Mathematics Stack Exchange
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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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How can I calculate the probability of multiple events assuming they may or may not be independent?

I have this problem in my work. Assume I have multiple events, say A, B, C. I want to calculate probability P(ABC) But among A,B,C some of them may or may not be dependent to each other. With regards to my work scenario, I have the values of…
CyberPlayerOne
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Probability with conditional substitution

An urn contains 6 balls, 1 purple, 2 blue, and 3 brown. When a ball is selected it is replaced with a green ball unless the ball drawn is green, in which case the green ball is simply returned to the urn. What is the probability of 3 green, 1…
user138763
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Computing $E\left[X\,\middle|\,X \leq \dfrac{Y}{2}\right]$

Let X be a uniformly distributed over $[0,2]$, and $Y$ to take values from $[0,\infty]$, how do we compute $E\left[X\,\middle|\,X \leq \dfrac{Y}{2}\right]$. My attempt: $$ E\left[X\,\middle|\, X \leq \dfrac{Y}{2}\right]= \int_0^2…
user90831
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conditional expectation of squares

Say we want $E_{I}(E(X|I))^2$ where $X$ is a continuous random variable and $I=0$ or $1$ is a discrete random variable. Can we write the following ? $E_{I}(E(X|I))^2 = (E(X|I=1))^{2}\times P(I=1) + (E(X|I=0))^{2}\times P(I=0)$
Pradipta
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Probability of finishing a course of treatment

You need to take a medicine 1 x per day for 5 days. What is the probability of completing the treatment expressed in terms of the probability of taking each dose given that the probability of taking a dose (other than the first dose) is conditional…
mariey
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finding the conditional density function slight problem

The joint density function of $X$ and $Y$ is given by $f(x,y)=xe^{-x(y+1)}$ for $x>0,y>0$. find the conditional density of $Y$ given $X$. I am close to get the answer but with a little problem $\displaystyle f_{Y|X}(y|x)=\frac{f(x,y)}{f_x(x)}…
natsu
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Law of total probability applied to conditional probability

Sorry for the really simple question. I am reading a book about psychophysiology and I got lost at one point. I will take out all the unrelevant information and just go straight to the point. The book present this probabilities: 1)…
Vaaal88
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Conditional probability for continuous function

let $f$ is a continuous probability density function. Then a conditional probability is $f\left( x|Y=y \right)=\frac{{{f}_{XY}}\left( x,y \right)}{{{f}_{Y}}\left( y \right)}$. For example if we want $Y=1$ we put this in ${{f}_{Y}}\left( y \right)$.…
mert
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Changing balls between 2 buckets

Let there be two buckets. Bucket I contains balls of 4 white, 1 red. Bucket II contains balls of 3 white, 5 red and 1 black. Consider the following procedure: You draw a ball uniformly at random from Bucket I, put it in Bucket II. Then, you draw a…
jbb
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Fundamental doubt on independent events

I have a general doubt on independent events. After studying some basics on probability from A first course in probability, I don't see exactly why the relation for independent events comes from, specially if we derive it from the conditional…
pdaranda661
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Conditional Probabilities and independence question

Could someone please help me solve the following question: Say we have 3 coins which have probabilities 1/3, 1/2, 2/3 respectively of getting heads. We choose a random order in which to flip the coins (all possible orderings being equally likely),…
MFFF
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Conditional probability about biased coin

Suppose there are two coins, with coin 1 landing heads when flipped with probability 0.3 and coin 2 with probability 0.5. Suppose also that we randomly select one of these coins and then continually flip it. Let $H_j$ denote the event that flip…
Jiajun Li
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Conditional probability of a sum of independent random variables

I have a question about conditional probability. Let's say $y = x + e$ where $x$ ~ $U[0,1]$ and $e$ ~ $N(0,1)$ and both are independent. What is $p_x(x | y = 10)$ ? In particular why isn't it the same as $p_e(10 - x)$, i.e. $N(10-x, 1)$? I'm trying…
user3882729
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Finding the joint density function of two non independent random variables

Given two random variables $X$ and $Y$ that are independent and exponentially distributed with the same parameter $\lambda$, and given that $Z = X + Y$, the joint density $f_{X,Z}(x,z)$ can be expressed as: $$ f_{X,Z}(x,z) = f_X(x) f_Y(z - x) $$ Why…
Heng Wei
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Is $P(A|h)(x,\cdot) = P_2(A(x)|h(x,\cdot))$ a.s. for a product space?

Let $X, Y$ and $Z$ be measurable spaces. Let $P_1$ and $P_2$ be probability distributions on $X$ and $Y$ respectively and $P=P_1 \times P_2$. Let $h : X \times Y \to Z$ be a random variable. Let $A \subseteq X \times Y$ be measurable and let $A(x)$…
Matthias Georg Mayer
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