Questions tagged [conditional-probability]
In probability, conditional probability, is the probability that an event occurs given something else has already occurred.
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Probability of specific card being in the deck vs your players hand?
If there are 7 cards left in a deck of cards, and 5 cards in the other players hand and you've seen all the cards except for the following:
Spades: ♠ 10 K
Hearts:♥ 10 Q K A
Clubs: ♣ 10 J Q K A
...
1
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0answers
18 views
Multivariate Conditional Entropy as a test of correlation between random variables
I use the word columns to mean the data from which a random variable can be estimated. It is a sample of a random variable.
I am working with $N$ columns of weakly correlated data. Furthermore, I ...
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1answer
38 views
How does the knowledge of one's preference for A over B affect the probability of one's preference for C over B if one's preference is transitive?
Suppose you know someone has preferences between the three pizza toppings pepperoni, olives and mushrooms. If you are told that they prefer pepperoni over olives then what is the probability that ...
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1answer
31 views
Conditional probability of $X$ being even given joint pmf $f_{XY}$
I want to calculate:
$$ \mathbb{P}[ 2Y = 3X ~\vert~ \text{$X$ is even} ]$$
given this joint pmf:
$$ \mathbb{P}(X = k, Y = n) = e^{-2} \frac{(1-e^{-1})^k}{(n-k)!}$$
From the joint I found the ...
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1answer
40 views
What is the probability that heads or tails shows twice in a row exactly once in the ten coin tosses [on hold]
What is the probability that heads or tails shows twice in a row exactly once in the ten coin tosses. I also want to understand the process followed to achieve it.
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1answer
31 views
What is the probability that in 10 coin tosses head never follows head and tail never follows tails [on hold]
I just want to learn the probability that in 10 coin tosses head never follows head and tail never follows tails. Also I want to understand the process by which the result is achieved.
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1answer
34 views
Conditional variance when there is uncertainity about the distribution of condition
Assume that $X \sim N(0,\sigma_x^2)$. Y has the following form
\begin{align}
Y &=
\begin{cases}
Y_1 \sim N(0, \sigma_1^2), & \text{w.p.} \quad \mu \\
Y_2 \sim N(0, \sigma_2^2), & \text{w....
1
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1answer
31 views
Probability of union using conditional probabilities
I'm struggling trying to calculate the probabilities associated with a random variable $Z$ whose value depends on the realizations of two other random variables $X,Y$. I have :
$$
Pr(X=A)=0.25, Pr(Y=...
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0answers
17 views
Truncated conditional expectation of multivariate normal distribution
Assume that $y_1 = \alpha_1 s_1 + u_1$ and $y_2 = \alpha_2 s_1 + \alpha_3 s_2 + \alpha_4 u_1 + \alpha_5 u_2$ where $s_1 \sim N(0,\Sigma_1)$, $s_2 \sim N(0,\Sigma_2)$, $u_1 \sim N(0,\sigma^2)$, $u_2 \...
1
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1answer
24 views
Using Bayes Law leads me to a seemingly nonsensical result
I am trying to prove that, given events $A$ and $B$, if $P(A) > P(B) > 0$, then
$P(A|B) > P(B|A)$
Now, here is my proof:
$P(A|B) = \dfrac{P(A)P(A|B)}{P(B)} \tag*{(by Bayes' Theorem)}$
$P(...
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1answer
13 views
Determine Computability of Joint and Conditional Probabilities Given Few Tables
Problem
I've encountered the following problem during the introduction lecture of my Machine Learning class. I haven't taken a formal probability course yet, so any help would be appreciated.
...
1
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1answer
26 views
Donkey hitting his head on the same stone - a question about probability
I am practising for an exam that has a section on introductory probability theory, we have covered basic Markov chains and conditional probability + binomial distribution, not much more. In these ...
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1answer
19 views
The conditional expected value of a two-dimensional variable
I have problem:
We have two-dimensional variable: ($X,$Y), which has density:
$$
f(x,y) = \left\{ \begin{array}{ll}
2x^2 & \textrm{when $-1<x<1$ and 0<y<|x|}\\
0 & \textrm{in ...
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0answers
28 views
Clarifications on the definition of $P(\cdot|X=x, Y\neq y)$?
I have a question on the following: take two random variables $Y,X$ with support $\mathcal{Y}, \mathcal{X}$ respectively.
Let $y\in \mathcal{Y}$, $x\in \mathcal{X}$.
Let $P(\cdot | X=x)$ be the ...
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0answers
11 views
Conditional probability of Negative Binomial R.V. given the SUM of its values
Suppose $\{z_{ij}\}$ are independent Negative Binomial random variables with means $\{\mu_{ij}\}$, with $i=1\dots I$ and $j=1\dots J$.
How do you find the (expectation of) conditional probability ...
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0answers
24 views
Show that sum of these two Random Variables is conditionally normal distributed (from IGARCH model)
According to Tsay's book (Analysis of Financial Time Series) in Chapter 7, for the Risk Metrics model, the following sum, $r_{t+1} + r_{t+2}$, should be conditionally normal distributed.
$σ_t^2 = ...
