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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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Flipping an unfair coin $N$ times while $N \sim \operatorname{Poi}(\lambda)$

Let be $U$ an unfair coin. Head with probability $p$. You flip this coin $N$ times while $N \sim \operatorname{Poi}(\lambda), \lambda>0$. You get $X$ times head, $Y$ times tail. What is the ...
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Elementary probability - subtle and naively looking question

Still having some time prior to the beginning of academic year, I take a look at different posts at Math Stack Exchange. One post attracted my attention and decided to open individual post related to ...
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Is $f(x) = F_{Y|X=x}^{-1}(0.5) $ measurable? [on hold]

f(x) = E[Y|X=x] is measurable on $X$ based on definition of conditional expectation. Is $f(x) = F_{Y|X=x}^{-1}(0.5) $ also measurable? i.e. Given $X=x$, the median value of $Y$. $F$ is the cdf ...
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coin is tossed and based on its outcome a number is shown

Two integers are chosen from $(-\infty,\infty)$. A fair coin is tossed and its outcome is not revealed. If Head comes, the larger number is shown and if Tail comes, the smaller one is shown. What is ...
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Formulate computation of certain probabilities based on the joint distribution and conditional probabilities

I would like to know if there is way to formulate the following problem in a generic way using the joint distributions, conditional probabilities, etc. Let $I$ be a set of customers indexed by $i$. ...
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Question on coin tossing - probability

When considering an infinite sequence of tosses of a fair coin, how long will it take on an average until the pattern H T T H appears? I tried to break the problem into cases where ultimately the ...
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Evaluating conditional probability for a cipher text given a plain text message

I'm not posting this on Cryptography as I have not even understood the math for this, and this had nothing to do with the cryptographic definition inside. Caesar cipher should give enough context, if ...
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1answer
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Expected value of number of trials to get k SUCCESSIVE successes

Independent trials, each of which is a success with probability p, are performed until there are k consecutive successes. What is the mean of the number of the necessary trials? Let $N_k$ be the ...
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How to measure (estimate) conditional independence?

I've been reading a few papers on causal inference with the PC algorithm and other things proposed; They always use conditional independencies in their algorithm. So given a graph (let us assume ...
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1answer
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Is this a good way to show that $I$ and $J$ are not independent?

Given $$P(I\mid J)=0.7$$$$P(I^c\mid J^c)=0.3$$ does the following bit of working out show that $I$ and $J$ are not independent events? $$P(I\mid J)P(J)=P(I \cap J)=0.7P(J)$$ $$P(I^c\mid J^c)P(J^c)=P(...
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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 ...
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Statistics: Probability of two cases under specific conditions.

A rain system lights up when it detects water. It performs differently under heavy and light rain. The detector fails to detect the rain 15% of the time under light rain and it fails 10% of the time ...
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Coupon Collecting Problem: $4$ coupons with $p_1 = p_2 = \frac{1}{8}$ and $p_3 = p_4 = \frac{3}{8}$

This is from Ross. I know how to solve everything but (d). The book answer is $\frac{123}{35}$. There are $4$ different types of coupons, the first $2$ of which compose one group and the second $2$...
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1answer
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Bayes, three tests, probability patient has disease if exactly two come out positive

I know how to apply Bayes Theorem to determine the likelihood a patient has a given disease if multiple trials return a positive result. In this case, if two trials both return positive: $$ P(\text{...
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Independence of two random variables Y/X and X

Consider two random variables $X$ and $Y$ such that X ∼ U(0, 1) and $Y|X=x$ ∼ N($x$, $x^2$), where $U$ and $N$ denotes an Uniform and Normal distributions, respectively. Prove that $Y/X$ and $X$ are ...
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How to formalize this problem on conditional probability involving $P(A|B\cap C)$?

