Mathematics Stack Exchange
Mathematics Stack Exchange

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 $$ P(A|B)P(B)=P(A\cap B)$$

5658 questions
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find $P(Y|X)$ from $P(X=x,Y\leq y)$

Suppose $$P(X=n,Y\leq y) = b\int_0^y \frac{(at)^n}{n!}e^{-(a+b)t}\,dt.$$ How do you actually find $P(Y|X)$? I tried from bayes that $P(Y=y|X=x)=\frac{d}{dy}\frac{P(X,Y\leq y)}{P(X)}$ and after finding the integral, it seems that differentiating…
smaillis
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If $\mathbb P(A)=0.61, \mathbb P(B)=0.56$ and $\mathbb P(A\cup B)=0.81$. Find $\mathbb P(A \mid A\cap B)$

I’ve figured out what $(A\cap B)$ is, which is $0.36.$ I tried putting 0.61 over 0.36 but the answer is over 1 which is obviously incorrect. How do I answer this question?
Somxr
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Conditional Probability of Binomial Distribution

If $Y_{1}\sim \text{Bin}(n_{1},\pi)$ and $Y_{2}\sim\text{Bin}(n_{2},\pi))$ are independent. Then find the conditional distribution of $Y_{1}$ given $Y_{1}+Y_{2} = m$. How do I calculate $\textbf{P}(Y_{1} = k|Y_{1}+Y_{2} = m)$?
Kyle Zhang
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Conditional probability and recursion

There are n unstable molecules in a row, $m_1,m_2,...,m_n$. One of the $n− 1$ pairs of neighbours, chosen at random, combines to form a stable dimer; this process continues until there remain $U_n$ isolated molecules. a) Show that the probability…
phw.
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Can $\Pr(B|-A)$ be written as $1-\Pr(-B|-A)$?

Can $\Pr(B|-A)$ be written as $1-\Pr(-B|-A)$? Would be helpful if someone could confirm this.
G2_2023
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Conditional Probability problem on single coin toss

A fair coin is tossed 4 times. Provided that at least 2 heads occurred within the 4 tossings, what's the probability that at least one tail also occurred?
Ashutosh Pandey
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Nerdy medical doctor trying to create a mathematical model without the mathematical ability to create the proper notation. Help?

How would I express the following concept in accepted mathematical notation? : As variable a plus variable b approach infinity the probability of event c occurring approaches 1. and, the probability of b is directly proportional to a I want to…
Yama Raja
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Verification of a Conditional expectation

I need to check if the following formula is correct. $$ \mathrm{E}(Z \vert X=x)=\sum_{y} \mathrm{E}(Z \vert X=x,Y=y)\mathrm{P}(Y=y \vert X=x) $$ Just as a reference, Law of total expectation states that $\mathrm{E}(Z)= \sum_{x} \mathrm{E}[Z \vert…
user91759
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Law of total probablity

If I recieve an email as Poisson distribution with parameter k, poi(k) during a time with geometric distribution with parameter p, geo(p). Then the total emails depends on the time t. How do I find that distribution? Law of total probability?
Abbe
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