional probability is the probability that an event occurs given another (conditioning) event occurred.

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Probability of Invalid document in a large data set

I'am auditing a very large data set of documents. A document can be Valid or Invalid. Checking a document is computationally ...
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Help finding the probability of a pipe being blocked in a system of parallel pipes

There is a system of pipes from one point to another. Pipe A is the start point, and connects left to right to Pipe B, C & D, which are in parallel and connect to the end point. The pipes can flow ...
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Is showing that $P(A|B) \neq P(A|{B}^c)$ enough to prove that $A$ and $B$ are not independent?

At least intuitively, to me this means that event B happening/not happening affects the probability of event A happening, which may seem enough to show these two events are not independent. But is ...
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For a set of n Exponential iid RV, what is the probability that the maximum exceeds the sum of the others?

My question stems from self-study of question 91 in page 372 of Ross's Introduction to probability models, 12th edition. The answer is given in the textbook but I am trying to understand a specific ...
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"randomizing" a sequence of random variables

I have a (kind of) follow up question on this question. Let $I \subset \mathbb{R}$ and for any $a \in I$ let $X_a$ be a real-valued random variable on $(\Omega_a, \mathcal{F}_a, \mathbb{P}_a)$. ...
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If $X$ and $Y$ are conditionally independent given $Z$, and $X$ and $Z$ are conditionally independent given $Y$, are $X,Y,Z$ independent?

Let $X,Y,Z$ be discrete random variables, I'll use $\perp$ to denote independence. Suppose $X\perp Y|Z$ and $X\perp Z|Y$, does this imply any of the following, $X\perp Y$, $X\perp Z$ or $Y\perp Z$?
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Conditional distribution of one of the two exponential random variables, given one is smaller than the other

Let $X$ be a random variable with exponential distribution with parameter $a$, i.e. $X\sim Exp(a)$. See https://en.wikipedia.org/wiki/Exponential_distribution for the definition of exponential ...
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Conditioning in Event Language VS Proposition Language

According to this video, one can freely decide to conceptualize probabilities in terms of either event language or proposition language. It states, "the mathematical rules are applied the same ...
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Find estimates for $\alpha_0$ and $\alpha_1$ and covariancematrix

I have that $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}$$ I have the conditional distribution: $$f_{\beta}(y|x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}(y-\beta_0-\beta_1x-\beta_2x^2)^2}$$ and we ...
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Given a person's age x, population's average life span u, predict life span.

Given a person's age x, population's average life span u, predict how long person can live? e.g. Say x is 6 and u = 76 one would expect life span of 74. But when x is 90, for same u = 76, one would ...
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Expected value of square conditional expected value

I need to prove that $\mathbb{V}(Y)=\mathbb{E}(\mathbb{V}(Y\mid X))+\mathbb{V}(\mathbb{E}(Y\mid X))$. Using the fact that $\mathbb{V}(Y\mid X)=\mathbb{E}\left (\left (Y-\mathbb{E}(Y\mid X)\right )^2\...
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Doesnt seem right; Conditional Probability question

So heres my working out for the question. I figured that I need to find three probabilities: P(c|x), P(y|c) and P(z'|c) as it is the probability they ate contaminated food given they have symptoms X ...
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Likelihood of censored exponential random variables

Consider $X_1, \dots, X_n \stackrel{\text{iid}}{\sim} \text{Exp}(\lambda)$ and define \begin{equation*} Y_i = \begin{cases} X_i & X_i \leq c \\ c & X_i > c \end{cases} \end{equation*} for ...
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Probability Algebra With a Wordle Example

Define $X$ as the proposition "you will win today's Wordle," $A$ as expressing the information returned by the coloring of your first guess (in a vacuum), $B$ as expressing the information ...
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Unsure about Conditional Probability question

I'm really not understanding this question So for Exercise 1, I got 0.794 which is correct. My reasoning was that there are only 2 ways to test positive, a "true" positive and false ...
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Probability Expression With Nested Given That Operators

Does the expression $\Pr((X\ |\ A)\ |\ B)$ make any sense? I want to say that $\Pr(X\ |\ (A \cap B))$ is equivalent to $\Pr((X\ |\ A)\ |\ B)$, and thus also to $\Pr((X\ |\ B)\ |\ A)$, but I'm not sure ...
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The gambler chooses dice at random, and rolls it $six$ times. What is the probability that fair die was chosen?