I'm attempting a solution to the following problem, copied verbatim from my problem set. (Apparently the problem comes from this book.) Every evening, two weather stations issue weather forecast ...
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Conditional expectation with 2 events

Using the definition of conditional expectation given an event I need to show $$ E (I_B | A) = P(B|A).$$ So far I have \begin{align*} E (I_B | A) &= \sum_x 1\cdot P(I_B = 1 |A) P(A) + 0 \cdot P(...
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1answer
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Conditional probability (example)

Suppose that the probability of a random traffic light displaying light of a specific color when the previous traffic light is displaying the same color is $p$. The first light is red with ...
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1answer
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Conditional probability for dependent events

There are two bags, the first bag contains 2 white balls and 5 black balls and the second contains 3 white balls and 4 black balls. What is the probability that the first bag was chosen given a white ...
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1answer
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Application of Bayes Formula

The question : Paul checks the weather report in order to decide whether to carry an umbrella. On any given day, with probability 0.2 the forecast is “rain" and with probability 0.8 the forecast is “...
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How to find the conditional distribution when we have the summation of two Poisson random variables?

We assume that $X_1\mid Y=y\sim \operatorname{Bin}(y,p_1)$ and $X_2\mid Z=z\sim \operatorname{Bin}(z,p_2)$, $Z$ and $Y$ are independent, and also $Y\sim \operatorname{Pois}(\lambda_1)$ , $Z\sim \...
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conditional probability: game with three different bags, balls in bags

We play a game with three bags of balls, where the game is best of five, and one wins when they select three winning balls. A bag is selected at the beginning of a game, and it does not change between ...
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2answers
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What is the conditional probability that the number of heads equals the number showing on the die, conditional on knowing that the die showed 1?

Suppose we flip two fair coins and roll one fair six-sided die. What is the conditional probability that the number of heads equals the number showing on the die, conditional on knowing that the die ...
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proof or counterexample — conditional probability & two events

Let $$P(A), P(B) > 0$$ Prove or give a counterexample: $$P(A|B) > P(A) \rightarrow P(A|B^c) < P(A)$$ I have been able to do neither. It seems like the statement should be true by intuition, ...
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Simplifying a probability conditioned on a disjunction

I'm looking for a simplified form of $$P(C\mid A \cup B)\,,$$ which expands to $$ \frac{P(A,C) + P(B,C) - P(A,B,C)}{P(A) + P(B) - P(A,B)}$$ (similar posts can be found here or here). Can this be ...
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2answers
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Find $P[r≤p|s≤p]$ where $r,s\in N$

Two natural numbers $r, s$ are drawn one at a time, without replacement from the set $S=\{1, 2, 3, ...., n\}$ . Find $P[r\leq p|s\leq p]$ , where $p\in S$ The solution is given as $\frac{p-1}{n-1}$ ...
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1answer
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Conditional probability for a monkey to randomly write a sentence

We all know the statement that a monkey, typing random keys, given enough time, will type anything we want. Say what I want is the sentence: "This is cool". There are 12 characters, if the monkey's ...
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2answers
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Show that $p_M=1-\frac{1}{2!}+\frac{1}{3!}-\frac{1}{4!}+…-\frac{1}{52!}$

[enter image description here]1 I do not understand why $p_1=52(1/52)$, could someone explain to me please? Thank you very much.
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Conditioning a PDF in the intersection of two circles.

Given the diagram below (The first figure): $$f_r(r)= \frac{\lambda \pi r \exp{(-\lambda \pi r^2/2)}}{1-e^{-\lambda \pi R^2/2}}$$ where $$0<r\leq R$$ The pdf $$f_{\text{Z}}(\text{z})=\frac{2\...
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What is the conditional probability that the first die shows 5, conditional on the event that exactly three dice show 5?

Suppose that we roll four fair six-sided dice. What is the conditional probability that the first die shows 5, conditional on the event that exactly three dice show 5? Let $A=\{\text{first dice ...
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1answer
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Find the probability that the third sword appears in the sixth extraction

From a deck of playing cards, several cards are successively selected randomly and without replacement. Find the probability that the third sword appears in the sixth extraction. Define Let $A$: ...
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Is proving one set of independence sufficient to conclude other events?

