# 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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### Genius prediction

It's a long story, but trust me, it's worth it. Matt is a detective and there's news of 3 murders about to happen in the upcoming week. Matt's usual success rate is 60%, i.e. He'll be successful in a ...
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### Bayesian network problem: third day rainy, given first day is

I've tried searching for this problem online but could not find a solution, hopefully you can help me. I have three random variables [r1,r2,r3], these three variables shows the probabilities of it ...
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### How $\mathbb{P} (A | B \cup B^c)$ is $\mathbb{P} (A | B) \cdot P(B) + \mathbb{P} (A | B^c) \cdot \mathbb{P} (B^c)$?

P(A) = $P(A|\Omega)$         = $P(A|B \cup B^c)$ But how to reach P(A | B) * P(B) + P(A | B$^c$) * P(B$^c$) from P(A | B U B$^c$) ?
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### How is P(N|$H_3$) derived?

An exercise with a solution attached below. I do not understand how is, in ii), $P(N\mid H_3$) derived. Could somebody please explain me?
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### Question about how to obtain the value of f X|Y(x|y)

In the following example question (from Bertsekas, edition 1), i have one question: Why the value of fY|X(y|x) is 1/2? Is it because Y is Y|X is either 0 or 1/6 (50% probability), or because some ...
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### Probability for sampling at least one member of a set from a larger set

Let's assume the following: Population size: N Individuals with a particular feature: x% of N Sample size: y% of N What's the chance that we get at least one ...
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### Law of total variance and covariance given X and Y are normal

I have a problem which asks me to find $\Bbb E[Y]$ and $Var(Y)$ given that $Y\text{~}Normal(x,1)$ conditional on $X=x$. $X$ is standard normal. So I have worked out that $\Bbb E[Y]=0$ using the law of ...
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### Conditional distribution for two random variables

I recently came across and exercise from a past exam and I was wondering how to solve it. Two independent random variables $A$ and $B$ are given and they both follow the exponential distribution but ...
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### If $X\sim \mathrm{lognormal}$ then $Y:=(X-d|x\geq d)$ has approximately a Generalized Pareto distribution.

Let $X$ be a random variable with lognormal distribution. Show that when sufficiently large then $Y:=(X-d|x\geq d)$ is approximately a random variable with generalized Pareto distribution. Hint: Use ...
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