Assume that a gambler has two dice, one of which is fair, and the other is biased toward landing on $six$, so that $0.25$ of the time it lands on $six$, and $0.15$ of the time it lands on each of $1$, ...
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The gambler chooses a dice at random and rolls it once and the dice comes up with a 6. What is the probability that fair die was used?

Assume that a gambler has two dice, one of which is fair, and the other is biased toward landing on $six$, so that $0.25$ of the time it lands on $six$, and $0.15$ of the time it lands on each of $1$, ...
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Finding Conditional expectance, variance from joint pdf

X and Y have the joint pdf: $f(x,y)=e^{-y}$ for $0<x<y<\infty$ Compute $E(Y|X)$ and $Var(Y|X)$. I computed $f_X(x)= \int_{x}^{\infty} f(x,y)dy=e^{-x}$ $f_{Y|X}= e^{x-y}$ for $ x>0$ Then $...
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How to calculate mean of conditional probability?

If I have $E[(A-B)|C]$ to calculate mean of something, is it equal with $E[A|C]-E[B|C]$? If yes, where I can get the references? Or how to prove it?
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Maximum likelihood for two dependent variables

Suppose you have a box containing 10 balls, of which $\theta$ are white and the rest are green. Suppose we take two balls for without replacement and let $X_i = 1$ if the i-th drawn ball is white and $...
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Conditional expectation of geometrically distributed random variable

Given $Y$ as a geometrically distributed random variable with $p\in (0,1)$, what is $E[Y|Y\geq 10]?$ I got:$$E(Y|Y\geq 10)=\frac{E(\mathbf{1}_{Y\geq 10}Y)}{P(Y\geq 10)}=\frac{E(\mathbf{1}_{Y\geq 10}Y)}...
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Conditional continuous probability on a fixed point

I was asked to show the following statement: Let $X, Y$ both be equally distributed random variables on $[0,1]$, we define $$ P(\{Y \leq y\}|\{X=x\}) = \lim_{h \downarrow 0} P(\{Y \leq y\}|\{x \leq X \...
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Conditional Independence, Bayes Network and d-separation

I have a diagram of a Bayes network as shown below: $$\begin{array}{c}A&&&&B&&&&C\\&\searrow&&\swarrow\\&&D&&&&E\\&&&\...
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Probability of a two-state continuous Markov chain

Consider a continuous-time Markov process ($\epsilon_t$) which takes two values ($\epsilon_t=0$ or $\epsilon_t=1$). Let $p_0$ denote the probability of switching from state 1 to 0 and let $p_1$ denote ...
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Question on use of indicator function

For my Financial Mathematics course I have the following exercise (with solutions): I don't really understand the start of the solution of (B). More specifically I do not understand why the $1_{\{X=\...
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Conditional probability for path calculation in a markov chain model

This question extends a previous question I asked with respect to the markov chain model. I post another question here since I'm trying to follow the Stack Exchange's guidelines. So, based on the ...
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Prove $P(B|A) = P(B)$, if $A$ and $B$ are independent [closed]

How can I show that $P(B|A) = P(B)$, given that $A$ and $B$ are independent?
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4 votes
1 answer
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Best betting strategy for an unfair random walk with a skewed payoff

Say you start with bankroll $B$ and i.i.d. random variables $U_i$ with distribution $p=P(U_i=r)>.5$ and $q=1-p=P(U_i=-1)$. Your earnings from bet $i$ is $W_iU_i$, where $W_i$ is your wager at step ...
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Simple question about conditional probability of joint distributions

I have this maths problem that I'm trying to figure out. It looks really simple but it confuses the heck out of me. The question: Let $X$ and $Y$ be two independent random variables so that $X \sim ...
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About Probabilities and Alcohol

Here I have the following exercice : Show that $P(A|\overline B)=\dfrac{P(A)-P(A| B)\times P(B)}{1-P(B)}$ This question is quite easy since $A=(A\cap B)\cup(A\cap\overline B)$ thus: \begin{align} P(...
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Simple Problem using Bayes Rule

This is Exercise 1 in Chapter 2 of the Probabilistic Robotics book by S. Thrun etal. Problem. A robot uses a range sensor that can measure ranges from $1$m to $3$m. For simplicity, assume that actual ...
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probability question, P(A|B) The conditional probability that the unit has at most three rooms, given that it has at least two rooms

I'm having issues with the following problem, Here is my dataset. ...
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Computing variance given bivariate normal

Suppose (X, Y ) has a bivariate normal distribution with means equal to zero, standard deviations equal to 1, and a correlation $0.5$. We want to find the variance of $XY$. To do so, I used law of ...
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Marie has been taking note of the time she leaves for work and the length of her morning commute. She decided to model the number of hours $X$ [closed]

Marie has been taking note of the time she leaves for work and the length of her morning commute. She decided to model the number of hours $X$ after 6:30 a.m that she leaves for work with a $uniform(2)...
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Overlap Probability Calculation

Consider a 1 second frame with two users having 2 milli-second(ms) symbol length transmission each, now each of the user can randomly select any 2 ms interval within the frame, what is the probability ...
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Does a continuous joint probability distribution conditioned on one of the variables have a meaning given the Borel-Kolomogorov paradox?