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votes
2answers
175 views
~Conditional Probability~ rate(A|B) and rate(not A|B)
Example if:
rate(A | B) = 10/180 x 100% ≈ 6%, rate(not A | B) = 170/180 x 100% ≈ 94%
Since rate(A | B) < rate(not A | B), is there association between A and B ...
1
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1answer
23 views
Are these numbers sufficient for calculating $P(\alpha|\beta,\gamma)$?
Given three probabilities $Pr(\beta|\alpha)$, $Pr(\gamma|\alpha)$ and $Pr(\alpha)$; and that $\beta$ and $\gamma$ are conditionally independent given $\alpha$, can $Pr(\alpha|\beta, \gamma)$ be ...
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1answer
37 views
Finding the probability of fair coin [closed]
A hat contains 100 coins, where at least 99 are fair, but there may be one that is double-headed (always landing Heads); if there is no such coin, then all 100 are fair.
Let D be
the event that ...
2
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1answer
58 views
What is the probability that after removing all red balls, you have exactly 1 blue and at least 1 green ball remaining?
You have a box with 10 red, 20 blue, and 30 green balls. You take out balls from the box, without returning them.
What is the probability that after you've taken out the last red ball there is ...
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1answer
16 views
Conditional probability of continuous random variable
I understand that the formula for calculating a conditional probability is the following $P(A \mid B) = \frac{P(A \cap B)}{P(A)}$
I have this probability to calculate: $P(2\le X \le 3 \mid X \ge1)$.
...
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0answers
22 views
What is the probability of next person has ace of diamond given that you don't have it [closed]
4 people draw 2 deck of cards. What is the probability of the person next to you have at least 1 ace of diamond(2 ace of diamond in the entire deck) given that you don't have any.
Also what is the ...
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0answers
20 views
Modelling conditional distribution based on multiple variables of various types?
I have a looking basic statistics problem: basing on a large sample of multivariate data, model conditional probability distribution (continuous) of one variable based on the remaining ones: a few ...
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0answers
21 views
adapted, increasing, (locally) integrable variation process is a (local) submartingale
I read a theorem that an adapted, increasing, (locally) integrable integrable, variation process is a (local) submartingale. (here increasing includes right continuity).
Definition: A process is $\...
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0answers
45 views
Let $M$ ~ $Geometric(p)$ and $X \vert M = m$ ~ Uniform on integers 1…m
Let $M$ ~ $Geometric(p)$ and $X _{\vert M = m}$ ~ Uniform on integers 1....m.
I want to find the density of X and its expected value.
This is what I have done so far, but I am not sure it is right:
...
0
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0answers
18 views
calculating Probability and prediction.
I am trying to calculate the probability for admission to a restaurant, in foot forward to have a rough idea to predict how many admissions would be there in 5 years from now.
The question is: is 18 ...
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0answers
17 views
How to describe sigma algebra for both conditional and unconditional probabilities
This question is not about random variables and stochastic processes, I'm talking about 101 Kolmagorov Probability definitions.
I have a problem with determining a correct probability space for this ...
1
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1answer
32 views
How to drive the CDF of random variable $Z$ define by $ Z=\frac{X_iX_j}{2X_i+X_j} $
I am working in wireless communication and some times we use the PDFs and CDFs of random variables.
So I have read a paper and I found the derivation of CDF and PDF of random variable, but I did not ...
2
votes
2answers
59 views
What is the distribution of $X|W=w$?
Let $X$ and $Y$ be independent random variables with uniform distribution between $0$ and $1$, that is, have joint density $f_{xy}(x, y) = 1$, if x $\in$ $[0,1]$ and y $\in [0,1]$ and $f_{xy} (x, y) = ...
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1answer
30 views
Three cities, four roads, one railway: a probabilistic traveling problem
Two roads join Ayton to Beaton, and two further roads join Beaton to the City. Ayton is directly connected to the City by a railway. All four roads and the railway are each independently blocked by ...
2
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1answer
33 views
Dick throws a die once. If the upper face shows $j$, he then throws it a further $j−1$ times and adds all $j$ scores shown. If this sum is $3$ . . .
Dick throws a die once. If the upper face shows $j$, he then throws it a further $j − 1$ times and adds all $j$ scores shown. If this sum is $3$, what is the probability that he only threw the die
(a)...
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0answers
33 views
You roll two fair dice. What is the probability to win the game based on restrictions over the sum of the results?
You roll two fair dice. If the sum of the numbers shown is $7$ or $11$, you win; if it is $2$, $3$, or $12$, you lose. If it is any other number $j$, you continue to roll two dice until the sum is $j$ ...
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0answers
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Is my results true in Conditional probability?
I'm new in Conditional probability, Is my results true??
if we have
$\text{Universal Set}= \{1,2,3,...,100\}$,
$A=\{1,2,3,...,20\}, B=\{1,2,3,...,8\}, C=\{9,10,11\}$ .
$P(A)= \frac{20}{100}$,
$P(B)=...
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votes
0answers
33 views
Finding probability of 5staples in a stapler [on hold]
An office employee has TWO staplers on his table, one purple and another
green. Each time he needs to staple two documents, he is equally likely to choose either of
the purple or green staplers. ...
