# Questions tagged [conditional-probability]

The probability that an event A will occur, when another event B is known to occur or to have occurred. It is commonly denoted by P(A|B).

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### If we know A is independent of B, why isn't P(A|B,C) = P(A|C) necessarily true?

Let's say we know that A is independent of B, or mathematically: $$P(A|B) = P(A)$$ Then how come we can't say the following is necessarily true: $$P(A|B,C) = P(A|C)$$ If the outcome of B doesn't have ...
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### Wikipedia entry on likelihood seems ambiguous

I have a simple question regarding "conditional probability" and "Likelihood". (I have already surveyed this question here but to no avail.) It starts from the Wikipedia page on likelihood. They say ...
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### Can anyone explain conjugate priors in simplest possible terms?

I have been trying to understand the idea of conjugate priors in Bayesian statistics for a while but I simply don't get it. Can anyone explain the idea in the simplest possible terms, perhaps using ...
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### Definition of Conditional Probability with multiple conditions

Say I have two events, $A$ and $B$, and some distribution parameters $\theta$, and I'd like to look at $P(A | B,\theta)$. The simplest definition of conditional probability is, given two events $A$ ...
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### What is the probability that this person is female?

There is a person behind a curtain - I do not know whether the person is female or male. I know the person has long hair, and that 90% of all people with long hair are female I know the person has a ...
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### Two dice rolls - same number in sequence

I am currently studying Statistical Inference class on Coursera. In one of the assignments, the following question comes up. ...
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### Is there any difference between Frequentist and Bayesian on the definition of Likelihood?

Some sources say likelihood function is not conditional probability, some say it is. This is very confusing to me. According to most sources I have seen, the likelihood of a distribution with ...
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### Problem with proof of Conditional expectation as best predictor

I have an issue with the proof of $E(Y|X) \in \arg \min_{g(X)} E\Big[\big(Y - g(X)\big)^2\Big]$ which very likely reveal a deeper misunderstanding of expectations and conditional expectations. The ...
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### Why is a normalizing factor required in Bayes’ Theorem?

Bayes theorem goes $$P(\textrm{model}|\textrm{data}) = \frac{P(\textrm{model}) \times P(\textrm{data}|\textrm{model})}{P(\textrm{data})}$$ This is all fine. But, I've read somewhere: Basically, ...
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### Coin flip game: HH vs HT in a sequence of flips

An interesting thought experiment involving flipping a fair is going around X/Twitter: Flip a fair coin 100 times—it gives a sequence of heads (H) and tails (T). For each HH in the sequence of flips, ...
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### difference between conditional probability and bayes rule

I know the Bayes rule is derived from the conditional probability. But intuitively, what is the difference? The equation looks the same to me. The nominator is the joint probability and the ...
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### Why is P(A,B|C)/P(B|C) = P(A|B,C)?

I understand $P(A\cap B)/P(B) = P(A|B)$. The conditional is the intersection of A and B divided by the whole area of B. But why is $P(A\cap B|C)/P(B|C) = P(A|B \cap C)$? Can you give some intuition?...
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### Monty Hall Problem with a Fallible Monty

Monty had perfect knowledge of whether the Door had a goat behind it (or was empty). This fact allows Player to Double his success rate over time by switching “guesses” to the other Door. What if ...
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### The paradox of i.i.d. data (at least for me)

As far as my aggregate (and scarce) knowledge on statistics permits, I understood that if $X_1, X_2,..., X_n$ are i.i.d. random variables, then as the term implies they are independent and identically ...
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### How should I mentally deal with Borel's paradox?

I feel a bit uneasy with how I've mentally dealt with Borel's paradox and other associated "paradoxes" dealing with conditional probability. For those who are reading this that aren't familiar with it,...
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### A meeting has 12 employees. Given that 8 of the employees are female, what is the probability that all employees are female? [closed]

If I use Bayes' theorem here , event A denoting that 12 employees are female and event B denoting that 8 employees are female, (assuming that an employee has equal chances of being a male or female) I ...
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### How to develop intuition for conditional probability?

In the video lectures from Harvard's Statistics 110:Probability course that can be found on iTunes and YouTube, I encountered this problem. I tried to summarize it here: Suppose we are given a ...
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### What is the expected number of children until having the same number of girls and boys?

A couple decides to keep having children until they have the same number of boys and girls, and then stop. Assume they never have twins, that the "trials" are independent with probability 1/...
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### Explanation of I-map in a Markov/Bayesian network

I am finding the concept of an I-map (Independency-map) in the context of Markov networks and Bayesian networks difficult to understand. From Probabilistic Graphical Models, Koller and Friedman, 2009: ...
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### Does independence imply conditional independence?

If two or more variables A, B, C, etc. are jointly mutually independent of one another, does this imply that that they are also conditionally independent given some set of conditioning variables X, Y, ...
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### If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$

Question If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$. Attempt: Please check if the below is correct. Let ...
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### Is "not independent" the same as "dependent" in English?

When I learned about conditional probability, I found this statement: if A is not independent of B then also B is not independent of A. Formally, if P(A) ≠ P(A|B) then P(B) ≠ P(B|A). I think "...
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### How to compute conditional expectations with respect to a sigma field?

Example: Toss a coin twice. Letting $\mathbb P$ be a probability measure, suppose $\mathbb P(HH)=p^2,\mathbb P(HT)=\mathbb P(TH)=p(1-p), \mathbb P(TT)=(1-p)^2.$ I would like to answer the following ...
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### What are the differences between stochastic and fixed regressors in linear regression model?

If we have stochastic regressors, we are drawing random pairs $(y_i,\vec{x}_i)$ for a bunch of $i$, the so-called random sample, from a fixed but unknown probabilistic distribution $(y,\vec{x})$. ...
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### The more important statistic: '90 percent of all women survived' or '90 percent of all those who survived were women'?

Consider the following statements w.r.t the Titanic: Assumption 1: Only men and women were on the ship Assumption 2: There were a large number of men as well as women Statement 1: 90 percent of all ...
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### A three dice roll question

I got this question from an interview. A and B are playing a game of dice as follows. A throws two dice and B throws a single die. A wins if the maximum of the two numbers is greater than the throw of ...
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### What are the differences between "Marginal Probability Distribution" and "Conditional Probability Distribution"?

While studying probability, I am kind of having difficulties in understanding marginal probability distribution and conditional probability distribution. To me, they look much the same and cannot find ...
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### Why are density functions sometimes written with conditional notation?

I keep seeing density functions that don't explicitly arise from conditioning written with the conditional sign: For example for the density of the Gaussian $N(\mu,\sigma)$ why write:  f(x| \mu, \...
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### How to obtain $p(x)$ given samples from $p(y|x)$ and $p(y)$?

Here, assume both $p(y\mid x)$ and $p(y)$ are too complicated to get closed forms, and we can only draw samples from them. Is there any way to estimate or draw samples from $p(x)$?
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### How to define a distribution such that draws from it correlate with a draw from another pre-specified distribution?

How do I define the distribution of a random variable $Y$ such that a draw from $Y$ has correlation $\rho$ with $x_1$, where $x_1$ is a single draw from a distribution with cumulative distribution ...
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### Bayes Theorem with multiple conditions

I don't understand how this equation was derived. $P(I|M_{1}\cap M_{2}) \leq \frac{P(I)}{P(I')}\cdot \frac{P(M_{1}|I)P(M_{2}|I)}{P(M_{1}|I')P(M_{2}|I')}$ This equation was from the paper "Trial by ...
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