Suppose $f(X_1,X_2)\sim MVN(\vec0,\Sigma)$, and to simplify let $\Sigma= \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} $. Is $f(X_1|X_2=0)$ defined? It seems to me that the Borel-Kolomogorov ...
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Simple question on discrete uniform distribution

We have a random variable $X$ uniformly distributed on the set $\{1,\ldots,n\}$. Assume $s<<n$. Can anyone please advise, how to find the conditional probability $P\{X = k | X\le s \}$, where $k\...
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Extending conditional probability to more than 2 events

Quick question, In the context of conditional probability if we can define $$P(A\cap B) = P(A|B) P(B) $$ Then what is a similar way to write $$P(A\cap B \cap C)?$$
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Balls are randomly taken out of a box of 4 red and 6 green balls, without replacement, until we have a ball of each colour.

What is the probability of Needing exactly two draws Needing two or three draws Needing to make exactly three draws if the first two balls drawn are of the same colour For part 1. I computed that ...
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10 good and 3 bad batteries are mixed and then 5 are chosen. What is the probability of the fifth one being dead given that the first 4 aren't?

I approached this as follows: $A$: First four are good, $B$: Fifth one is bad. Then $$|A|={10\choose4}$$ because this is the number of ways we can choose 4 good batteries from the available 10. Then $$...
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Conditional distribution of $X$ given that $X+Y > t$. [closed]

If $X$ and $Y$ are independent exponential random variables with the same mean, then what is the conditional distribution of $X$ given that $X+Y > t$? Form comment: I was able to get that the ...
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1 answer
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Conditional distribution of $x_1\mid \sum_{i=1}^{n} x_{i}=t$ [closed]

If $x_1, x_2, \ldots , x_n \sim \operatorname{Exp}(\lambda)$, then find $P(x_1 \mid \sum_{i=1}^{n} x_{i}=t)$. My approach is, $t=\sum_{i=1}^{n}x_i \sim \operatorname{Gamma}(n, \lambda)$, and $v=\sum_{...
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Probability of finding a specific colored plant in a row.

In a packet of flower seeds 3/4 are yellow flowering and rest are white. If 200 rows of each plants are planted, how many will contain (i) all yellow flowers. (ii) all white flowers (iii) two yellow ...
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1 vote
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Why is solution 2 correct?

Question: N(t) is a Poisson process with parameter $\lambda$, let $S_n$ denotes the time when the $n$-th event happens. Try to calculate $E(S_4|N(1)=2)$. Solution 1: Let $F(x|N(1)=2)$ be the CDF. $$...
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Gibbs sampling - finding probabilities of joint exponential random variables

Say I have two IID exponential random variables, both with mean $\lambda$. I would like to find p(x < a | x+y > b) for two positive integers a and b using Gibbs sampling. I understand that for ...
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Help with joint density probabilities and conditional density functions in limited domais

I'm trying this excercise: Let the joint density function of random variables X and Y be: $$f_{x,y}(x,y)=\left\{\begin{array}{l} \frac{3}{16}(4-2x+y) & for \,\,\, x \geqslant 0, y \geqslant 0 \,\,...
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-1 votes
1 answer
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Conditional probability (Bayes theorem)

One of the 256 subsets of {1,2,3,4,5,6,7,8} is chosen uniformly at random. Let X be the number of elements of this subset. Let Y be 0 if the subset is empty and be the least element of the subset ...
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Is the bounded uniform distribution the only continuous parametric bounded distribution with a mean invariant under truncation?

Suppose I have a continuous parametric bounded distribution PDF $f_{a, b}$ with support $[a, b]$. It has mean $$\mu_f({a,b}) = \int_a^b xf_{a,b}(x)dx.$$ We can truncate a distribution to a smaller ...
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Integrating Conditional Probabilities

I have a function : F(a,b,c) I want to take the Expected Value of this function in the following form: E[ F(a,b,c) | a, c) I know that for a single variable, the Expected Value is E(x) = integral(x*p(...
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