2
votes
2answers
43 views
What is the probability that no cup is on a saucer of the same colour?
A tea set comprises four cups and saucers in four distinct colours.
If the cups are placed at random on the saucers, what is the probability that no cup is on a saucer of the same colour?
MY ATTEMPT
...
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0answers
17 views
How to represent conditional distribution of a given Markov Chain?
Let $\mathbf{X} = (X_1,\dots, X_n) \sim P_{\mathbf{X}}$ be i.i.d samples and
$\mathbf{x}:=(x_1,\dots, x_n)$ be its realizations, respectively.
Let $T:= t(\mathbf{X})$ be a statistic, where
$t(\cdot)$...
0
votes
1answer
20 views
Summation over an Event given Conditional Probability
Consider two events $G$, $A \in E(\Omega)$, where $E(\Omega)$ is an event space.
Consider P(G|A) is the posterior probability distribution over $G$. Now if you sum over $G$, then why is it equal to $...
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1answer
73 views
Show that $\max\{\textbf{P}((A\cup B)^{c}),\textbf{P}(A\cap B),\textbf{P}(A\triangle B)\}\geq\frac{4}{9}$
Let $A$ and $B$ be independent events. Show that
\begin{align*}
\max\{\textbf{P}((A\cup B)^{c}),\textbf{P}(A\cap B),\textbf{P}(A\triangle B)\}\geq\frac{4}{9}
\end{align*}
MY ATTEMPT
Since $\textbf{P}...
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0answers
32 views
Show that two processes have the same distribution knowing that their paths up to time $τ$ and their distributions conditioned on $\mathcal F_τ$ agree
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(E,\mathcal E)$ be a measurable space
$(Y_n)_{n\in\mathbb N_0}$ and $(\tilde Y_n)_{n\in\mathbb N_0}$ be time-homogeneous Markov ...
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votes
2answers
28 views
Should I use conditional probability/prior probability here?
I have been debating the following problem with my friends over the past few days:
“I'm going on a holiday to Mumbai. I'm curious about the weather there so I call my friend who lives there and ask ...
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2answers
37 views
Four red balls and two blue balls are placed at random into two urns so that each urn contains three balls
Four red balls and two blue balls are placed at random into two urns so that each urn contains three balls. What is the probability of getting a blue ball if
(a) You select a ball at random from the ...
2
votes
0answers
25 views
What can we say about a regular version of the conditional distribution given a random variable $X$ on the set $\left\{X=x\right\}$?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(E_i,\mathcal E_i)$ be a measurable space
$X_i:\Omega\to E_i$
Assume $X_2$ is $(\mathcal A,\mathcal E_2)$-measurable and that there ...
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0answers
29 views
Pick 5 numbers - calculate probability that respective 4-number combinations appear for $n-th$ time (given probability for their $n-th$ time)?
We pick 5 unique (distinct) numbers out of [1..36] numbers (like playing a lottery). Each time we pick 5 unique numbers ("play one ticket") thus we select 5 distinct combinations of 4 numbers out of ...
1
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0answers
21 views
Integral of multivariate probability function divided by its marginal?
I do not know how properly establish the problem in words, I will show it in equations.
$$ \int f_{Y|X}(y|x) dx = \int \frac{f_{X,Y}(x,y)}{f_X(x)} dx $$ where $ f_{Y|X}(y|x) $ is conditional ...
0
votes
1answer
19 views
Conditional probability with choosing two cards from a deck
Two cards are picked randomly from a deck. What is the probability that if the first card is Spades, the second card will be a Queen?
What I have so far:
Let $A$ be the event the first card is ...
1
vote
1answer
48 views
What is the conditional probability density function of a statistic given its samples?
I want to find a probability density function (pdf) of
a statistic $T:=t(X_1,\dots, X_n)$ given its
samples $(X_1,\dots, X_n)=: \mathbf{X}$, where
$t(\cdot)$ is a function such as
$t(x_1,\dots, x_n)...
0
votes
1answer
32 views
Probability of two aces when we get at least one ace
"In drawing two cards from a deck (without returning the first card) what is the probability of two aces when you get at least one ace?"
I am aware there is exact same question answered here.
...
1
vote
1answer
39 views
What is the expected value from these two different coin tossing games?
Consider these two games:
Game $1$: Toss $4$ coins. If coins $1$ and $2$ are heads, you win $\$5$. If coins $3$ and $4$ are heads, you win an additional $\$5$.
Game $2$: Toss $3$ coins. If coins $...
0
votes
1answer
30 views
Does a zero conditional expectation imply pairwise covariance is 0?
Suppose in econometrics,
$$ y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + ... + \beta_{k}x_{k} + u$$
In Gujarati's book, it says that the following equation (1)
$$ E[u | x_{1}, x_{2},..., x_{k}] = ...
0
votes
1answer
40 views
Finding $Var(X)$ from conditional PDF
Let $X$ and $Y$ be random variables such that $X \vert Y=y$ is normal distributed as $N(y,1)$ and Y is a continues random variable with PDF:
$f_Y(y)=3y^2$
for $0<y<1$
and $0$ otherwise.
...