The table below shows the proportions of males and females who prefer tennis or do not prefer tennis. Suppose we wish to determine whether preference for tennis is dependent on, or independent of ...
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Beginner probability question about the phrase “order doesn't matter”

This is a lame question but it's been bugging me for a while... A father and his son are at a diner and each make one selection (randomly and independently) from a list of $10$ different dishes on ...
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1answer
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Conditional probability in gambling game

I think I'm not understanding how it works: Let $(X_n)_n$ a simple random walk in $\mathbb{Z}$, $\mathbb{P}(X_n=+1)=\mathbb{P}(X_n=-1)=.5$ and $S_n=\sum^n_{k=1} X_k$, $T=\min\{n\in \mathbb{N}: S_n \in ...
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1answer
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Probability of trials without replacement using conditional probability

A bag contains $5$ red marbles and $3$ black marbles. Three marbles are drawn one by one without replacement. What is the probability that at least one of the three marbles drawn be black, if the ...
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2answers
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conditional probability when throwing balls multiple times from a bag

Suppose a box contains 20 balls: each ball has a distinct number in {1, . . . , 20} written on it. We pick 10 balls (without replacement) uniformly at random and throw them out of the box. Then we ...
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1answer
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Elegant way to calculate probability of a randomly selected person from a test group beeing sick, given specificity, sensitivity, and test results

Given a decease and a test that identifies: 99.9% of sick patients as sick (sensitivity) 99.8% of healthy patients as healthy (specificity) 15 of 5000 patients were tested positive, what is the ...
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If I draw 2 balls from the box with 6 white balls and 4 red balls, what is probability that second ball is red?

There is a question bugging me about counting / probability. So the question goes: If I draw 2 balls from the box with 6 white balls and 4 red balls, what is the probability that 2nd ball is red? ...
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Proving the conditional probability [closed]

Can someone help me with this? For any three events $A$, $B$ and $C$, prove the following identity: $$Pr(A \cap B\vert C) = Pr(A\vert B \cap C)Pr(B\vert C)$$ I really have no idea how to do this and ...
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What is the probability of drawing a spade again if I already have $4$?

I would like to know how many spades an English deck has, I am confused, the English deck not only contains hearts, spades, clubs, diamonds? I believe that if you know how many there are, the ...
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1answer
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Number of flips $X$ of a fair coin to win a game

You win by getting $10$ heads in a row (fair coin). Any other result and you lose immediately. For example if you start flipping and get $T$ or $HT$ or $HHHT$ you lose. Let $X$ be the number of coin ...
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Why defining regular conditional probability?

Given probability space $(\Omega, \mathcal{F}, \mathbb{P})$, I can understand the definition of conditional expectation $\mathbb{E}[X\mid \mathcal{G}]$, where $\mathcal{G}$ is a sub $\sigma$-algebra ...
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1answer
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Having problems seeing how two events are conditionally independent (Craps dice game)

Background: $$\text{Show}: \quad E[R\mid\text{win},S=4] = E[R\mid S=4]$$ where $R$ is the number of rolls of the dice in a game of craps $S$ is the initial sum and for this question let's say it's $...
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1answer
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uniform cutting problem

We cut [0,1] interval into 6 pieces by using uniformly chosen 5 separate points. Then, what is the probability that the lengths of all pieces are less than half? I approach this problem by a standard ...
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1answer
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Sum of Conditional Probabilities of two discrete random values

While studying probability theory, I came across this truth statement. In my lecture notes, it has been depicted as False. However, according to my reasoning, it is supposed to be true. Could someone ...
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1answer
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Probability of $A$ given $A$ intersection $B$ [closed]

Two dice are rolled, and $A$ and $B$ stand for: $A=$ first of the numbers is an odd number, $B=$ the sum of the numbers is $4$. How do I write down this probability?
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1answer
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How do we know that the expression in the law of total probability converges to a finite value?

Let $<A_{i}>$ be a countably infinite partition of sample space $\Omega$. Then the probability of an event $A$ is given by, $$P(A) = \sum_{n = 1}^{\infty} P(A_{i})\cdot P(A|A_{i})$$ The ...
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1answer
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Probability of a device having anomaly type A

$A$ = component has anomaly A $B$ = component has anomaly B. $P(B)=0.09$ $P(A|B) = 0.5$ $P(A|B^c) = 0.01$ I figured out that the probability of device having both anomalies is equal to $P(B \cap ...
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Mind Boggled : Not exactly a fixed number of Trials, but Independent Events?

And so, I have a math question for my stats homework that goes like this : A car dealer has a list of 15 cars. The probability of selling one car during a typical week is 40%. The chance of selling ...
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Find the conditional probability that a ball of certain color is chosen from an urn among several urns containing variable number of colored balls.

There are $n$ urns of which the $i$-th urn contains $i − 1$ red balls and $n − i$ blue balls. You pick an urn at random and remove two balls at random without replacement. Find the probability that (a)...